**The Faculty**

Janet L. Beery

James Bentley

Joanna Bieri

Richard N. Cornez

Elizabeth Doolittle

Deon Garcia

Alexander Koonce

Steven Morics

Tamara Veenstra

**The Major**

The Bachelor of Science degree in mathematics offers both breadth and depth in mathematical preparation appropriate for graduate studies or careers in a variety of fields. The requirements for the major in mathematics include both theoretical and applied courses. Students should plan with their advisors as they select elective courses appropriate to their educational interests and goals.

Students declaring a mathematics major are required to have a 2.3 cumulative GPA in the sophomore core mathematics sequences (MATH 201, or MATH 204, MATH 221, MATH 241). In order for students to maintain satisfactory progress toward graduation in four years, students should have successfully completed MATH 201, or MATH 204, MATH 221, and MATH 241 by the end of their sophomore year. In order for transfer students to graduate in two years after matriculation to the University of Redlands, they need to have completed the Calculus sequence (I, II, and III) and Linear Algebra prior to transferring to Redlands; additionally, a course in discrete mathematics or introduction to proofs is highly recommended.

Learning outcomes for this program may be found at http://www.redlands.edu/BS-MATH/learning-outcomes.

**Bachelor of Science**

*Please note: MATH 122 is required except for those mathematics majors who begin the calculus sequence with MATH 221. Math 201 or MATH 204 or MATH 205 may be taken. *

Riemann sums and the definite integral; techniques of integration and application of integrals; introduction to differential equation; sequences and series.

Prerequisite: MATH 121 or MATH 119 or by permission.

Study of discrete mathematical topics important in both mathematics and computer science, including combinatorial techniques, sets and relations, algorithms, and graph theory.

Prerequisite: MATH 122.

Offered as needed.

Introduction to the nature and structure of mathematics. Through active study and exploration of a selected area of discrete mathematics, students develop problem-solving skills, as well as skills in proving mathematical theorems. A different topic is selected each year based on student and faculty interest. May be repeated for up to 6-degree credits with departmental permission.

Prerequisite: A minimum grade of 2.7 in MATH 221.

Mathematical techniques in cryptology with a focus on problem solving and forming conjectures. Monoalphabetic ciphers and frequency analysis, polyalphabetic ciphers, public key cryptography and block ciphers. Incorporates results from discrete mathematics, number theory, probability, and permutations. Repeating the course for grade replacement is allowed only once and requires permission.

Topics in multivariable calculus related to differentiation and integration. Sequences, series and Taylor approximations.

Study of vector spaces. Topics include systems of linear equations, matrices, the geometry of vectors, vector spaces, linear transformations, eigenvalues and eigenvectors, determinants, and selected applications.

Prerequisite: MATH 221.

Selected topics are assigned for individual students to research and present to mathematics majors and faculty. A paper is submitted prior to presentation of the topic.

Prerequisite: Senior standing and successful completion of a minimum of seven mathematics courses at the 200 level or higher that must include at least one from the Mathematical Reasoning Requirements. At least one of MATH 321 or MATH 341 is strongly recommended.

*Please note: students must take at least two courses from those listed, with at least one numbered 321 or above.*

Study in two related areas: number theory and history of mathematics. Number theory topics include primes, Diophantine equations, congruences, number theoretic functions, modern applications, and unsolved problems of number theory. Readings include primary and secondary historical sources.

Prerequisite: An average GPA of 2.3 in MATH 241 and one of MATH 201, or MATH 204, or MATH 205.

Offered in alternate years.

A modern approach to classical geometries such as Euclidean, non-Euclidean, and projective. Sets, logic and synthetic and analytic proof techniques in geometry are studied.

Prerequisite: An average GPA of 2.3 in MATH 241 and one of MATH 201, MATH 204, or MATH 205 or by permission.

Offered in alternate years.

Rigorous approach to the concepts underlying the calculus, building on the fundamental idea of the limit within the real number system. Topics include metric spaces, continuity, the derivative, the Riemann integral, and series of constants and functions.

Prerequisites: An average GPA of 2.3 in MATH 241, and at least one of MATH 201, MATH 204, or MATH 205. Students must have junior standing or by permission.

Offered in alternate years.

Study of significant algebraic structures and their properties, with particular attention given to groups, rings, and fields.

