*Department of Mathematics*

### Department Faculty

## The Mathematics Program

The interests and expertise of the mathematics faculty cover a broad range of mathematical areas, including algebra, analysis, topology (modern geometry), discrete mathematics (mathematics of computer science), number theory, statistics, and applied mathematics. With this spectrum of faculty knowledge, the student is afforded an opportunity to learn the contemporary view of mathematics. Inside the classroom, student comprehension is the main concern of the faculty. Outside the classroom, the faculty offers opportunities for independent study, undergraduate research, and internship supervision.

Courses in mathematics vary from the theoretical to the applied. Thus, a Bachelor of Science degree in Mathematics can be a foundation for a career in industry, government, teaching, or the pursuit of a higher degree in graduate school. The department faculty encourages double majors, giving students entrance to a wide variety of fields upon graduation. Majors in other disciplines can be enhanced with one of our minors in mathematics, applied mathematics, or actuarial science.

The University of Mary Washington hosts a chapter of Pi Mu Epsilon, a national honorary mathematics society, and a chapter of the Mathematical Association of America. The Oscar Schultz Award in Mathematics represents the department’s top academic honor and is given annually to a junior or senior in the department. Four additional scholarships are available. The recipients of the Meredith C. Loughran ’94 Scholarship are selected based on their meritorious academic record, citizenship and leadership in public service. The Merrilyn Sawyer Dodson/class of 1968 Scholarship and the Mary Farley Talley ’66 Scholarship each recognize the scholastic achievements of mathematics majors, while the Louise W. Robertson, M.D. ’56 Scholarship is awarded to a student majoring in mathematics or a health field.

Qualified mathematics majors having at least a 3.5 GPA in mathematics courses and an overall GPA of at least 3.0 may graduate with Honors in Mathematics by completing a directed study or undergraduate research which culminates in an approved Honors thesis.

Majors are encouraged to fulfill the general education experiential learning requirement by completing URES 197, MATH 491, MATH 492, or MATH 499. Alternatively, majors may meet this requirement by participating in an approved supervised on-campus or off-campus experiential learning activity developed in consultation with the department (such as the UMW Summer Science Institute or a similar program at another college or university). To complete the experiential learning requirement through a summer research experience, contact the department chair for more details.

## Requirements for the Mathematics Major

A minimum of 41 credits are required. Twenty-six (26) credits must be from the following mathematics courses: MATH 122, 201, 224, 300, 330, 431, 471 and either 432 or 472. An additional twelve (12) credits must be taken from MATH or STAT courses at the 300-/400-level with at least three (3) credits from 400-level MATH or STAT courses. Three (3) additional credits must be from MATH or STAT courses numbered 207 or above; computer science (CPSC) courses numbered 220 or above (except CPSC 302); physics (PHYS) courses numbered 105 or above (except PHYS 108); or PHIL 306. Mathematics majors must meet the department’s computer programming requirement by taking one of the following courses: MATH 351, 421, CPSC 110, 219, or 220. Courses used to satisfy the programming requirement may also be used elsewhere in the major. At most six (6) credits of directed study (MATH or STAT 491/492) will count for the major. No internship (MATH or STAT 499) credits will count for the major.

## Mathematics Course Offerings

**Mathematics course offerings will be found under the 4 letter code of MATH in the course listings.**

**110 – Finite Mathematics with Applications (3)**

Includes topics such as sets, logic, probability, statistics, and counting. Other topics are at the discretion of the instructor. Designed for the non-major.

**111 – Precalculus (3)**

Emphasis on elementary functions including rational, exponential, logarithmic and trigonometric functions. Designed for students who intend to take calculus.

**115 – Introduction to Mathematical Modeling (3)**

Mathematical topics include linear functions, linear regression, curve fitting, probability models, and difference equations. Emphasis on environmental issues such as population growth, pollution, natural disasters, epidemics, genetics, and patterns in nature.

**120 – Quantitative Reasoning for the Sciences (3)**

Designed to prepare students for success in the sciences by providing them with appropriate mathematics and quantitative reasoning skills. Course topics include measurement and estimation, growth and decay phenomena, scaling transformations, and an introduction to probability and statistics.

