Allegheny College: Mathematics

Professors Werner, Barry, Carswell, Ellers, Haytock, Holmgren, Lakins, Leech, Lo Bello, Lundberg, Weir, Hollerman, A. Wolfe

The Department of Mathematics offers a wide range of courses designed to introduce students to major areas of mathematical thought, formal reasoning processes, general methods of problem solving, applications of mathematics to diverse areas, the history of mathematics, and the effective communication of mathematics. Our courses emphasize the activity of thinking with ideas, as opposed to learning content by rote memorization. They develop the analytical and reasoning skills that not only prepare students to be mathematicians, but also serve students well no matter what they do in life. We strive to give students an appreciation for the culture of mathematics as revealed through its history, the beauty of its ideas, and its particular way of knowing, which sets mathematics apart from all other disciplines.

All students new to Allegheny must take the Mathematics Placement Exam, unless exempted by taking an AP Calculus course in high school or by transferring an appropriate calculus course from another college or university. Students should begin their study of Mathematics as indicated by their performance on this exam.

The Major

The major program in Mathematics leads to the Bachelor of Science degree and requires the completion of at least 43 semester hours of coursework numbered above Mathematics 160, including Mathematics 170, 210 (unless exempted through advanced placement), 205 (which should be completed by the end of the sophomore year), 320, 325, 340, 585 and 601. FS 201 MAT may be counted toward the major in Mathematics. No course may be taken on a Credit/No Credit basis for the major program in Mathematics. Transfer students majoring in Mathematics must complete at least 24 semester hours of coursework at Allegheny.

Computer Science 360, Numerical Analysis, or Computer Science 230, Theory of Computation and Formal Languages, may be counted toward the major in Mathematics. However, a student who elects to include these courses in the Mathematics major program may not also count them within a major or minor program in Computer Science. The student can prepare for several career areas in the mathematical sciences. Below is a list of the areas followed by courses recommended by the Mathematics Department in addition to the required courses for the major.

Actuarial Mathematics
Mathematics 290, 345, 346, and 440, as well as coursework in Computer Science and mathematical economics.

Applied Statistics
Mathematics 345, 346, and 365, and Computer Science courses in data structures, operating systems and computability.

Computational and Applied Analysis Mathematics 270, 290, 341, 380, 440, and strong work in the physical sciences.

b>Pure Mathematics
(recommended for those who plan to do graduate study) Mathematics 315, 330, 341, 350, 400, 425 and 440.

Scientific Computing
Mathematics 270, 290, 380 and Computer Science courses in programming languages, data structures, parallel and vector processing, computer graphics, computer simulation and software design.

Teaching (secondary)
Courses required are Mathematics 205, 210, 220, 290, 320, 325, 345, and 350. Mathematics 290 satisfies the modeling requirement. At least one course taken must make use of computing. Be aware that specific content requirements vary from graduate school to graduate school. Please consult with the liaison for Teacher Preparation Programs.

Students in cooperative programs who want to major in Mathematics must meet all of the requirements for a major as described above except for Mathematics 601. At least 28 of the 47 semester hours required must be taken at Allegheny. Students in cooperative engineering programs are required to take Mathematics 290. Students in cooperative programs who want to minor in Mathematics must meet the requirements for a minor listed below.

Each Mathematics major, whether concentrating in applied mathematics or not, should be familiar with applications of mathematics to at least one other field. For this reason, the Department of Mathematics strongly recommends that majors pursue a sequence of three or more courses in at least one of the following departments: Biology, Chemistry, Computer Science, Economics, Geology or Physics.

Mathematics majors are required to have a GPA of at least 2.0 in Mathematics at graduation. All Mathematics courses taken at Allegheny having a number higher than 159 must be taken on a letter grade basis and are included in the calculation. In the case of repeated courses, only the most recent grade will be included.

Normally, the Department of Mathematics will only award the honor citation in Mathematics to students who have completed at least four courses in Mathematics numbered between 250 and 500 (including Computer Science 230 and 360); at least one course must be selected from among Mathematics 400, 425 and 440.

The Minor

The minor in Mathematics requires at least 20 semester credit hours at the 170 level or higher. At least four semester hours must include a course numbered 300 or above. FS 201 MAT may be counted toward the minor in Mathematics. Students should note that Mathematics 310 is also cross-listed as Computer Science courses. If either of these courses is counted within the minor in Mathematics, it may not also be counted for a major or minor in Computer Science. No course may be taken on a Credit/No Credit basis for the minor.

An algebra-based elementary modeling course. Linear, polynomial, exponential, and logarithmic functions are studied from numerical, graphical, and analytical points of view. The emphasis is on modeling real-world problems and rates of change. May not be taken for credit if credit for any calculus course has already been received. Does not count toward a major or minor in mathematics.

