# Mathematics and Computer Science

Chair: Associate Professor Fraboni

Professor: Schultheis; Associate Professors: Coleman, Hartshorn, Shank; Assistant Professor: Talbott; Instructor of Mathematics: Moser; Instructor of Computer Science: Mota; Visiting Instructor: Sommer; Adjunct Faculty: Ahmadizadeh, Brooks, Kovarik, Lebo, Moller, Rolon, Yudt

The program in mathematics seeks to develop in students the excitement of learning and discovering mathematics, and has three major objectives: to prepare mathematics majors for graduate study, for teaching mathematics, or for work in business and industry; to offer students in natural, social, or behavioral sciences or the humanities an introduction to mathematical concepts and skills necessary to use mathematics in those fields of interest; and to provide the non-specialist with an understanding of the contributions of mathematics to cultural development and the importance of mathematics in modern society.

Graphing calculators and computer programs are used to promote understanding of concepts and to investigate applications and modeling of real-world situations. Emphasis is placed on connections between various areas of mathematics and interpretation of results.

Computer science is the study of information processes and the creative application of abstraction and formal reasoning to solve problems. With the ever-increasing ubiquity of computational devices, computer science is an important field of study with diverse applications. From the natural and social sciences to the arts and humanities, computer science has become woven into the fabric of business, research, and everyday life.

At Moravian, the computer science program prepares students for professional life or graduate study. The core curriculum integrates a study of the theoretical underpinnings of the discipline with the practice of programming. Elective offerings explore the breadth of the discipline and expose students to the applications of computer science.

### The Major in Mathematics

The major in mathematics consists of ten course units in mathematics: nine course units in mathematics plus a capstone experience are required. All mathematics majors are required to take the following five courses: Mathematics 170 (or its equivalent sequence Mathematics 106-166), Mathematics 171, 211, 216, and 220. In addition, each mathematics major will engage in a capstone experience. Mathematics 370 will serve as the capstone experience for most majors. Successful completion of Mathematics 400-401 (Honors) can serve as an alternative capstone experience, although students who plan to pursue an Honors project are encouraged to take Mathematics 370 in their junior year.

For the additional four mathematics courses that students take to complete the major (other than the five required courses and the capstone experience), students must choose at least one from each of the groups described below. In addition, students must have at least three courses numbered 310-380, 390-399, or 400-401. (One of these three may be Mathematics 370.)

In order that students may understand and experience the breadth of mathematics, the department's major courses (other than the required courses and Mathematics 370) have been grouped into three areas: algebra/geometry topics; analysis topics; and applied mathematics topics. Current catalog courses in each of these areas are as follows:

- Algebra/geometry: Mathematics 313 and 340.

- Analysis: Mathematics 221, 327, 328, and 329.

- Applied mathematics: Mathematics 214, 225, 231, and 332.

As special topics or new courses are offered, they will be placed in the appropriate group.

Students must also choose two co-requisite courses from the following group of four courses: Physics 111, Physics 112, Computer Science 120, and Computer Science 121. Substitutions for this requirement may be made only with the approval of the Mathematics and Computer Science Department.

In fulfilling the above requirements, students planning to teach mathematics in secondary schools must complete the following courses: Mathematics 170 (or its equivalent sequence Mathematics 106-166), 171, 211, 216, 220, 231, 313, 340, 370, and one of the following: Mathematics 221, 327, 328, or 329.

Students who are seeking certification in early childhood education with a major in mathematics are required to complete Physics 111. The second co-requisite course is waived for these students. Students who are pursuing early childhood teacher certification with a major in mathematics do not need to complete Mathematics 125.

Courses in mathematics are listed below.

### The Minor in Mathematics

The minor in mathematics consists of five course units in mathematics: Mathematics 170 (or the equivalent sequence Mathematics 106-166), 171, and three courses numbered 210 or above, including at least one of the following: Mathematics 216, 220, 231.

### The Interdepartmental Major in Mathematics

The six mathematics courses that meet Set I requirements are Mathematics 170 (or its equivalent sequence Mathematics 106-166), 171, 211, and three additional courses chosen by the student with the approval of the advisor. Mathematics courses to be taken to satisfy Set II requirements will be determined by the student's prior preparation in mathematics and his or her educational objectives.