Prerequisites: An average GPA of 2.3 in MATH 241 and at least one of MATH 201, MATH 204, or MATH 205. Students must have junior standing or by permission.

*Please note: students must take at least one course from the following: *

Introduction to the theory of probability with applications in management science and the physical and social sciences. Topics include combinatorial probability, densities, mathematical expectation, moment-generating functions, and the central limit theorem.

Prerequisite: MATH 221.

The theory and application of numerical methods for solving mathematical problems. Topics include numerical methods for solving algebraic equations and ordinary differential equations, interpolation and approximation, and numerical integration.

Prerequisite: MATH 235 or MATH 241.

Offered in alternate years.

Partial Differential Equations theory and applications. We will explore solution methods for parabolic, hyperbolic, and elliptic equations. Topics include separation of variables, transforming nonhomogeneous equations, Eigenfunction expansions, Integral Sine and Cosine transformations, Fourier and Laplace Transforms, the Method of Characteristics, and an introduction to Green’s Functions.

Prerequisite: MATH 235.

Offered in alternate years.

*Please note: student must take four additional courses from the ones listed. Students may take MATH 201 or above. Both MATH 201 and MATH 204 may be counted toward the major only with departmental approval. At the most, two of MATH 208, MATH 231, or MATH 212 may count toward the major. If taken as part of a second major, one of MATH 208, MATH 231, or MATH 212 can be replaced with ECON 202, ECON 400, ECON 401; PHYS 331, PHYS 332, PHYS 341, PHYS 344; CHEM 331, CHEM 332; any CS course at the CS 111 level and above (except CS 301); PHIL 151 (4 credits).*

Prerequisite: MATH 122.

Offered as needed.

Prerequisite: A minimum grade of 2.7 in MATH 221.

Games are used to model competition in economics, politics, and conflict. The mathematical techniques used to analyze these games are explored. Topics include zero-sum and nonzero-sum games, Nash equilibria, pure and mixed strategies, and cooperative games. Combinatorial games are also considered.

Offered in alternate years with ECON 202.

Investigation of the process of modeling. Special emphasis placed on how to build, test, and refine models; how to analyze assumptions and results; and defining model limitations. Deterministic and stochastic models, rate equations and population dynamics, and statistical analysis. Final project tied to outside interests.

Prerequisite: MATH 119 or MATH 120 or MATH 121 or MATH 122 or MATH 221 or by permission.

Application of mathematical techniques to real-world problems. Groups of students act as consultants on problems solicited from university departments, local businesses, and/or charitable organizations. Additional material may be included as needed. May be repeated for degree credit, but 4 credits maximum may be applied toward the math major or minor.

Evaluation grade only.

Prerequisite: CDIS 208 or MATH 111 or POLI 202 or PSYC 250, or by permission.

Offered as needed.

Application of the analytical tools of mathematics and probability to the study of behavior in strategic interactions. Topics include simultaneous move games, pure versus mixed strategies, Nash equilibrium, sequential-move games, subgame perfection, repeated games, and evolutionary games. Applications include pricing, advertising, cooperation, bargaining, and conflict.

Prerequisite: ECON 100 or ECON 101 or by permission.

Offered in alternate years.

Application of descriptive and inferential statistics to the measurement and testing of various economic models. Diagnosis and correction of various problems with empirical research: specification errors, multicollinearity, heteroskedasticity, serial correlation, and simultaneity. Statistical software used to implement estimation techniques.

Prerequisites: ECON 350, MATH 111, or POLI 202, or CDIS 208, or by permission.

Introduction to mathematical methods in economics. Topics include matrices, linear algebra, systems of equations; univariate and multivariate differential calculus; comparative statistics, Taylor series approximations, unconstrained and constrained optimization; integral calculus; differential and difference equations.

Prerequisite: ECON 350 or by permission.

Offered in alternate years.

Applications of Newtonian mechanics to various systems, and introduction of calculus of variations and Lagrangian mechanics. This course acquaints students with mathematical techniques used to solve more realistic and complex problems.

Prerequisite: PHYS 231.

Pre- or corequisite: MATH 235.

Development and application of Maxwell’s equations describing electromagnetic fields. Topics include boundary value problems, and dielectric and magnetic materials.

Prerequisites: PHYS 232 and MATH 235.