An examination from a college perspective of mathematical topics related to the elementary school curriculum with an emphasis on development of problem-solving strategies. Mathematical concepts, their history, and their connections to the real world are studied. The course is intended for students who are seeking certification for elementary school teaching. Prerequisite: Permission of the instructor.

An introduction to the differential calculus of algebraic, logarithmic, and exponential functions. The emphasis is on the concept of the derivative and applications of calculus to the life and social sciences. Precalculus topics are covered as needed. May not be taken for credit if credit for any calculus course has already been received. Does not count toward a major or minor in mathematics.

A continuation of the study of differential calculus begun in Math 157 and an introduction to integral calculus and the multivariate calculus involving algebraic, logarithmic, and exponential functions. In addition to further applications of the derivative, the concepts of the integral of a function of one variable and differentiation of multivariable functions are applied to the life and social sciences. Precalculus topics are covered as needed. May not be taken for credit if credit has been received for Math 156 or Math 160. Does not prepare students for Math 170, and does not count toward a major or minor in mathematics. Prerequisite: Math 157.

A study of the mathematical concepts which are a prerequisite to the study of calculus: functions, domains, ranges, graphs, equations, and inequalities. Specific functions include algebraic, exponential, logarithmic, and trigonometric functions. The purpose is only to prepare students to take Math 160, Calculus I. May not be taken for credit if credit for any calculus course has already been received. Does not count toward a major or minor in mathematics or toward distribution in the natural science division. Prerequisite: Permission of instructor is required.

A study of real numbers, functions, limits, continuity, differentiation, and integration. All entering students planning to enroll in this course must take the Mathematics Placement Examination. Prerequisite: Placement in the course based on examination performance. Students who have received credit for Mathematics 155, 156, or 158 will not receive credit for Mathematics 160. Four 50-minute lectures per week.

A study of the applications of the definite integral; logarithmic, exponential, and trigonometric functions; techniques of integration; sequences and series; and indeterminate forms. Students completing Mathematics 170 with less than a “C” grade must request permission of the instructor to enroll in subsequent courses in Mathematics. Prerequisite: Placement in the course based on consultation with the department chair or the completion of Mathematics 160 with the grade of “C” or better. Four 50-minute lectures per week.

An introduction to concepts encountered in the study of abstract mathematics. Topics covered include logic, mathematical proofs, set theory, relations, functions, mathematical induction, and introductory number theory. The concepts of injectivity, surjectivity, and inverses are discussed as well as elementary computational tools such as the Division Algorithm and Euclid’s algorithm for the greatest common divisor. Additional topics may include cardinality, combinatorics, graph theory, algebraic structure, the real number system, and concepts of mathematical analysis. It is recommended that a major complete this course before the end of the sophomore year. Prerequisite: Completion of Mathematics 160 with a grade of “C” or better and sophomore standing, or permission of instructor.

A study of two- and three-dimensional vectors, vector-valued functions, continuity and differentiation of functions of several variables, multiple integration, and line integrals. Prerequisite: Placement in the course based on consultation with the department chair or the completion of Mathematics 170 with a grade of “C” or better. Four 50-minute lectures per week.

A survey of the progress of mathematics from ancient to modern times. Attention is given to the philosophy of mathematics and to the bearing of mathematics on other branches of knowledge. Prerequisite: Mathematics 210 or permission of instructor.

An examination of techniques for applying mathematical methods to real world problems. Emphasis is on analyzing the problem, constructing a suitable mathematical model, analyzing the model, transforming it into a form suitable for computation, and verifying the results. Applications are chosen from several different areas. Prerequisite: Mathematics 170 or permission of instructor.

An examination of methods of solving ordinary differential equations with emphasis on the existence and uniqueness of solutions of first order equations and second order linear equations. Topics may include Laplace transforms, systems of linear differential equations, power series solutions, successive approximations, linear differential equations, and oscillation theory with applications to chemistry and physics. Prerequisite: Mathematics 210.

(Also listed as Computer Science 230)
An introduction to the theories of finite-state machines, pushdown automata and Turing machines as well as the relation between automata and the formal languages they recognize. Students explore computational theory and its practical applications in lexical analysis and language parsing. Prerequisites: Computer Science 102 and Mathematics 205 or permission of instructor. Offered in alternate years.

An introduction to symbolic logic as a mathematical model of deductive thought. Topics covered include propositional logic, models, formal proofs, and the Completeness, Compactness, and Incompleteness Theorems. Additional topics from Computability theory or set theory may be included. Prerequisites: Mathematics 205 or permission of instructor.

