# Mathematical Sciences

Associate Professors: Peluso, Sprechini (Chair)

Assistant Professors: deSilva, Smith

Instructors: Abercrombie, Brandon, Laird, Mifsud, Reed

- Major: Mathematics
- Courses required for major: 10 or 11 (depending on choices)
- Math prerequisite (not counted in major): placement out of or C- or better in Math 127
- Non-credit Colloquium: 4 semesters
- Capstone requirement: 4 semesters of MATH 449
- Minors: Computer Science, Computational Science, Mathematics

The Department of Mathematical Sciences offers a major program in mathematics and minor programs in computer science, computational science, and mathematics. Interested students may want to investigate the interdisciplinary actuarial mathematics major as well.

## Mathematics (MATH)

### Major Requirements

A major in mathematics consists of CPTR 125, MATH 128 (or exemption by examination), 129, 130, 234, 238, 432, 434, and one of the following three options: 1) MATH 332 and one other mathematics course numbered 216 or above; 2) MATH 214 and one other mathematics course numbered 220 or above; 3) MATH 123 and two other mathematics courses numbered 220 or above. In addition, four semesters of MATH 449 are required. All majors are advised to elect PHIL 225 (in the freshman year); PHIL 333; and PHYS 225, 226.

Students interested in teacher certification should refer to the Department of Education listings.

### Capstone Requirement

All majors must successfully complete four semesters of MATH 449.

### Writing Courses

The following course, when scheduled as a W course, counts toward the Writing Requirement: MATH 234 and 434.

### Minor Requirements

A minor in mathematics consists of MATH 128 (or exemption by examination), 129, and 238; either 216 or 234; one additional course selected from 130, 214, or any course numbered 200 or above; and two semesters of MATH 449. The two semesters of MATH 449 may be replaced by any course numbered 220 or above.

**100 **

INDIVIDUALIZED LABORATORY INSTRUCTION IN BASIC ALGEBRA

A computer-based program of instruction in basic algebra including arithmetic and decimals, fractions, the real number line, factoring, solutions to linear and quadratic equations, graphs of linear and quadratic functions, expressions with rational exponents, algebraic functions, exponential functions, and inequalities. This course is limited to students placed therein by the

Mathematics Department. *2 credits.*

**106 **

COMBINATORICS

An introduction to the analysis of counting problems. Topics include permutations, combinations, binomial coefficients, inclusion/exclusion principle, and partitions. The nature of the subject allows questions to be posed in everyday language while still developing sophisticated mathematical concepts. *Prerequisite: Credit for or exemption from MATH 100.*

**109 **

APPLIED ELEMENTARY CALCULUS

An intuitive approach to the calculus concepts with applications to business, biology, and social-science problems. Not open to students who have completed MATH 128. *Prerequisite: Credit for or exemption from MATH 100. *

**112**

FINITE MATHEMATICS FOR DECISION-MAKING

An introduction to some of the principal mathematical models, not involving calculus, which are used in business administration, social sciences, and operations research. Includes both deterministic models such as graphs, networks, linear programming and voting models, and probabilistic models such as Markov chains and games.

*Prerequisite: Credit for or exemption from MATH 100.*

**115**

APPLIED DISCRETE MATHEMATICS

Introduction to discrete structures and their applications in computer science. Topics include elementary logic, discrete number systems, elementary combinatorial theory, finite automata, formal language constructs, and general algebraic structures including Boolean algebras, graphs, and trees. Laboratory experience is included using current software. *Prerequisite: Credit for or exemption from MATH 100 .*

**123 **

INTRODUCTION TO STATISTICS

Topics include tabular and graphical descriptive statistics, discrete and continuous probability distributions, Central Limit Theorem, one- and two-sample hypotheses tests, analysis of variance, chi-squared tests, nonparametric tests, linear regression, and correlation. Other topics may include index numbers, time series, sampling design, and experimental design. Also includes some use of statistical software. *Prerequisite: Credit for or exemption from MATH 100.*

