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# Mathematical Sciences (CPTR, MATH)

Associate Professors: Sprechini

Assistant Professors: Brandon, Pillai, Smith (Chair)

Lecturer: Mifsud, Reed

Instructors: Fuller, Mikula

- Major: Mathematics
- Courses required for both majors: 10 (not including zero or 1 credit courses)
- Math prerequisite for Math majors (not counted in major): placement out of or C- or better in MATH 127
- Math prerequisite for Applied Computer Science majors (not counted in major): placement out of or C- or better in MATH 127
- Non-credit Colloquium for Math: At least 3 semesters of MATH 449
- Capstone requirement for Math: One course from MATH 441, 442, 443, 444, or 445
- Capstone requirement for Applied Computer Science: CPTR 400
- Minors: Computational Science, Computer Science, Mathematics

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

**Computer Science (CPTR)**

#### Major Requirements

The Department of Mathematical Sciences offers a major in Applied Computer Science. The Applied Computer Science major balances theory and practice. Students majoring in this field are prepared for jobs in software engineering, applications development, web development, mobile development, machine learning / data science, or database administration. The major consists of 10 courses:

**Seven Core Courses**

CPTR 125 | Introduction to Computer Science |

CPTR 126 | Operating System Concepts |

CPTR 246 | Principles of Object-Oriented Programming |

CPTR 247 | Data Structures and Algorithms |

CPTR 400 | Software Engineering |

MATH 128 | Calculus with Analytic Geometry I |

MATH 216 | Discrete Mathematics |

**Three Required Electives**Students must take three additional courses from the lists of approved courses below, at least one of which must be a CPTR course. Interdisciplinary electives allow students to explore the business, social, political, and design principles inherent in the field.

**Computer Science Electives**Any additional CPTR course numbered 200 or higher

**Interdisciplinary Electives (may select up to two)**

ACCT 320 | Accounting Information Systems |

ART 213 | Introduction to 3D Character Animation |

ART 240 | Digital Photography I |

ART 243 | Introduction to Digital Media |

ART 313 | Advanced 3D Character Animation |

ART 340 | Digital Photography II |

BIO 215 | Introduction to Geographic Information Systems (GIS) |

BUS 352 | Decision Making |

BUS 438 | Financial Modeling |

ECON 241 | Introduction to Economic Research |

ECON 340 | Econometrics |

ECON 347 | Game Theory |

FVA 200 | Digital Film and Video I |

FVA 220 | Sound for the Screen |

FVA 300 | Digital Film and Video II |

MATH 231 | Differential Equations |

MATH 234 | Foundation of Mathematics |

MATH 434 | Abstract Algebra |

MATH 442 | Numerical Analysis |

MATH 443 | Linear Algebra |

MATH 444 | Partial Differential Equations |

MATH 445 | Graph Theory |

PHIL 225 | Symbolic Logic |

PHIL 340 | Philosophy of Cognitive Science |

PHYS 331 | Classical Mechanics |

PHYS 435 | Nonlinear and Complex Systems |

PHYS 336 | Math Methods of Physics |

PHYS 341 | Electronics |

PHYS 345 | Experimental Physics |

PHYS 445 | Data Analysis for Physics |

### Minor

The Department of Mathematical Sciences offers two computing minors: Computational Science and Computer Science. 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 thinking in their chosen discipline.

A minor in Computational Science consists of MATH 216; CPTR 125, 246, and 247; one of CPTR 345, 339, MATH 231, 342, or 442; 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.

A minor in Computer Science consists of MATH 216; CPTR 125, 246, 247, and two courses chosen from computer science courses numbered 220 or above or MATH 342 or 442.

### Writing Courses

A list of** **courses that, when scheduled as W courses, count toward the Writing Requirement, can be found on the Registrar’s website and in the GENERAL EDUCATION REQUIREMENTS section of the catalog.

**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: Math placement of level 4 or credit for MATH 115 or 127.*

**126**OPERATING SYSTEM CONCEPTS

An introduction to the design and implementation of modern operating systems. The topics include processes, threads, mutual exclusion, synchronization, memory management, virtual memory, processor scheduling, disk management, input and output, virtualization, multiprocessor systems, and security issues. This is a projects-based course where students implement programming assignments using a procedural language.

*Prerequisite:*

*A grade of C- or better in CPTR 125*.

**246**PRINCIPLES OF OBJECT-ORIENTED PROGRAMMING

Principles of effective programming based on concepts of inheritance, encapsulation, abstract classes, enumerations, inner classes, interfaces, and generic types and lists. Specifies abstract data types using various constructs provided by the programming language. Topics include functional programming using lambda expressions, polymorphism using various features provided by the collections framework, recursion and its applications, and exception handling frameworks. Includes laboratory experience.

*Prerequisite: A grade of C- or better in CPTR 125.*

**247**DATA STRUCTURES AND ALGORITHMS

Covers data abstraction and analysis of algorithms associated with data structures and emphasizes efficiency of algorithms. Topics includes bags, queues, stacks, Elementary and linearithmic sorting algorithms, elementary symbol tables, binary search trees, balanced search trees, and hash tables.