Offered in alternate years.

Fundamentals of quantum theory. Includes development of mathematical formalism, application to two- and three-dimensional models, and a detailed treatment of the one-electron atom.

Prerequisites: PHYS 233 and MATH 235.

Offered in alternate years.

Fundamental principles of classical thermodynamics and kinetic theory. Classical and quantum statistical properties of matter and radiation.

Prerequisites: PHYS 233 and MATH 221.

Offered in alternate years.

Quantum chemistry, spectroscopy, and statistical thermodynamics applied to the study of the physical and chemical properties of matter. The laboratory involves the systematic study of the theory and practice of modern spectroscopic methods. Four hours lecture and three hours laboratory.

Prerequisites: CHEM 132 and MATH 122 or higher math course and PHYS 220 or higher physics course or by permission.

Equilibrium thermodynamics and kinetics applied to the study of the physical and chemical properties of matter. CHEM 331 need not be taken before CHEM 332.

Prerequisites: CHEM 132 and MATH 122 or higher math course and PHYS 220 or higher physics course or by permission.

Advanced topics concerning data and algorithm representation using C++/Java. Topics include stacks and recursion, dynamic memory, pointers, linked lists, queues, trees, searching, sorting, and object-oriented programming (OOP) and classes.

Prerequisite: CS 110 and MATH 121 Calculus (or Math 118 and Math 119).

Co-requisite MATH 121.

Features a topic of current interest in computer science not otherwise offered in the curriculum.

Prerequisite: by permission. May be repeated for degree credit for a maximum of 8 credits, given a different topic.

Offered as needed.

Notions of complexity analysis, along with the mathematical underpinnings of efficient algorithm design will be studied. Techniques will incorporate divide-and-conquer and searches and sorts. Additional topics will include graph theory and simulations.

Prerequisites: CS 111 and one of MATH 119, 121, 122, or 221.

Introduction to principles of operating systems. Topics include processes (sequential and concurrent), tasks, task management, processor scheduling, memory management, file handling, device management, command languages, interrupts, I/O, and security.

Prerequisite: CS 111.

This course experiments with programming concepts and techniques used in interactive visual environments such as games. Students will explore strategies for solving recursive backtrack problems, design intelligent animations, and deconstruct physical worlds. Students will produce interactive projects, incorporating graphics, text, video, audio, and object-oriented programming, using multimedia, industry-standard authoring software.

Prerequisite: CS 111 and one of MATH 119, 121, 122, or 221.

Offered as needed.

Numeric grade only.

The usage of languages like HTML, JavaScript, and XML will form the core of this course. Syntax and semantics of HTML and XML that enable creation of web pages with a variety of textual and graphical information units will be studied in depth. Client-server programming and Windows applications will also be covered.

Prerequisite: CS 111 and one of MATH 119, 121, 122, or 221.

Offered every year.

Basic principles of problem-solving and algorithm development are studied. Various statements of the programming language Visual Basic will be presented and used in this context. A fairly rapid pace of coverage will occur in this course, as this is not the first course in programming; complex and demanding assignments will form part of the coursework.

Prerequisite: CS 111.

Offered as needed.

Exploration of the Java language for students familiar with object-oriented programming. Topics include multimedia programming, threads, exception handling, and network communications.

Prerequisite: CS 111.

Offered as needed.

Prerequisite: by permission. May be repeated for degree credit for a maximum of 8 credits, given a different topic.

Offered as needed.

Prerequisite: by permission. May be repeated for degree credit for a maximum of 8 credits, given a different topic.

Offered as needed.

This course provides the opportunity for a senior in Computer Science to design, develop, and implement a reasonably-sized software project as a capstone experience. This implementation work integrates the knowledge acquired from earlier computer science courses and the principles of project management and delivery.

Prerequisite: Senior standing.

Introduction to the new and maturing field of software engineering. Topics include the management of expectations, computer technologies, people and their skills, time, cost, and other resources needed to create, test, and maintain a software product that meets the needs of computer users.

Prerequisite: One of CS 222, 223, or 240.

Introduction to programming language concepts and representatives of several different programming language techniques. Topics include data, operations, sequence control, data control, storage management, operating environment, syntax, and comparison of various programming paradigms.

Prerequisite: CS 111.