**127**

PRECALCULUS MATHEMATICS** **The study of polynomial, rational, exponential, logarithmic, and trigonometric functions, their graphs and elementary properties. This course is an intensive preparation for students planning to take Calculus (MATH 128-129) or Matrix Algebra (MATH 130) or those whose major specifically requires Precalculus. This course is taught solely as a review of topics which must be mastered by students who intend to take MATH 128 or MATH 130. Prerequisite:

*Credit for or exemption from MATH 100.*

*May not be used to satisfy Distribution requirements.*

**128**

CALCULUS WITH ANALYTIC GEOMETRY I

Differentiation and integration of algebraic functions, conic sections and their applications, graphing plane curves, applications to related rate and external problems, areas of plane regions, volumes of solids of revolution, and other applications. *Prerequisite: Exemption from or a grade of C- or better in MATH 127. *

**129 **

CALCULUS WITH ANALYTIC GEOMETRY II

Differentiation and integration of trigonometric, exponential, logarithmic, and transcendental functions and their inverses; volumes, arc-length, surface-area, and other applications; techniques of integration including integration by parts, partial fractions, trigonometric substitutions first order differential equations; numerical integration; L’Hôpital’s Rule, improper integrals and their convergence, parametric equations and plane polar coordinates; infinite sequences and series, and tests for convergence. *Prerequisite: exemption from or a grade of C- or better in MATH 128. *

**130 **

INTRODUCTION TO MATRIX ALGEBRA

A study of systems of linear equations and matrix arithmetic, points and hyperplanes, infinite dimensional geometries, bases and linear independence, matrix representations of linear mappings, the fixed point problem, special classes of matrices. *Prerequisite: MATH 127 or its equivalent. *

**214**

MULTIVARIABLE STATISTICS** **The study of statistical techniques involving several variables. Topics include confidence intervals and hypothesis tests about means and variances, confidence intervals and hypothesis tests with simple and multiple linear regression and correlation, assessing appropriateness of linear regression models, one- and two-way analysis of variance with post hoc tests, analysis of covariance, and analysis of contingency tables. Other topics may include discriminant analysis, cluster analysis, factor analysis, and canonical correlations, repeated measure designs, time series analysis, and nonparametric methods. Also includes extensive use of a statistical package (currently SPSS).

*Prerequisite: A grade of C- or better in MATH 123, or a grade of C- or better in both MATH 128 and any mathematics course numbered 129 or above, or consent of instructor.*

**216 **

DISCRETE MATHEMATICS

An introduction to discrete structures. Topics include equivalence relations, partitions and quotient sets, mathematical induction, recursive functions, elementary logic, discrete number systems, elementary combinatorial theory, and general algebraic structures emphasizing semi-groups, lattices, Boolean algebras, graphs, and trees. *Prerequisite: CPTR 125 or consent of instructor. *

**231 **

DIFFERENTIAL EQUATIONS

A study of ordinary differential equations and linear systems. Solution techniques include reduction of order, undetermined coefficients, variation of parameters, Laplace transforms, power series, and eigenvalues and eigenvectors. May also include an introduction to numerical methods. *Prerequisite: A grade of C- or better in MATH 129. MATH 130 recommended.*

**233 **

COMPLEX VARIABLES

Complex numbers, analytic functions, complex integration, Cauchy’s theorems and their applications. *Corequisite: MATH 238. Alternate years.*

**234 **

FOUNDATIONS OF MATHEMATICS

Topics included are the nature of mathematical systems, essentials of logical reasoning, and axiomatic foundations of set theory. Other topics may include approaches to the concepts of infinity and continuity, and the construction of the real number system. The course serves as a bridge from elementary calculus to advanced courses in algebra and analysis. *Prerequisite: A grade of C- or better in MATH 129 **or 130; both courses recommended. Corequisite: MATH** 449.*

**238 **

MULTIVARIABLE CALCULUS

Algebra, geometry, and calculus in multidimensional Euclidean space; n-tuples, matrices; lines, planes, curves, surfaces; vector functions of a single variable, acceleration, curvature; functions for several variables, gradient; line integrals, vector fields, multiple integrals, change of variable, areas, volumes; Green’s theorem. *Prerequisites: A grade of C- or better in MATH 129 and either MATH 130 or 231. *