### The Major in Computer Science

The major in computer science consists of nine course units: Computer Science 120, 121, 222, 234, 244, 334, one of the following: Computer Science 320, 333, 364; and two additional courses in computer science, one of which must be numbered 310-380 or 390-399. Courses numbered 286, 288, or 381-388 and courses from other schools may not be used to satisfy the major requirements without prior written departmental approval. The major also requires Mathematics 170 (or its equivalent sequence Mathematics 106-166), 171, 216, and one additional mathematics course numbered 210 or higher, or a two-semester laboratory sequence in science. Because analytic and abstract reasoning is important to the study and application of computer science, majors are encouraged to take additional coursework in science, mathematics, and logic.

Courses in computer science are listed below.

### The Minor in Computer Science

The minor in computer science consists of Computer Science 120, 121, and three other course units numbered above 110. One of the following courses may, with departmental consent, be counted toward the computer science minor: Mathematics 214, 216, 225, 231; Philosophy 211. With departmental consent, one course with significant computing content from another program may be counted as one of the three elective course units towards the computer science minor.

### The Interdepartmental Major in Computer Science

The six courses that compose Set I of the interdepartmental major in computer science include Computer Science 120, 121, and four other courses numbered above 110, at least one of which is expected to be numbered 310-380 or 390-399. The additional courses in computer science and the six courses of Set II are selected by the student with the approval of the advisor.

### Courses in Mathematics

**100.2. Applications in Mathematics.** Investigation of a variety of mathematical models. Models to be investigated will be chosen from the areas of game theory, network models, voting theory, apportionment methods, fair division, and probability and statistics. We will apply these models in such diverse fields as biology, sociology, political science, history, and psychology. Does not count towards the mathematics major or minor. One 100-minute period.

*Fraboni*

**101.2. A History of Infinity.** Human beings have always struggled with the concept of infinity. Philosophers and mathematicians have gone mad contemplating its nature and complexity—and yet it is a concept now routinely used by school children. We will trace the history of this mind-boggling concept from Archimedes to Cantor through the eyes of the mathematician. Does not count towards the mathematics major or minor.

*Schultheis*

**102.2. Mathematics and Origami.** In this course, we will use origami (paper-folding) to explore topics in mathematics such as trisecting angles, solving cubic equations, and creating 3-dimensional polyhedra. In the process, we will see how mathematics has revolutionized origami over the past 50 years. Does not count towards the mathematics major or minor.

**104. Quantitative Reasoning and Informed Citizenship.** Quantitative reasoning skills to interpret and assess numerical arguments, with emphasis on issues relevant for informed and effective citizenship. Topics include creating and interpreting graphs and charts; single- and multiple-variable functions; linear, exponential, and logarithmic growth; indexes; inductive and deductive reasoning; decision theory; measures of center and spread of data; correlation; probability; expected value; experimental design; sampling and surveys. Three 70-minute periods. (F2)

*Staff*

**106. Analytic Geometry and Calculus I with Review, Part 1.** Beginning calculus with extensive review of algebra and elementary functions. Topics include Cartesian plane, algebraic functions, limits and continuity, introduction to the concept of derivative as a limit of average rates of change, theorems on differentiation, and the differential. Continued in Mathematics 166. The sequence Mathematics 106-166 is equivalent to Mathematics 170; credit may be earned for 106-166 or 170 but not both. (F2) Prerequisite: Placement by the Mathematics and Computer Science Department.

*Staff*

**107. Elementary Statistics.** Introduction to statistical concepts and methods without the use of calculus. Topics include descriptive statistics, elementary probability, discrete and continuous probability distributions, correlation and regression, estimation, and hypothesis testing. Mathematics 107 may not be taken for credit by students who have earned credit for Economics 156 or Mathematics 231. Three 70-minute periods. (F2)

*Staff*

**108. Functions and Derivatives with Applications.** Emphasis on concepts and applications to business and social and natural sciences. Use of graphing calculators. Topics include linear functions, polynomial functions, exponential functions, average rate of change, instantaneous rate of change, the derivative, interpretations of the derivative, rules of differentiation, and applications of the derivative. Includes review of algebra and elementary functions. May not be taken for credit by students who have completed Mathematics 106 or 170. (F2) Prerequisite: Placement by the Mathematics and Computer Science Department.

