# Mathematics

**Faculty **

**David A. Reimann, **chair and associate professor.

B.S., 1986, University of Toledo; M.A., 1990, Ph.D., 1998, Wayne State University. Appointed 1996.

**Paul L. Anderson, **professor.

B.S., 1976, M.S., 1979, Ph.D., 1989, Colorado School of Mines. Appointed 1990.

**Mark E. Bollman, **associate professor.

B.A., 1986, Northwestern University; M.A., 1988, The University of Michigan; Ph.D., 2001, Central Michigan University. Appointed 1999.

**Harold S. Connamacher,** instructor.

B.A., 1991, Oberlin College; M.A., 2000, University of Oregon; Ph.D. candidate, University of Toronto. Appointed 2005.

**Darren E. Mason, **associate professor.

B.S., 1991, Ph.D., 1996, University of Minnesota. Appointed 2001.

**Cayley A. Pendergrass, **assistant professor.

B.A., 2000, Swarthmore College; M.A., 2001, Ph.D. 2006, University of California, San Diego. Appointed 2006.

**Introduction**

The Mathematics and Computer Science Department at Albion College includes the disciplines of pure and applied mathematics, computer science and statistics.

The courses are structured to meet the overlapping needs of students who fall in one or more of the following categories: (1) those who wish to develop their appreciation of the power and beauty of mathematics; (2) those who wish to explore the dynamic field of computer science; (3) those who intend to pursue graduate work in mathematics, computer science or other related fields; (4) those who will exploit the applications of mathematics in the natural sciences, social sciences and other areas of quantitative studies; and (5) those who plan to enter the teaching profession in mathematics or computer science.

The Mathematics and Computer Science Department offers two majors: the mathematics major described below and the computer science major described under that heading in a separate section of the catalog.

Department of Mathematics and Computer Science Web site

**Career Opportunities**

There has long been a demand in both industry and government for people with training in mathematics and statistics. The mathematics major who takes courses in computer science or statistics will enter an extremely favorable job market. There is also a need for secondary school teachers who are certified to teach mathematics or computer science. A major in mathematics provides a good foundation for further study in mathematics or for teaching on the secondary school level. With a degree in mathematics, it is also possible to gain admission to graduate school in other fields such as public policy, management and operations research.

**Special Features**

The Mathematics and Computer Science Department annually awards approximately $30,000 in scholarships in honor of E. R. Sleight, a beloved mathematics professor who taught at Albion from 1908 to 1948. Prospective students with strong interests in mathematics are encouraged to contact the department to apply for these scholarships. Additional awards are made to outstanding upperclass students in mathematics and computer science.

Each year the Mathematics and Computer Science Department nominates five mathematics majors to membership in the Mathematical Association of America. The J. R. Lancaster Award is presented to the student who best exemplifies the liberally educated mathematics student. The E. R. Sleight Prize is awarded to the outstanding senior in the mathematical sciences. Each summer several students receive stipends as Kresge Fellows and from other sources for independent research projects in the mathematical sciences. The Michigan Alpha chapter (established at Albion in 1937) of the mathematics honorary Kappa Mu Epsilon promotes mathematical lectures, films and social events. Students participate in the Michigan Autumn Take-Home Challenge, the Lower Michigan Mathematics Competition, and at the national level, in the William Lowell Putnam Competition and the Mathematical Contest in Modeling. Students are encouraged to attend and present papers at departmental colloquia and at regional conferences in undergraduate mathematics. Internships and the Oak Ridge Science Semester provide additional opportunities for intensive study in the mathematical sciences.

The Math/Stat Computing Laboratory is designed especially for students in mathematics, statistics and computer science courses. This computer laboratory features microcomputers running Windows and a laser printer for high-resolution graphics and typesetting. Statistics students routinely analyze data with the Minitab statistical analysis program; graphing calculators and the Mathematica computer algebra system are integrated into precalculus, calculus and higher-level mathematics courses. This lab is part of Albion's campus-wide computer network connecting faculty offices, classrooms, laboratories, public computer areas, printers, the library automation system and residence hall rooms. From computers on the network, students can access their files, run software on the campus network, interact with other computers, send electronic mail and browse the World Wide Web.

**Requirements for Major in Mathematics**

There are four tracks for a mathematics major, as described below. **The mathematics curriculum is highly sequential with a rigid and necessary prerequisite structure, and not all courses are offered each year. **Students planning an academic program that includes a mathematics major, especially one including teacher certification (Tracks III and IV), are urged to consult with a member of the mathematics faculty early in their Albion career so that a proper sequence of courses may be arranged.