Introduction to artificial intelligence designed to introduce the basic ideas about search and control strategies, heuristics, problem-solving, constraint exploitation, and logic. Rule-based systems and expert systems techniques and the process of generating intelligent behavior for computers using these information processing strategies are also discussed.

Prerequisite: CS 111.

Offered as needed.

Introduction to principles of database design and management for information systems. Discussion of file design leads to study of logical and physical database concepts relating to three models of database organization: hierarchical, network, and relational. Includes issues relating to query processing, integrity and security of data, and distributed database systems.

Prerequisite: CS 111.

Offered as needed.

Introduction to the development of mobile device applications with an emphasis on programming for the latest Android platform. Topics will include the implementation of multi-touch gestures, sensor and camera events, threads and background tasks, and working with location services. Current development issues are also examined.

Prerequisite: One of CS 222, 223, or 240.

Offered in alternate years.

Practical introduction to logic and critical thinking, with emphasis on developing the ability to detect fallacious arguments and construct sound ones in a variety of practical contexts.

**Upper Division Requirements (3 courses/ 12 credits)**

Beyond the mathematical core requirements, a total of 3 courses taken in the major must be numbered above MATH 300, not including MATH 459.

**Related Field Requirements (4 courses/ 16 credits)**

At least 16 credits in courses outside mathematics that involve quantitative or logical reasoning, or a minor or second major in any field. These courses must include CS 110, Introduction to Programming, or a course in a structured programming language, or the student must demonstrate proficiency in a structured programming language. Courses from EDUC 401, 402, 404, 406 and 408 may be counted toward the related fields requirement.

**Single-Subject Teaching Credential**

Students interested in obtaining the California Single-Subject Teaching Credential in mathematics should meet with advisors in the Mathematics Department and the School of Education for information regarding admissions to the School of Education, coursework pathways, and requirements to earn a California Teaching Credential. Students discuss their plans with their CAS advisor and work closely with them to customize an academic plan.

All teacher candidates in California must demonstrate competency in the subject matter they will be authorized to teach. Students interested in earning a credential to teach mathematics can be prepared to pass the California Subject Examination for Teachers (CSET) by completing a Bachelor’s in Mathematics.

The following courses are recommended for students to prepare for the CSET exams:

1. CSET: Mathematics Subtest (Number and Quantity, Algebra)

MATH 241 Linear Algebra (Spring) and MATH 245 Number Theory (Fall) and MATH 341 Abstract Algebra (Spring)

2. CSET: Mathematics Subtest II (Geometry, Probability, and Statistics)

MATH 251 Geometry (Spring – alternate years) and MATH 311 Probability (Fall) and MATH 312 Statistics (Spring)

3. CSET: Mathematics Subtest III (Calculus)

MATH 221 Calculus III (Fall and Spring) and MATH 321 Real Analysis (Fall – alternate years)

The Department of Mathematics recommends that students seeking a California Single-Subject Teaching Credential in mathematics complete a minor in another academic area commonly taught in secondary schools.

**The Minor**

*Please note: students must take MATH 121, 122, and 221 and will choose at least one from MATH 201, 204, 241, or 300 level *or* above. Students must also take two additional courses from MATH 201 and above, to include at most one of MATH 208, MATH 212, MATH 231. Both MATH 201 and MATH 204 may be counted toward the major only with departmental approval.*

Functions and their graphs; successive approximation and limits; local linearity and differentiation; applications of differentiation to graphing and optimization; and the definite integral, antiderivatives, and differential equations.

Prerequisite: Permission based on Mathematics Placement Exam.

Prerequisite: MATH 121 or MATH 119 or by permission.

Prerequisite: MATH 122.

Offered as needed.

Prerequisite: A minimum grade of 2.7 in MATH 221.

Prerequisite: MATH 221.

Offered in alternate years with ECON 202.

Evaluation grade only.

Prerequisite: CDIS 208 or MATH 111 or POLI 202 or PSYC 250, or by permission.

Offered as needed.

Prerequisite: MATH 119 or MATH 120 or MATH 121 or MATH 122 or MATH 221 or by permission.

**Advanced Placement in Mathematics** *Calculus AB or BC*

Students who attain BC scores or AB scores or AB sub-scores of four or five automatically receive 4 credits for MATH 121. Students with a BC score of five receive 4 credits for MATH 121 and 4 credits for MATH 122. *Statistics*

Statistics students who attain scores of four or five receive 4 credits for MATH 111.