*Staff*

**109. Mathematics for Design.** Provides mathematical background and techniques useful to aspects of artistic design in the plane and in space. Essential mathematical concepts and tools applied to solve design problems. Topics include ratio and proportion, similarity, geometric constructions with Euclidean tools and dynamic geometry soft ware, properties of polygons and polyhedra, isometries and other geometric transformations in the plane and space, symmetry, and periodic designs, projections from space onto a plane. Spring. Three 70-minute periods. (F2)

*Hartshorn*

**125. Topics in Mathematics for Teaching.** Problem-solving, communication, and reasoning. Topics include estimation, geometry and spatial sense, measurement, statistics and probability, fractions and decimals, patterns and relationships, number systems, number relations, and number theory. Designed for prospective early childhood and middle level education teachers. Three 70-minute periods. (F2)

*Staff*

**166. Analytic Geometry and Calculus I with Review, Part 2.** Topics include exponential and trigonometric functions and their derivatives, related rates, extremum problems, logarithmic curve sketching, antidifferentiation, the definite integral, the fundamental theorem of calculus, area under a curve, and applications to business and economics. The sequence Mathematics 106-166 is equivalent to Mathematics 170; credit may be earned for 106-166 or 170 but not both. (F2) Prerequisite: Completion of Mathematics 106 with a grade of “C-” or better.

*Staff*

**170. Analytic Geometry and Calculus I. **Review of real numbers, analytic geometry and algebraic and transcendental functions. Limits and continuity. Definition, interpretations, and applications of the derivative. Definite and indefinite integrals, including the fundamental theorem of calculus. May not be taken for credit by students who have earned credit for Mathematics 166. (F2) Prerequisite: Placement by the Mathematics and Computer Science Department.

*Staff*

**171. Analytic Geometry and Calculus II.** Applications of the definite integral. Techniques of integration of both algebraic and transcendental functions. Indeterminate forms and improper integrals. Separate differential equations. Infinite sequences and series. (F2). *Prerequisite*: Placement by the Mathematics and Computer Science Department or completion of Math 170 or 166 with a grade of “C-” or better. *Note: *Students who are placed by the department into Math 171 and complete it with a grade of “B” or better will automatically receive credit for Math 170 if their transcript does not show credit for an equivalent course.

*Staff*

**211. Analytic Geometry and Calculus III. **Vectors in the plan and three-space. Parametric equations and space curves. Polar, cylindrical and spherical coordinates. Calculus of functions of more than one variable, including limits, partial derivatives, directional derivatives, multiple integration, and applications. Prerequisite: Completion of Math 171 with a grade of “C-” or better.

*Staff*

**214. Mathematical Methods in Operations Research.** Introduction to mathematical techniques to model and analyze decision problems. Linear programming, including sensitivity analysis and duality, network analysis, decision theory, game theory, queuing theory. Prerequisites: Mathematics 171. Spring, alternate years.

*Staff*

**216. Discrete Mathematical Structures and Proof.** Elementary mathematical logic and types of mathematical proof, including induction and combinatorial arguments. Set theory, relations, functions, cardinality of sets, algorithm analysis, basic number theory, recurrences, and graphs. Writing intensive. Prerequisite: Mathematics 171. Fall.

*Staff*

**220. Linear Algebra.** Vector spaces and linear transformations, matrices, systems of linear equations and their solutions, determinants, eigenvectors and eigenvalues of a matrix. Applications of linear algebra in various fields. Prerequisite: Mathematics 171. Spring.

*Schultheis*

**221. Differential Equations.** Various methods of solution of ordinary differential equations, including first-order techniques and higher-order techniques for linear equations. Additional topics include applications, existence theory, and the Laplace transform. Prerequisite: Mathematics 211. Spring.