**Failure to consider carefully the implications of course enrollment** **decisions may result in delayed graduation.**

FOUNDATION COURSES

Mathematics 141: Calculus of a Single Variable I

Mathematics 143: Calculus of a Single Variable II

Mathematics 236: Linear Algebra

Mathematics 239: Discrete Structures

Mathematics 245: Multivariate Calculus

The department may waive one or more of the foundation course requirements for students with advanced high school mathematics preparation.

TRACK I

Track I leads toward graduate work in the mathematical sciences.

- Nine units in mathematics, including the five foundation courses, plus four additional courses: Mathematics 331, 335; one from 309, 360; and one other 300-level mathematics course.
- Computer Science 171.

Students contemplating graduate study in mathematics should also take as many other 300-level mathematics courses as their schedules will allow, as well as course work in French, German or Russian.

TRACK II

Track II leads toward immediate employment or further study in applied mathematics or a related area.

- Nine units in mathematics, including the five foundation courses, plus four additional courses: Mathematics 247; two from 309, 316, 326, 360, 380; and one from 331, 335.
- Computer Science 171.

Students in this track should also take advanced course work in an area of application.

TRACK III

The third track leads to secondary teacher certification. See "Requirements for Mathematics Major With Secondary Education Certification" below.

TRACK IV

The fourth track leads to elementary teacher certification. See "Requirements for Mathematics Major with Elementary Education Certification" below.

**Requirements for Minor in Mathematics**

- Five units in mathematics, including the three foundation courses, Mathematics 141, 143, 239, plus one from 236, 245, 247 and one from 331, 335, 342.
- Computer Science 171.

Not open to mathematics majors.

### R**equirements for Minor in Applied Mathematics **

- Five units in mathematics, including Mathematics 141, 143, 245, 247, plus one from 316, 326, 360, 380.
- Computer Science 171.

Not open to mathematics majors.

**Requirements for Minor in Statistics **

- Six units in mathematics, including Mathematics 141, 143, 245, 309, 310, plus one from 109, 210.

Not open to mathematics majors.

**Requirements for Mathematics Major with** **Secondary Education Certification**

- Nine units in mathematics, including the five foundation courses, Mathematics 141, 143, 236, 239, 245, plus 309, 331, 335, 342.
- Computer Science 171.
- Completion of all other requirements for teacher certification.

**Requirements for Mathematics Minor with** **Secondary Education Certification**

- Five units in mathematics, including the three foundation courses, Mathematics 141, 143, 236, plus 335 and 342. The department may waive one or more of the foundation course requirements for students with advanced high school mathematics preparation.
- Mathematics 210, 239 and Computer Science 171 are recommended.
- Completion of all other requirements for teacher certification.

**Requirements for Mathematics Major with** **Elementary Education Certification**

- Nine units in mathematics, including the five foundation courses, Mathematics 141, 143, 236, 239, 245, plus 309, 335, 342, 345.
- Computer Science 171.
- Completion of all other requirements for teacher certification.

**Other Requirements for All Mathematics Majors and Minors**

- While a student may begin with Mathematics 125 and still complete a major, it is recommended that prospective majors take a similar course in high school if at all possible.
- Students are required to earn one-half unit of credit by enrolling in Mathematics 299, 399, Colloquium in Mathematics.
- No course to be counted toward a major or minor in mathematics may be taken on a credit/no credit basis, except Mathematics 299 and 399, which are only offered as credit/no credit courses.
- Students majoring or minoring in mathematics are expected to furnish the Department of Mathematics and Computer Science with information about their course work and activities related to the department. The department faculty will use this information when nominating students for awards, scholarships and membership in professional societies, and as the basis for letters of recommendation. Students are encouraged to include this information on their personal World Wide Web pages or to develop a portfolio Web page for their activities related to their major.

**Courses**

Initial course placements in mathematics and computer science are generally determined by the Mathematics Placement Test. After students take their first course, they must take courses in sequence as determined by the departmental prerequisites. Any exceptions must be approved by the course instructor and department chair.

**104** **Mathematics for Elementary Teachers** **(1)** **Spring**

Prerequisite: Three years of college-preparatory mathematics (or its equivalent). Priority given to students in the elementary education program. An investigation of mathematics (arithmetic, geometry, algebra, problem solving) for elementary school teachers. Topics are selected from: sets, relations and functions; numeration systems; whole numbers and their operations; number theory; rational numbers and fractions; decimals and real numbers; geometry and measurement; and probability and statistics. Emphasizes doing mathematics, using manipulatives, and developing intuition and problem-solving skills. Laboratory. *Bollman.*

**109** **Statistical Methods** **(1)** **Fall, Spring**

Prerequisite: Permission of instructor.