**121 – Calculus I (4)**

First course in calculus. Topics include limits, derivatives and their applications, antiderivatives, definite integrals, the fundamental theorems of calculus, the substitution rule for integrals, and transcendental functions

**122 – Calculus II (4)**

Prerequisite: MATH 121. Topics include techniques and applications of integration, sequences and series.

**201 – Introduction to Discrete Mathematics (3)**

Designed to prepare prospective mathematics majors for advanced study in the field by introducing them to a higher level of mathematical abstraction. Topics include sets and logic, functions and relations, methods of mathematical proof including mathematical induction, and elementary counting techniques. (Prospective mathematics majors should take this course during their freshman year.)

**204 – Mathematical Concepts and Methods I (4)**

Prerequisite: EDUC 203. Mathematical concepts and methods of teaching for the elementary school. Topics include number systems and their properties, problem solving, and topics in number theory. Course intended for students certifying to teach grades PreK-6. Significant field experience required. (3 lecture credits, 1 practicum credit).

**205 – Selected Topics in Mathematics (1-3)**

Prerequisite: Course dependent. Opportunity for additional study of lower-level topics in mathematics.

**207 – History of Mathematics (3)**

The history of mathematics begins with the early numbering systems and mathematics of the Egyptians and the Babylonians. The course then turns to the Greeks and their emphasis on logical deduction and geometry. The Arabs develop algebra in the Middle Ages, and calculus is created during the Age of Reason. The development of individual branches of mathematics then is studied (probability, number theory, non-Euclidean geometry, set theory, and topology). The course ends with the Computer Age and implications for the future.

**223 – Calculus III (3)**

Prerequisite: MATH 122. Includes analytic geometry, parametric equations, polar coordinates, improper integrals, L’Hôpital’s rule, sequences, and infinite series.

**224A – Multivariable Calculus (4)**

Prerequisite: MATH 122. Topics include parametric equations, vectors, polar, cylindrical, and spherical coordinates, vector-valued functions, functions of several variables, partial derivatives, multiple integrals, and vector calculus.

**300 – Linear Algebra (3)**

Prerequisites: MATH 122 and either MATH 201 or CPSC 284. An introduction to linear algebra. Usually includes matrix algebra, systems of equations, vector spaces, inner product spaces, linear transformations, and eigenspaces.

**312 – Differential Equations (3)**

Prerequisite: MATH 122. Ordinary differential equations which may include Laplace transformations, linear differential equations, applications, approximations, and linear systems of equations.

**321 – Number Theory (3)**

Prerequisite: MATH 201 or CPSC 284. An elementary, theoretical study of the properties of the integers.

**325 – Discrete Mathematics (3)**

Prerequisite: MATH 201 or CPSC 284. Includes topics such as discrete probability, graph theory, recurrence relations, topics from number theory, semigroups, formal languages and grammars, finite automata, Turing machines, and coding theory.

**330 – Foundations of Advanced Mathematics (3)**

Prerequisite: MATH 122 and 201. Introduction to mathematical reasoning and rigor. Includes topics such as basic logic, set theory, mathematical induction, relations, functions, sequences, cardinality, elementary number theory, and axiomatic construction of the real numbers. Emphasis placed on reading mathematics, understanding mathematical concepts, and writing proofs.

**351, 352 – Numerical Analysis I. II (3, 3)**

Prerequisite: MATH 300 or 312. Mathematics 351 introduces the theory and applications of the basic computational techniques of numerical approximation. Topics include an introduction to computer programming and algorithms, root finding, interpolation, polynomial approximation, numerical differentiation and integration, and numerical linear algebra. MATH 352 expands on the basic approximation techniques to include scientific computing. Topics include methods of simulation, initial value problems and boundary value problems for ordinary/partial differential equations, and applications in science and engineering. Only in sequence.

**361 – Topics in Mathematics (3)**

Prerequisite: Course dependent. Opportunity for additional study of mathematical topics.

**372 – Modern Geometry (3)**

Prerequisite: MATH 300. Axiomatic development of various geometries including modern Euclidean and non-Euclidean geometry, finite geometries, hyperbolic geometry, and elliptic geometry. Topics could also include convexity, transformational geometry, projective geometry, and constructability.