**321 **

INTRODUCTION TO NUMERICAL ANALYSIS

Topics from the theory of interpolation, numerical approaches to approximating roots and functions, integration, systems of differential equations, linear systems, matrix inversion, and the eigenvalue problem. *Prerequisites: CPTR 125 and MATH 129. MATH 130 strongly recommended. Cross-listed as CPTR 321.*

**324 **

AUTOMATA, FORMAL LANGUAGES, AND COMPUTABILITY

The study of finite state machines, pushdown stacks, and Turing machines along with their equivalent formal language counterparts. Topics include results on computability, including results regarding the limits of computers and specific problems that cannot be solved. *Prerequisite: MATH 216 or 234. Cross-listed as CPTR 324. Alternate years. *

**325**

THEORY OF INTEREST WITH APPLICATIONS

Explores the mathematical theory of interest in both finite and continuous time, with some applications to economics and finance. Specifically, these concepts are applied in the use of the various annuity functions and in the calculation of present and accumulated value for various streams of cash flows as a basis for future use in reserving, valuation, pricing, duration, asset/liability management, investment income, capital budgeting, and contingencies. *Prerequisite: A grade of C or better in MATH 129.*

**330**

TOPICS IN GEOMETRY

An axiomatic treatment of Euclidean geometry with an historical perspective. *Prerequisite: **A grade of C or better in either MATH 129 or 130. Alternate years.*

**332 **

MATHEMATICAL STATISTICS I

A study of probability, discrete and continuous random variables, expected values and moments, univariate distributions, joint distributions, marginal distributions, correlation. *Corequisite: MATH 238. Alternate years. *

**333 **

MATHEMATICAL STATISTICS II

A study of conditional distributions, least squares line, sampling, point estimation, sampling distributions, interval estimation, test of hypotheses, regression and linear hypotheses, experimental design models. *Prerequisites: MATH 332*. *Alternate years. *

**338 **

OPERATIONS RESEARCH

Queuing theory, including simulations techniques, optimization theory, including linear programming, integer programming, and dynamic programming; game theory, including two-person zero-sum games, cooperative games, and multiperson games. *Prerequisite: MATH 112 or 130. Alternate years. *

**400 **

TOPICS IN ACTUARIAL MATHEMATICS

Study of topics selected from those covered on the examinations administered by the Society of Actuaries, with the exception of the topics already covered in MATH 332, 333. *Prerequisite: Varies depending on the topic being taught. May be repeated for credit with consent of instructor when topics are different,*

**432 **

REAL ANALYSIS

An introduction to the rigorous analysis of the concepts of real variable calculus in the setting of normed spaces. Topics from: topology of the Euclidean plane, completeness, compactness, the Heine-Borel theorem; functions on Euclidean space, continuity, uniform continuity, differentiability; series and convergence; Riemann integral. *Prerequisites: MATH 238 and a grade of C- or better in MATH 234. *

**434 **

ABSTRACT ALGEBRA

An integrated approach to groups, rings, fields, and vector spaces and functions which preserve their structure. *Prerequisites: MATH 130 and a grade of C- or better in MATH 234.*

**438 **

SEMINAR

Topics in modern mathematics of current interest to the instructor. A different topic is selected each semester. Designed to provide junior and senior mathematics majors and other qualified students with more than the usual opportunity for concentrated and cooperative inquiry. *Prerequisite: Consent of instructor. 2 credits. May be repeated for credit when topics are different.*

**449 **

MATH COLLOQUIUM

This required non-credit course for mathematics majors and minors and actuarial mathematics majors offers students a chance to hear, prepare, and give presentations on topics related to, but not directly covered in formal math courses. Students either prepare or present a lecture on some appropriate topic in mathematics. Mathematics majors present two lectures, typically one during the junior year and one during the senior year. Actuarial mathematics majors and mathematics minors present one lecture. A letter grade is given based on attendance and on either presentation preparation or the presentation given.*One hour per week. Non-credit.*

**470-479 **

INTERNSHIP

**N80-N89 **

INDEPENDENT STUDY

**490-491 **

INDEPENDENT STUDY FOR DEPARTMENTAL HONORS

## Computer Science (CPTR)

### Minor

The Department of Mathematical Sciences offers two computing minors: Computer Science and Computational Science.