Descriptive statistics, probability, sampling distributions, hypothesis testing, parameter estimation, confidence intervals, linear regression, curve fitting and analysis of variance are discussed. Uses the Minitab statistics package to display and analyze data. Students may not receive credit for both Mathematics 109 and 210. Usually not open to students who have had Mathematics 141. *Anderson.*

**119** **Finite Mathematics for Decision Making** **(1)** **Spring**

An introduction to discrete mathematics. Applications are drawn from diverse areas including biological sciences, economics, political science and personal finance. Topics typically include graph theory, management science, statistics, the mathematics of social choice, game theory and the logical foundations of mathematics. Investigation and creation of mathematical models. Intended for non-majors. *Staff.*

**125** **Functions** **(1)** **Fall, Spring **Prerequisite: Permission of department.

A modern, unified approach to algebra, trigonometry, logarithms and analytical geometry based on the concept of a function. Linear equations and inequalities, quadratic equations and inequalities, polynomials and rational functions, logarithms and exponential functions, trigonometric and inverse trigonometric functions, and analytic geometry (the circle, the parabola, the ellipse and the hyperbola) are normally covered. Emphasizes the use of graphing calculators and the use of mathematics as a problem-solving tool. Covers applications in natural science, social science and business. Serves as a preparation for calculus. Well-prepared students who already have a strong working knowledge of algebra, trigonometry and logarithms should elect Mathematics 141 in place of Mathematics 125. A graphing calculator is required.

*Staff.*

**141** **Calculus of a Single Variable I** **(1)** **Fall, Spring**

Prerequisite: Mathematics 125 or permission of department.

Mathematics 141 and 143 constitute a thorough introduction to calculus for students who intend to continue in mathematics and for those who will use calculus in other fields such as science and engineering. Mathematics 141 covers limits, continuity, derivatives and a brief introduction to integration, as well as applications to problems in related rates, optimization, solid geometry and elementary mechanics. Requires a strong working knowledge of algebra and trigonometry. Students who are weak in these areas should elect Mathematics 125. A graphing calculator is required. *Staff.*

**143** **Calculus of a Single Variable II** **(1)** **Fall, Spring**

Prerequisite: Mathematics 141 or permission of department.

Second half of the standard one-year calculus sequence (see Mathematics 141 above). Mathematics 143 covers techniques of integration, applications of the integral, simple differential equations with their associated mathematical models, and sequences and series. Requires a strong working knowledge of algebra, trigonometry, derivatives, and some familiarity with integration, including Riemann sums and the Fundamental Theorem of Calculus. Students with a calculus background who are weak in these areas should elect Mathematics 141. A graphing calculator is required. *Staff.*

**210** **Introduction to Statistical Analysis** **(1)** **Spring**

Prerequisite: Mathematics 141 or equivalent.

Topics include descriptive statistics, principles of probability, random variables, sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, regression and non-parametric statistics. Uses Minitab statistics program to analyze data. Students may not receive credit for both Mathematics 109 and 210. *Anderson.*

**236** **Linear Algebra** **(1)** **Fall, Spring**

Prerequisite: Mathematics 143 or 239, or permission of instructor.

Vector spaces, matrices, Gauss-Jordan reduction, products, dimension, linear transformations, eigenvalues and eigenvectors, and a selection of applications of linear algebra to other disciplines. Develops skills in mathematical writing and creating mathematical proofs. Properties of equality, logical implication, proof by contradiction, quantification and proof by induction are illustrated in context. *Messer.*

**239** **Discrete Structures** **(1)** **Spring**

Prerequisite: Mathematics 141.

A survey of discrete mathematics with topics selected from set theory, functions and relations, number theory, combinatorics, graph theory, logic (predicate calculus, quantifiers), introduction to proof techniques, and probability. *Staff.*

**245 Multivariate Calculus (1) Fall **

Prerequisite: Mathematics 143.

Vectors, inner and cross products, and vector-valued functions including parametric representations of curves and surfaces in space. Partial differentiation, the chain rule, function gradients, implicit differentiation, multivariate optimization, and Lagrange multipliers, multiple integrals and vector analysis, including divergence and curl of vector fields, as well as the theorems of Green, Stokes and Gauss. * Mason. *

**247 Differential Equations and Linear Algebra (1) Spring **

Prerequisite: Mathematics 245.

First-order differential equations and numerical algorithms of Euler and Runge-Kutta. Linear algebraic systems, Gaussian elimination, row-echelon form matrix algebra, inverses and determinants. Vector spaces, subspaces, linear independence, bases, span, dimension, linear mappings and function spaces. Second and higher-order linear differential equations. Eigenvectors, eigenvalues and spectral decomposition methods. First-order linear differential systems, including solutions methods using matrix exponentials. Applications focus on problems in physics, chemistry, biology, economics and engineering. Additional topics may include nonlinear dynamical systems, stability theory, transform theory and power series solutions. *Mason. *

**299 Colloquium in Mathematics and Computer Science (1/4) Fall, Spring**

Prerequisite: Mathematics 143 or Computer Science 173.