*Prerequisite: A grade of C- or better in CPTR 246 or consent of instructor.*

**300**COMPUTER ARCHITECTURE

An introduction to computer abstractions and language, representation of instructions and data, arithmetic in hardware, the structure of the processor, memory, and cache design. Also covers parallel processors, RISC and CISC architectures, and graphical processing units.

*Prerequisite:*

*A grade of C- or better in CPTR 125*.

**310**PRINCIPLES OF 2-D GAME DESIGN

An introduction to the design and implementation of a typical game engine and concepts of 2-D animation. Topics include event-driven programming, multi-tasking, introduction to a modern windowing toolkit, coding to a framework, design patterns, sprite creation, animation, and design and implementation of fully functional 2-D games. This course is lab-based where students implement projects covering relevant topics.

*Prerequisite: CPTR 246 or consent of instructor.*

**320**MOBILE APPLICATION DEVELOPMENT

A project-oriented course introducing students to an industry standard framework. Topics covered include interface design, Application Programmer Interface usage, and the design of fully functional mobile applications using a modern object-oriented programming language.

*Prerequisite: CPTR 246.*

**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.*

**340**ARTIFICIAL INTELLIGENCE

An introduction to generic intelligent agents, techniques in problem solving, knowledge representation and reasoning, problem solving by searching, the structure of adversarial games, and constraint satisfaction problems. Topics include first order logic, probabilistic reasoning, and the philosophical and ethical aspects of AI. This course introduces the student to machine learning, computer vision, and robotics and provides an overview of deep learning for natural language processing.

*Prerequisite: CPTR 246.*

**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.*

**400**SOFTWARE ENGINEERING

Covers principles of software metrics and identification of key performance indicators and provides a comparative study of various software development life cycles. Topics include software testing and frameworks used for integration and unit testing, frameworks used for revision control and management of cooperative work environments, generating requirements specifications, Unified Modeling Language (UML), interactive graphs, and design patterns. Students develop an application as the capstone project.

*Prerequisite: CPTR 247.*

**410**PARALLEL AND CONCURRENT COMPUTING

Offers an introduction to patterns of concurrent computations and theoretical models. This course covers processes, threads, concurrent architectures, Amdahl’s law, Gustafson-Barsis’s law, multi-threading, and thread pools concepts. Topics include shared variables, race conditions, concurrent computing paradigms, and synchronization issues in concurrent programming. Explores Critical Section, Deadlock, Livelock, resource starvation and theoretical models to study situations. This course covers additional topics such as the dining philosopher’s problem, Lamport’s Bakery Algorithm, Byzantine general’s problem, design of Semaphores, Mutexes, Monitors, and similar models.

*Prerequisite: CPTR 247.*

**470**INTERNSHIP

**N80-N89**INDEPENDENT STUDY

**490-491**INDEPENDENT STUDY FOR DEPARTMENTAL HONORS

## Mathematics (MATH)

### Major Requirements

A major in mathematics consists of CPTR 125, MATH 128 (or exemption by examination), 129 (or exemption by examination), 130, 234, 238, 432, 434, one from 441 442, 443, 444, or 445, and one additional MATH course numbered 200 or above. In addition, students are required to take at least three semesters of MATH 449,

Students interested in teacher certification should refer to the Department of Education listings. These students should take MATH 230 as their additional course.

### Capstone Requirement

All majors must successfully complete one course from 441, 442, 443, 444, or 445.

### Writing Courses

A list of** **courses that, when scheduled as W courses, count toward the Writing Requirement, can be found on the Registrar’s website and in the GENERAL EDUCATION REQUIREMENTS section of the catalog.

### Minor Requirements

A minor in mathematics consists of MATH 128 (or exemption by examination), 129 (or exemption by examination) 130, 234, 238, and one additional 4-credit course selected from CPTR 125, MATH 123, or any mathematics course numbered 200 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. Open only to students with math placement level of 1 or 2. *2 credits.*

**104**STATISTICAL LITERACY

A conceptual survey of sampling methods, descriptive statistics, and inferential statistics with an emphasis on active learning and simulation. This course is intended for students in MATH 100 who need a two-credit companion course, teacher certification candidates who need an additional two-credit math course, and social science majors who will eventually take introductory statistics.