A minor in computer science consists of either Math 115 or 216; CPTR 125, 246, 247, and two other computer science courses numbered 220 or above.

A minor in computational science consists of either Math 115 or 216; CPTR 125, 246, and 247; one of CPTR 321, 345, or 339; and an approved computational research project in the student’s major discipline which can be fulfilled through ASTR/PHYS 448, BIO 447, CHEM 449, Independent Study, Honors Project, Research Experience for Undergraduates (REU), or other research experience. Computational science is the study of the application of computation to the sciences. The minor in computational science provides students with a core understanding of computer-based problem solving and prepares them to apply that computational power in their chosen discipline.

### Writing Course

The following course, when scheduled as a W course, counts toward the Writing Requirement: CPTR 247.

**101 **

MICROCOMPUTER FILE MANAGEMENT

An introduction to a file-management system, i.e. a database system that uses a single file, in the Windows environment. *2 credits. *

**102 **INTRODUCTION TO VIRTUAL WORLDS

Using Carnegie Mellon’s

*Alice*software, students create 3-D animations for both storytelling and gaming applications. Class time in this project-based course is roughly split one-third demonstration/lecture and two-thirds hands-on project development. Topics include storyboarding, object-oriented modular construction, decision and repetition control structures, and event handling.

*2 credits.*

**125 **

INTRODUCTION TO COMPUTER SCIENCE

Introduction to the discipline of computer science with emphasis on programming utilizing an object-oriented high-level programming language. Topics include algorithms, program structure, and problem solving techniques. Includes laboratory experience. *Prerequisite: Credit **for or exemption from MATH 100. *

**246 **

PRINCIPLES OF ADVANCED PROGRAMMING

Principles of effective programming, including structured and object oriented programming, stepwise refinement, debugging, recursion, inheritance, polymorphism, pointers, and linked data structures. Includes laboratory experience. *Prerequisite: A grade of C- or better in CPTR 125.*

**247 **

DATA STRUCTURES

Representation of data and analysis of algorithms associated with data structures. Topics include representation of lists, trees, graphs, algorithms for searching and sorting. Emphasizes efficiency of algorithms.** ***Prerequisite: A grade of C- or better in CPTR 246 or consent of instructor. *

**321 **

INTRODUCTION TO NUMERICAL ANALYSIS

Topics from the theory of interpolation, numerical approaches to approximation of roots and functions, integration, systems of differential equations, linear systems, matrix inversion, and the eigenvalue problem. *Prerequisites: CPTR 125 and MATH 129. MATH 130 strongly recommended. Cross-listed as MATH 321.*

**322**

INTRODUCTION TO WEB-BASED PROGRAMMING

Intermediate programming on the World Wide Web. Topics include client/server issues in Web publishing and current programming languages used in Web development. Includes laboratory experience. *Prerequisite: CPTR 125. *

**324 **

AUTOMATA, FORMAL LANGUAGES, AND COMPUTABILITY

The study of finite state machines, pushdown stacks, and Turing machines along with their equivalent formal language counterparts. Topics include results on computability, including results regarding the limits of computers and specific problems that cannot be solved. *Prerequisite: MATH 216 or 234. Cross-listed as MATH 324. Alternate years.*

**339**

INTRODUCTION TO DATABASE SYSTEMS

An introduction to the relational database model and SQL. Topics include but are not limited to relational model of data; ER diagrams; schema; SQL commands for table construction, updating, and querying; transaction processing; and database integrity. Includes laboratory experience. *Prerequisite: CPTR 125. *

**345 **

INTRODUCTION TO COMPUTER GRAPHICS

An introduction to graphics software with emphasis on the algorithms, data structures, and application programming interfaces that support the creation of two and three dimensional image generation and animation. *Alternate years.*

**470 **

INTERNSHIP

**N80-N89 **

INDEPENDENT STUDY

**490-491 **

INDEPENDENT STUDY FOR DEPARTMENTAL HONORS