*Schultheis*

**225. Numerical Analysis.** Numerical techniques for solving applied mathematical problems. Topics include interpolation and approximation of functions, solution of non-linear equations, solution of systems of linear equations, and numerical integration, with error analysis and stability. Prerequisites: Mathematics 171 and a course in computer science. Spring, alternate years.

*Fraboni, Hartshorn*

**231. Mathematics Statistics I.** A calculus-based introduction to probability and statistical concepts and methods. Topics include descriptive statistics, probability, discrete and continuous probability distributions, regression analysis, sampling distributions and the central limit theorem, estimation and hypothesis testing. Prerequisite: Mathematics 171.

*Shank*

**313. Modern Algebra.** Group theory, including structure and properties: subgroups, co-sets, quotient groups, morphisms. Permutation groups, symmetry groups, groups of numbers, functions, and matrices. Brief study of rings, subrings, and ideals, including polynomial rings, integral domains, Euclidean domains, unique factorization domains, and fields. Prerequisite: Mathematics 216 or permission of instructor. Fall.

*Schultheis*

**327. Advanced Calculus. **Differential and integral calculus of scalar and vector functions. Differential calculus includes differentials, general chain rule, inverse and implicit function theorems, and vector fields. Integral calculus includes multiple integrals, line integrals, surface integrals, and theorems of Green and Stokes. Prerequisite: Mathematics 211. Fall, alternate years.

*Hartshorn*

**328. Introduction to Analysis.** Rigorous study of real-valued functions, metric spaces, sequences, continuity, differentiation, and integration. Prerequisites: Mathematics 211 and Mathematics 216 or 220. Spring, alternate years.

*Fraboni, Hartshorn*

**329. Complex Analysis.** Analytic functions, complex integration, application of Cauchy's theorem. Prerequisite: Mathematics 211. Spring, alternate years.

*Fraboni, Schultheis*

**332. Mathematical Statistics II.** Development of statistical concepts and methods. Multivariate probability distributions, point and interval estimation, regression analysis, analysis of variance, chi-square goodness-of-fit and contingency table analysis, and nonparametric tests. Prerequisite: Mathematics 231. Spring.

*Shank*

**340. Higher Geometry.** Topics in Euclidean two- and three-dimensional geometry from classical (synthetic), analytic, and transformation points of view. Transformations include isometries, similarities, and inversions. Construction and properties of two- and three-dimensional geometric figures. Brief study of some non-Euclidean geometries. Prerequisite: Mathematics 216 or 220. Fall, alternate years. Writing-intensive.

*Hartshorn, Staff*

**370. Mathematics Seminar.** A capstone course designed to review, unify, and extend concepts developed in previous mathematics courses. Students will read historical, cultural, and current mathematical material. They will express their mathematical understanding through writings, oral presentations, and class discussions. Assignments will include both expository and research-oriented styles of writing, including a significant individual research project. Prerequisite: Mathematics 216 and any 300-level course in mathematics.

*Fraboni, Schultheis*

**190-199, 290-299, 390-399. Special Topics.
**

**286, 381-384. Independent Study.
**

**288, 386-388. Internship.
**

**400-401. Honors.**

### Courses in Computer Science

**105. Fundamental Ideas in Computer Science.** Emphasis on contributions that computer science has made to contemporary society. Topics include physical and logical aspects of computers, algorithms and problem-solving, introduction to programming, and simple computer architecture, supplemented by laboratory exercises in which students create programs or utilize existing programs. Recommended for those not intending a major or minor in the department. (F2)

*Staff*

**120. Computer Science I.** Introduction to the discipline with emphasis on algorithm design and program development. Emphasis on problem-solving activity of developing algorithms. Topics include computer organization, computer usage and application, programming languages, software engineering, data structures, and operating systems. Recommended for students intending to develop or maintain software in their own area of concentration. (F4)

*Coleman*

**121. Computer Science II.** Emphasis on data and procedural abstraction. Basic organizations of instructions and data in hardware design and software development. Topics include encoding schemes for instructions and data, representative machine architectures, data representations in computer memory and in high-level languages. Prerequisite: Computer Science 120.