**Departmental Honors**

A departmental honors program is available for exceptionally able and motivated students. Admission to the program may come by departmental invitation or, should students initiate their own applications, by affirmative vote of the mathematics faculty. Honors students are required to take a minimum of 4 courses above MATH 300, not including MATH 459, and must maintain a cumulative GPA of 3.45 or higher in their mathematics courses and overall. Interested students should consult during their junior year with a mathematics faculty member for information about procedures and requirements.

Unless otherwise indicated, all courses in the department are offered for either a numerical grade or evaluation with the permission of the instructor. To meet a course prerequisite, a minimum grade of 1.7 must have been earned in the prerequisite course. Placement in mathematics courses is determined based on either SAT/ACT scores (for MATH 100 and MATH 101) or a placement exam (for MATH 118, MATH 119 or MATH 121) completed prior to the start of classes. Students with AP Calculus credit are placed into a mathematics class based on a faculty interview. Students who do not place into a University of Redlands mathematics course required for a general education (LAF or LAI as appropriate) or academic program requirement will be placed in a 1-credit preparatory mathematics laboratory course as follows:

- MATH 001L prepares students for MATH 100 or MATH 101
- MATH 002L prepares students for the MATH 118–MATH 119 sequence

Placement in a course with prerequisites does not constitute a course challenge to any of the prerequisite courses.

Selections from both historical and current topics in mathematics are included in this general interest course. Topics may include the mathematics of voting and power, fair division and apportionment, population growth, finance, management science and art. MATH 101 is not a prerequisite to calculus.

Prerequisite: MATH 001L or placement at the MATH 100/101 level.

Introduction to modern ideas in finite mathematics. Topics may include probability, logic, combinatorics, functions, matrix algebra, linear programming, and graph theory. MATH 101 is not a prerequisite to calculus.

Prerequisite: MATH 001L or placement at MATH 100/101 level.

Topics include structure of mathematical systems, elementary number theory, operations in the real number system, and elementary problem-solving. Review of arithmetic, algebraic, and geometric topics to reinforce existing knowledge. Introduction to methods and tools currently recommended for use by K–8 educators. An 80-minute lab experience is required weekly. Not recommended for first-year students.

Prerequisite: MATH 101 and LBST 101.

Introduction to mathematical analysis of paper folding. Topics include geometric analysis of designs, constructing and analyzing polygons and polyhedra, Euler’s formula for polyhedra, three and four colorability criteria, Hamilton cycles, and theorems about when an origami crease pattern will fold flat.

Prerequisite: Mathematics placement at MATH 100 / 101 level or by permission.

Descriptive and inferential statistics for students from diverse fields. Distribution, correlation, probability, hypothesis testing, use of tables, and examination of the misuse of statistics and relation of statistics to vital aspects of life. Computer packages used as tools throughout the course.

Prerequisite: Mathematics placement at MATH 100 / 101 level or by permission.

Introduction to the history of mathematics, especially elementary mathematics. Topics include early uses of counting, number systems, arithmetic, fractions, geometry, algebra, probability, and infinite series in civilizations around the world.

Prerequisite: Mathematics placement at MATH 100 / 101 level or by permission.

Offered as needed.

For students whose programs require calculus but who, based on their background and placement examination scores, are not prepared for MATH 121. Topics from precalculus include properties of linear, rational, exponential, logarithmic, and trigonometric functions; and compositions, transformations, and inverses of these functions. Calculus topics include successive approximation and limits of functions; local linearity and differentiation; applications of differentiation to graphing and optimization; and the definite integral, antiderivatives, and differential equations.

Prerequisite: MATH 002L or Math Placement at MATH 118 level or by permission.

For students whose programs require calculus but who, based on their background and placement examination scores, are not prepared for MATH 121. Topics from precalculus include properties of linear, rational, exponential, logarithmic, and trigonometric functions; and compositions, transformations, and inverses of these functions. Calculus topics include successive approximation and limits of functions; local linearity and differentiation; applications of differentiation to graphing and optimization; and the definite integral, antiderivatives, and differential equations.

Prerequisite: MATH 118 or by permission.