**411- Chaotic Dynamical Systems (3)**

Prerequisite: MATH 122. Chaotic dynamical systems including iteration, graphical analysis, periodic points, bifurcations, the transition to chaos, fractals, Julia sets and the Mandelbrot set.

**412 – Complex Variables (3)**

Corequisite: MATH 471. Analytic functions, Cauchy-Riemann conditions, integration, power series, calculus of residues, conformal mappings and applications.

**421 – Applied Partial Differential Equations (3)**

Prerequisites: MATH 224 and 312. This course introductions three main types of partial differential equations (PDEs): parabolic, elliptic and hyperbolic as well as mathematical and computational tools for solving PDEs. It balances mathematical rigor, computational techniques, and real-world applications. Topics include heat equation, method of separation of variables, Laplace’s equation, Fourier series, wave equation, finite difference/element methods, and high-dimensional PDEs.

**431, 432 – Abstract Algebra (3, 3)**

Prerequisite: MATH 300 and at least one other 300- or 400-level mathematics course. Mathematical systems including groups, rings, fields, and vector spaces. Only in sequence.

**441 – Topology (3)**

Prerequisite: MATH 300 and at least one other 300- or 400-level mathematics course. Includes topics from point-set topology such as continuity, connectedness, compactness, and product and quotient constructions.

**461 – Topics in Mathematics (3)**

Prerequisite: Course dependent. Topics such as partial differential equations, optimization, Fourier series, ring theory, cryptology, algebraic number theory, coding theory, and modeling. May be taken up to three times for credit.

**471, 472 – Real Analysis (3, 3)**

Prerequisites: MATH 300, and at least one other 300- or 400-level mathematics course. A rigorous, real analysis approach to the theory of calculus. Only in sequence.

**481 – Theory of Interest (3)**

Prerequisite: MATH 122. This course introduces the mathematical concepts underlying the theory of interest. Topics include measurement of interest (including accumulated and present value factors), annuities, yield rates, amortization schedules and sinking funds, bonds and related securities, derivative instruments, and hedging and investment strategies.

**491, 492 – Directed Study (1-3, 1-3)**

Prerequisite: Departmental permission. Individual study beyond the scope of normal course offerings, done under the direction of a faculty member. May lead to graduation with Honors in Mathematics.

**499 – Internship (credits variable)**

Supervised off-campus experience, developed in consultation with the department. Does not count in the major program or minors.

## Statistics Course Offerings

#### Statistics course offerings will be found under the 4 letter code of STAT in the course listings.

*180 – Introduction to Statistics (3)*

First course in statistical methods. Includes descriptive and inferential techniques and probability, with examples from diverse fields. Topics vary with instructor and may also include sampling methods, regression analysis, and computer applications.

*205 –Selected Topics in Statistics (1-3)*

Prerequisite: Course dependent. Opportunity for additional study of lower-level topics in statistics.

*280 – Statistical Methods (3)*

Prerequisite: STAT 180. Second course in statistical methods. Include one-way and higher ANOVA, multiple regression, categorical data analysis, and nonparametric methods with examples from diverse fields. Topics vary with instructor and may also include time series and survival analysis.

*320 – Applied Regression Analysis (3)*

Prerequisite: STAT 180. Topics include simple linear regression, multiple linear regression, categorical predictors, model building principles, residual analysis, multicollinearity and other regression problems, robust regression, nonlinear regression, logistic regression, time series and generalized linear models.

*361 –Topics in Statistics (3)*

Prerequisite: Course dependent.** **Opportunity for additional study of statistical topics.

*381, 382 – Probability and Statistical Inference (3, 3)*

Prerequisite: MATH 122. An introduction to probability theory and calculus-based statistics including probability distributions of discrete and continuous random variables, functions of random variables, methods of estimation, and statistical inference. Only in sequence.

*461 –Topics in Statistics (3)*

Prerequisite: Course dependent. Topics such as time series analysis, computational statistics, design of experiments, probability theory, stochastic processes, and queuing theory. May be taken up to three times for credit.

*491, 492 – Directed Study (1-3, 1-3)*

Prerequisite: Departmental permission. Individual study beyond the scope of normal course offerings, done under the direction of a faculty member. May lead to graduation with Honors in Mathematics.

*499 – Internship (credits variable)*

Supervised off-campus experience, developed in consultation with the department. Does not count in the major program or minors.