*Coleman*

**217. Digital Electronics and Microprocessors.** (Also Physics 217) Laboratory-oriented course in computer hardware for science, mathematics, and computer-science students. Topics include logic gates, Boolean algebra, combinational and sequential logic circuits, register-transfer logic, microprocessors, addressing modes, programming concepts, microcomputer system configuration, and interfacing.

*Staff*

**222. Computer Organization. **A study of what happens when a computer program is executed. We examine the organization of a modern computer from the perspective of a programmer; our examination focuses on the layers of abstraction between a high-level language program and its execution. Topics include the set of instructions that a processor supports, how a high-level language program is translated into this instruction set, how a processor carries out instructions, concurrency, the memory hierarchy, and storage systems. Prerequisite: Computer Science 121.

Staff

**234. Introduction to Software Engineering.** An introduction to professional software development using object-oriented techniques. Topics include the use of object-oriented design as a tool for building correct and maintainable software systems, test-driven development, best-practices in object-oriented design and development informed by component-based engineering, advanced object oriented language features, and languages for communicating design. Prerequisite: Computer Science 244.

*Staff*

**244. Data Structures and Analysis of Algorithms. **Issues of static and dynamic aggregates of data. Topics include logical characteristics of various data organizations, storage structures implementing structured data, design and implementation of algorithms to manipulate storage structures, and classical applications of data structures. Representative data structures include stacks, queues, ordered trees, binary trees, and graphs. Implementation and performance issues of contiguous and linked storage. Prerequisites: Computer Science 121 and Mathematics 170 (or 106-166).

*Coleman*

**260. Artificial Intelligence. **Topics and methods for emulating natural intelligence using computer-based systems. Topics include learning, planning, natural-language processing, machine vision, neural networks, genetic algorithms. Prerequisite: Computer Science 120.

*Coleman*

**265. Database Systems. **Data file organization and processing, indexed data files and indexing techniques, database design; database applications; query languages; relational databases, algebra, and calculus; client-server models and applications; database system implementation and web programming. Prerequisite: Computer Science 120 or permission of the instructor.

*Staff*

**320. Networking and Distributed Computing.** Theory and practice of concurrent programming. We examine the difference between shared- and distributed-memory models of computation, what problems are computable in parallel and distributed systems, the principle differences between concurrent and sequential programming, as well as data structures and algorithms for concurrent programming. Prerequisite: Computer Science 244.*
Staff*

**330. Game Programming.** Focus on the mathematics and algorithms necessary to create computer games and the software engineering principles used to manage the complexity of these programs. Topics include advanced programming in an object-oriented language, the mathematics of game programming, artificial intelligence, event-loop programming, and 2D graphics. Prerequisite: Computer Science 244.

Staff

**333. Operating Systems. **The structure and organization of operating systems, how modern operating systems support multiprogramming (e.g., processes, threads, communication and synchronization, memory management, etc.), files systems, and security. Programming projects involve both using operating system services as well as the implementation of core operating system components. Prerequisites: Computer Science 222 and 244.

*Coleman*

**334. Systems Design and Implementation.** Project-oriented study of ideas and techniques for design and implementation of computer-based systems. Topics include project organization, interface design, documentation, and verification. Prerequisites: Computer Science 234 and senior standing. Writing-intensive.

*Coleman*

**335. Simulation. **When real-world experiments are either too dangerous or too expensive to perform, computer simulation is used as an alternative. In addition to considering how to model real-world problems using computer simulation, this course studies other relevant topics including how to generate random data using a deterministic machine and how to collect and display data in a meaningful way.

**364. Foundations of Computing. **Theoretical aspects of computing. Topics include formal languages (regular, context-free, and context-sensitive grammars), automata (finite-state machines, push-down automata, and Turing machines), limitations of respective computational models, and unsolvable problems. Prerequisite: Computer Science 244.

*Coleman*

**190-199, 290-299, 390-399. Special Topics.
**

**286, 381-384. Independent Study.
**

**288, 386-388. Internship.
**

**400-401. Honors.**