Intended for business, environmental science, or other related fields. The following topics are presented with applications in the business world and applied science: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, and relevant applications of integration and optimization. This course is NOT a prerequisite for MATH 122.

Prerequisite: MATH 002L or Math Placement at MATH 118 level or by permission.

Offered every year.

Numeric grading only.

Prerequisite: Permission based on Mathematics Placement Exam.

Prerequisite: MATH 121 or MATH 119 or by permission.

Practice in the mathematical area of problem-solving in preparation for the Putnam Examination. Material and problems are chosen from prior Putnam Exams, Mathematics Olympiads, and other sources; and from across mathematics, including basic strategies, combinatorics, geometry, induction, series, number theory, algebra, and calculus.

Credit/no credit grade option.

Prerequisite: permission of the instructor.

Introductory topics of current interest in mathematics not otherwise covered in the curriculum.

Prerequisite: completion of Mathematics Placement Exam at MATH 100 or MATH 101 level.

Offered as needed.

Prerequisites: Mathematics Placement at MATH 100/101 level.

Prerequisite: MATH 122.

Offered as needed.

Prerequisite: A minimum grade of 2.7 in MATH 221.

Offered in alternate years with ECON 202.

Evaluation grade only.

Prerequisite: CDIS 208 or MATH 111 or POLI 202 or PSYC 250, or by permission.

Offered as needed.

Investigation of vector calculus with an emphasis on applications in physics. Parametrized curves and surfaces; vector fields; line integrals and Green’s Theorem; flux integrals; divergence and curl; the Divergence Theorem and Stokes’s Theorem.

Prerequisite: MATH 221.

Offered as needed.

Prerequisite: MATH 119 or MATH 120 or MATH 121 or MATH 122 or MATH 221 or by permission.

Differential equations theory and applications. First-order linear and nonlinear differential equations with analytic and numerical techniques. Higher-order linear differential equations and complex algebra. Phase trajectory and stability analysis. Systems of linear differential equations with constant coefficients. Matrix methods, Gauss-Jordan, and iterative techniques.

Prerequisite: MATH 221.

Prerequisite: MATH 221.

Prerequisite: An average GPA of 2.3 in MATH 241 and one of MATH 201, or MATH 204, or MATH 205.

Offered in alternate years.

Prerequisite: An average GPA of 2.3 in MATH 241 and one of MATH 201, MATH 204, or MATH 205 or by permission.

Offered in alternate years.

A group of students pursue topics in mathematics not otherwise covered in the curriculum. Prerequisite: by permission. May be repeated for degree credit, but maximum of 8 credits allowed for the degree from MATH 260, MATH 360, and MATH 460.

Offered as needed.

Offered as needed.

Offered as needed.

Prerequisite: MATH 221.

Principles of statistical decision theory. Estimation and hypothesis testing, regression, and parametric and non-parametric tests. Mathematical theory and applications of above.

Prerequisite: MATH 311 or by permission.

Prerequisites: An average GPA of 2.3 in MATH 241, and at least one of MATH 201, MATH 204, or MATH 205. Students must have junior standing or by permission.

Offered in alternate years.

Analytic functions and their properties, including contour integrals, residues, transforms, and conformal mappings.

Prerequisite: MATH 321.

Offered in alternate years.

Prerequisite: MATH 235 or MATH 241.

Offered in alternate years.

Techniques for mathematical modeling of continuous, discrete, and stochastic systems are presented. Topics include purpose and validation, continuous systems, random numbers and variables, and discrete systems.

Prerequisite: MATH 235.

Recommended: MATH 311.

Offered as needed.

Prerequisite: MATH 235.

Offered in alternate years.

Prerequisites: An average GPA of 2.3 in MATH 241 and at least one of MATH 201, MATH 204, or MATH 205. Students must have junior standing or by permission.

Metric spaces, topological spaces, continuous mappings and homeomorphisms, connectedness, and compactness.

Prerequisite: MATH 321.

Offered every third year.

Preliminary background and research will be conducted to lay the groundwork for the senior capstone project in mathematics. A final proposal will be submitted outlining the project to be completed during senior year.

Prerequisite: Senior standing and successful completion of a minimum of seven mathematics courses at the 200 level or higher that must include at least one from the Mathematical Reasoning Requirements. At least one of MATH 321 or MATH 341 is strongly recommended.