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Mathematical Sciences (CPTR, MATH)

Assistant Professors: Brandon, Pillai, M. Smith (Chair)
Lecturers: Reed, G. Smith
Instructor: Cowden, Rublein

Major: Applied Computer Science and 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.

Applied 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 or MATH 234	Discrete Mathematics or Foundations of 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.

Any course counted or substituted for one of the seven core courses may not be used to fulfill the major electives requirement for this degree plan.

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	Advanced Topics in 3D Animation
ART 240	Digital Photography I
ART 243	Introduction to Digital Media
ART 313	Advanced 3D Character Animation
ART 340	Digital Photography II
BUS 352	Decision Making
BUS 438	Financial Modeling
ECON 241	Introduction to Economic Research
ECON 340	Econometrics
ECON 347	Game Theory
ENV 215	Introduction to Geographic Information Systems (GIS)
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 345	Computational Graph Theory
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 336	Math Methods of Physics
PHYS 341	Electronics
PHYS 345	Experimental Physics
PHYS 435	Nonlinear and Complex Systems
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
BASIC ALGEBRA
Arithmetic with integers and rational numbers, solving linear equations, graphing linear equations, solving linear systems, exponents and polynomials, factoring, rational expressions and equations, radical expressions and equations, completing the square and the quadratic formula. Open only to students with math placement of level 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.

140
HISTORY OF MATHEMATICS
Explores the mathematics of past civilizations and cultures such as ancient Egypt, Babylon, China, Greece, and India along with Mayan, Incan, and Arabic mathematics leading up to the invention of calculus in Western Europe. Specific attention will be given to the various mathematical tools and techniques used by ancient and modern civilizations. This course includes learning logical thinking, analytical skills, and effective decision-making through practical or abstract applications. Prerequisite: Math placement of level 2, 3, or 4, or credit for MATH 100. Fulfills Global Cultural Diversity Requirement. Credit may not be earned for both MATH 140 and 240.

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 two of CPTR 125, MATH 128, 129, or 130.

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. Prerequisite: A grade of C- or better in MATH 129.

240
ADVANCED HISTORY OF MATHEMATICS
Explores the mathematics of past civilizations and cultures such as ancient Egypt, Babylon, China, Greece, and India along with Mayan, Incan, and Arabic mathematics leading up to the invention of calculus in Western Europe. Specific attention will be given to the various mathematical tools and techniques used by ancient and modern civilizations. This course is designed for students majoring or minoring in STEM fields and will have a specific focus on how mathematical advances from various civilizations have had a profound effect on modern mathematics, including ideas found in Calculus. Prerequisite: A grade of C- or better in MATH 128. Fulfills Global Cultural Diversity Requirement. Credit may not be earned for both MATH 140 and 240.

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.

345
COMPUTATIONAL GRAPH THEORY
Introduces students to the fundamental concepts and ideas in graph theory, including subgraphs, trees, connectivity, optimization, covers, connectivity, vertex and edge colorings, and embeddings. Applications discussed include topics such as tournament scheduling, spanning trees, Kuratowski’s Theorem, and embeddings on surfaces. This course emphasizes computational graph theory as it relates to modeling, computer science, and other fields. This course is an option for Mathematics majors who will take a capstone other than MATH 445. Prerequisites: A grade of C- or better in either MATH 238 or CPTR 247, or consent of the instructor. Credit may not be earned for both MATH 345 and MATH 445.

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.

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.

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. Credit may not be earned for both MATH 345 and MATH 445.

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