Selected topics in mathematics and computer science as presented by students, departmental faculty and visiting speakers. Requirements include written summaries of each presentation and a paper on a mathematics/computer science topic of personal interest. Same as Computer Science 299. *Staff. *

**309** **Mathematical Statistics** **(1)** **Fall**

Prerequisite: Mathematics 236 or 245.

A mathematical study of probability distributions, random sampling, and topics selected from statistical theory: estimation, hypothesis testing and regression. *Anderson.*

**310** **Applied Mathematical Statistics** **(1)** **Spring**

Prerequisite: Mathematics 309.

A continuation of Mathematics 309. In-depth studies of regression analysis, analysis of variance, experimental design and nonparametric statistics. Covers topics pertinent to actuarial mathematics. Offered in alternate years. *Anderson.*

**316 Numerical Analysis (1) Fall **

Prerequisites: Mathematics 236 or 247 and Computer Science 171.

Methods of obtaining numerical solutions to mathematical problems. Stresses the implementation and error analysis of algorithms. Topics include solution of non-linear equations, systems of equations, interpolating polynomials, numerical integration and differentiation, numerical solution to ordinary differential equations, and curve fitting. Offered in alternate years. Same as Computer Science 316. * Mason. *

**326 Operations Research (1) Spring **

Prerequisites: Mathematics 236 or 247, and Mathematics 245.

An introduction to computational methods in mathematical modeling including linear programming and Markov chains. Applications in business, economics and systems engineering. Knowledge of probability is helpful. Offered in alternate years. Same as Computer Science 326. * Mason.*

**331** **Real Analysis** **(1)** **Spring**

Prerequisites: Mathematics 245 and either 236 or 239.

A study of the concepts underlying calculus of a single variable: The completeness property of the real number system, convergence, continuity, properties of elementary functions, the derivative and the Riemann integral. *Bollman.*

**335** **Abstract Algebra** **(1)** **Fall**

Prerequisites: Mathematics 236 and 239.

Properties of the integers, real number system and other familiar algebraic entities are viewed abstractly in structures such as groups, semigroups, rings and fields. Homomorphisms and isomorphisms (functions compatible with the algebraic operations) illuminate the underlying similarities among these structures. Students will develop their skills in mathematical writing and presentations. *Staff.*

**342 Geometry (1) Spring **

Prerequisites: Mathematics 143 and 239.

The logical foundation of Euclidean geometry, including the axiom systems of Euclid and Hilbert, and their philosophical implications. An introduction to hyperbolic, elliptic and projective geometry. Employs software such as Geometer’s Sketchpad to illustrate course topics. * Bollman.*

**345** **History of Mathematics** **(1)** **Fall**

Prerequisite: Mathematics 141.

A study of the history and evolution of mathematical ideas and their significance, from approximately 3500 B.C.E. to the present. Topics include number systems, arithmetic, Euclidean and non-Euclidean geometry, algebra, calculus, probability, number theory and applied mathematics. Offered in alternate years. *Bollman.*

**360 Mathematical Modeling (1) Spring **

Prerequisites: Mathematics 247 and Computer Science 171.

An introduction to analytical methods in mathematical modeling, including nonlinear optimization, dynamical systems and random processes. Applications in physics, biology, economics and systems engineering. Knowledge of probability and statistics is helpful. Same as Computer Science 360. *Mason.*

**380** **Mathematical Physics** **(1)** **Spring**

Same as Physics 380. *Staff.*

**388, 389** **Selected Topics** **(1/2, 1)** **Fall, Spring**

Prerequisite: Permission of instructor.

Topics chosen to fit departmental interests, such as complex analysis, mathematical logic, geometric topology, chaos and fractals, algebraic coding theory, experimental design, nonparametric statistics and stochastic processes. Offered on demand. *Staff.*

**391, 392** **Internship** **(1/2, 1)** **Fall, Spring**

Offered on a credit/no credit basis. *Staff.*

**399 Colloquium in Mathematics and Computer Science (1/4) Fall, Spring**

Prerequisites: Mathematics 299 and senior standing.

Selected topics in mathematics and computer science as presented by students, departmental faculty and visiting speakers. Requirements include written summaries of each presentation, a departmental major assessment examination and an oral presentation on a mathematics/computer science topic of personal interest. Offered only on a credit/no credit basis. Same as Computer Science 399. *Staff.*

**401, 402** **Seminar** **(1/2, 1)** **Fall, Spring**

*Staff.*

**411, 412** **Directed Study** **(1/2, 1)** **Fall, Spring**

*Staff.*