A study of vector spaces, linear transformations, matrices, determinants, systems of linear equations, similarity, and characteristic values and vectors. This course may be applied toward the Mathematics requirement for a Computer Science major. Prerequisite: Mathematics 205 or 210.

An introduction to the notion of an algebraic structure concentrating on the simplest such structure, that of a group. Rings and fields are also discussed. Prerequisites: Mathematics 205 and 320, or permission of instructor.

A study of divisibility properties of integers, linear diophantine equations, the theory of congruencies, the Euler-Fermat Theorem, perfect numbers, elementary results on the distribution of prime numbers, quadratic residues and some non-linear Diophantine problems. Prerequisite: Mathematics 205 or permission of instructor.

An examination of the theory of calculus of a single variable. Topics include properties of the real numbers, topology of the real line, and a rigorous treatment of sequences, functions, limits, continuity, differentiation and integration. Prerequisites: Mathematics 205 and 210, or permission of instructor.

A study of differentiation and integration with complex variables, conformal representation, and the calculus of residues, with applications to geometry and physics. Prerequisites: Mathematics 205 and 210, or permission of instructor.

A study of mathematical models, sample space probabilities, random variables, expectation, empirical and theoretical frequency distributions, moment generating functions, sampling theory, correlation and regression. This course may be applied toward the Mathematics requirement for a major in Computer Science. Prerequisite: Mathematics 210.

A continuation of Mathematics 345 treating the testing of hypotheses and goodness of fit, small sample techniques, statistical design, non-parametric methods and sequential analysis. Prerequisite: Mathematics 345.

An introduction to modern geometry. Topics may be drawn from axiomatic, projective, affine or hyperbolic geometry. Related topics at the discretion of the instructor. Prerequisite: Mathematics 205.

An introduction to the theory of undirected and directed finite graphs. Topics include the Konigsberg Bridge Problem, planar and non-planar graphs, the five-color theorem and the four-color theorem, Hamiltonian circuits, shortest path algorithms, and problems of network flow. This course may be applied toward the Mathematics requirement for a major in Computer Science. Prerequisite: Mathematics 205 or permission of instructor.

The study of topics from combinatorics and discrete mathematical models including the pigeonhole principle, permutations and combinations of finite sets and multisets, binomial and multinomial coefficients, the inclusion-exclusion principle, recurrence relations, and generating functions. This course may be applied toward the Mathematics requirement for a major in Computer Science. Prerequisite: Mathematics 205 or permission of instructor.

A study of the theory and techniques of mathematical optimization with application to economics, scheduling problems, and other areas of interest to the class. Techniques may include linear, integer, and dynamic programming. This course may also be of interest to students majoring in Economics and Computer Science. Prerequisites: Mathematics 320 or permission of instructor.

A study of topological spaces and continuous maps, separation axioms, compactness, metric spaces, product spaces, connectedness and fixed point theorems. Proof techniques are emphasized. The course material ties together some ideas presented in the basic Mathematics courses. Prerequisite: Mathematics 340.

A study of rings and fields, including integral domains, polynomial rings, ideals, homomorphisms, and irreducibility of polynomials over prime fields. Other topics may include unique factorization domains, Euclidean domains, extension fields, automorphisms of fields and Galois theory, additional algebraic structures, or advanced topics in group theory. Prerequisite: Mathematics 325.

An extension of the material introduced in Mathematics 340. Topics may include sequences and series of functions, uniform convergence, power series and Taylor’s theorem, the topology of Euclidean space, the foundations of the calculus of several variables, the implicit function theorem, the inverse function theorem, and the Lebesgue integral. Prerequisite: Mathematics 340.

A study of topics in the foundations of mathematics, including elementary set theory, infinite sets, and the construction of the real number system from the rational numbers. Preparation for the Senior Project is also begun. Students learn to typeset mathematical writing in Latex, and all assignments are done in Latex. Articles from mathematical journals are read and portions of these articles are presented orally in class. Each student selects a topic and a project advisor for the Senior Project. Prerequisite: Mathematics 205 or permission of instructor.

The student completes research and writing for the Senior Project and gives an oral defense. Prerequisite: Permission of instructor.

Sophomore Seminar

The study of the process of mathematical problem solving. Common strategies used by mathematicians in discovering solutions to problems are covered. These strategies include induction, recursion, contradiction, exhaustion, generalization, counting, the pigeonhole principle, parity issues, drawing an appropriate figure, and geometric analysis. Effective oral and written communication of solutions is emphasized, culminating in a capstone project. This course may be counted toward the completion of a major or minor in Mathematics. Prerequisite: Mathematics 160 with a grade of at least C, or permission of instructor.