*This course does not satisfy the statistics requirements for any major or minor and does not count for mathematics Distribution. Prerequisite: Math placement of level 2, 3, or 4; MATH 100; or concurrent enrollment in MATH 100. 2 credits. Offered every spring.*

**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: Math placement of level 2, 3, 4, or credit for 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: Math placement of level 2, 3, 4, or credit for 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: Math placement of level 2, 3, 4, or credit for 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: **Math placement of level 2, 3, 4, or credit for 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: Math placement of level 2, 3, 4, or credit for 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 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. *Prerequisite: Math placement of level 3 or 4 or credit for 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: Math placement of level 4 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 Cor better in MATH 128. *

**130 **

INTRODUCTION TO MATRIX ALGEBRA

Topics include matrix and vector arithmetic, systems of linear equations, applications of linear systems, matrix and vector geometry, linear programming, and eigenvectors/eigenvalues. This course emphasizes applications over theory and includes substantial work with Microsoft Excel. This course is intended for first year math majors and minors, as well as students in all other majors. *Prerequisite: Math placement of level 3 or 4, or permission of the instructor.*

**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 formal logic, proofs, induction, number theory, recursion, analysis of algorithms, sets, combinatorics and probability, relations, functions and matrices, graphs and trees, graph algorithms, computer logic, and languages, as well as an introduction to declarative programming. This course introduces the students to regular expression processing using a standard application programming interface. *Prerequisite: CPTR 125 or consent of instructor. *

**230**GEOMETRY AND STATISTICS FOR TEACHERS

Designed for mathematics majors pursuing secondary certification in mathematics, this course, covers the geometry and statistics necessary to be able to teach these subjects. Topics in geometry include the basics of geometry, segments and angles, parallel and perpendicular lines, triangles, triangle congruence, quadrilaterals, similarity, area of plane figures, volume and surface area of three-dimensional figures, right triangle trigonometry, circles, and transformations. Topics in statistics include graphical displays, descriptive statistics, correlation and regression, data collection, basic probability, conditional probability and independence, the binomial and normal distributions, estimation, confidence intervals, hypothesis tests, two sample problems, and chi-square tests. Students prepare lessons and teach this material to their peers, as though they were a high school teacher.

*Prerequisites: A grade of B- or better in MATH 127 or Level 4 Placement. Alternate years.*

**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 include symbolic logic, elementary proof methods, combinatorics, set theory, and mathematical induction. Students also learn a state of the art markup language for typesetting mathematical documents. This 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. *

**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.*

**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.*

**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. *

**341, 441**TOPICS IN ACTUARIAL SCIENCE

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 325, 332, 333. Students who enroll in MATH 441 are expected to prepare and deliver a 30-minute capstone presentation in MATH 449.

*Prerequisites: MATH 325 and MATH 332. Corequisite: MATH 449.*

**342, 442**TOPICS IN 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. Students who enroll in MATH 442 are expected to prepare and deliver a 30-minute capstone presentation in MATH 449.

*Prerequisite: CPTR 125 and MATH 129. Corequisite: MATH 449.*

**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 introduction to the commutative, associative, well-defined, closure, identity and inverse properties, which are the six defining properties of an Abelian Group. Explores several examples and non-examples of groups of numbers, matrices, subsets, integers mod *n*, functions, permutations and symmetries, as well as algebraic concepts that are universally true in all groups. Places a substantial emphasis on writing and revising formal mathematical proofs. *Prerequisites: A grade of C- or better in both MATH 130 and 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.*

**443**LINEAR ALGEBRA

Revisits matrices at a much more abstract level than MATH 130. Topics include matrix multiplication, determinants, invertibility and inverses of square matrices, real vector spaces and subspaces, linear independence, span, basis and dimension of a real vector space, as well as matrix transformations and linear transformations between vector spaces. Students write mathematical proofs involving these ideas.

*Prerequisites: A grade of C- or better in both MATH 130 and MATH 234. Either successful completion of MATH 434 or concurrent enrollment in MATH 434. Concurrent enrollment in MATH 449.*

**444**PARTIAL DIFFERENTIAL EQUATIONS

Serves as an introduction to partial differential equations with focus placed on solution techniques, both numerical and analytical. Solution techniques that may be covered include separation of variables, finite difference methods, and iterative methods. Several classical partial differential equations and various coordinate systems are studied.

*Prerequisites: A grade of C or better in CPTR 125, MATH 231, and MATH 238. Concurrent enrollment in MATH 449.*

**445**GRAPH THEORY

Introduces students to the fundamental concepts and ideas in graph theory, including subgraphs, trees, connectivity, optimization, covers, con nectivity, vertex and edge colorings, and embeddings. Applications discussed include topics such as tournament scheduling, spanning trees, Kuratowski’s Theorem, and embeddings on surfaces. Students write mathematical proofs involving these ideas.

*Prerequisites: A grade of C- or better in either MATH 234 or CPTR 247, or consent of the instructor. Concurrent enrollment in MATH 449.*

**449**MATH COLLOQUIUM

This required non-credit course for mathematics and actuarial science majors offers students a chance to give capstone presentations or poster sessions which were prepared in MATH 441, 442, 443, 444, or 445 as well as a chance to hear capstone presentations of their fellow majors and talks from faculty and external speakers.

*Meetings are announced at the start of each semester. One hour per week. Pass/Fail. Non-credit course.*

**470-479 **

INTERNSHIP

**N80-N89 **

INDEPENDENT STUDY

**490-491 **

INDEPENDENT STUDY FOR DEPARTMENTAL HONORS