Santa Clara University Undergraduate Bulletin - Department-of-Mathematics-and-Computer-Science

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DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

Professor Emeritus: Paul R. Halmos
Professors: Gerald L. Alexanderson (Michael and Elizabeth Valeriote Professor), José Barría, Jean J. Pedersen, Edward F. Schaefer, Dennis C. Smolarski, S.J. (Department Chair)
Associate Professors: Glenn Appleby, Robert A. Bekes, Frank A. Farris, Leonard F. Klosinski, Tamsen McGinley, Daniel N. Ostrov, Richard A. Scott, Nicholas Q. Tran, Byron L. Walden
Assistant Professor: Aaron A. Diaz
Senior Lecturers: Laurie Poe, Peter Ross, Nedra Shunk

The Department of Mathematics and Computer Science offers major programs leading to the Bachelor of Science in Mathematics or the Bachelor of Science in Computer Science (Mathematics), as well as required and elective courses for students majoring in other fields. Either major may be pursued with any of three principal goals: preparation for graduate studies leading to advanced degrees in mathematics, computer science, statistics, operations research, or other fields; preparation for secondary school teaching of mathematics or computer science; or preparation for a research career in business, industry, or government. The major in mathematics may be taken with an emphasis in applied mathematics, financial mathematics, or mathematics education. The emphasis in mathematics education is designed to prepare majors to take the California Subject Examination for Teachers. The major in computer science may be taken with an emphasis in cryptography and security. Minors in mathematics or computer science are also available.

The Department of Mathematics and Computer Science maintains a program for the discovery, encouragement, and development of talent in mathematics or computer science among undergraduates. This program includes special sections, seminars, individual conferences, and directed study guided by selected faculty members. Students are also encouraged to participate actively in research projects directed by faculty.

REQUIREMENTS FOR THE MAJOR

In addition to fulfilling University Core Curriculum requirements for the Bachelor of Science degree, students majoring in mathematics and computer science (mathematics) must complete the following departmental requirements for the respective degree:

Major in Mathematics

CSCI 10 (or demonstrated equivalent proficiency in computer programming)
MATH 11, 12, 13, 14, 22, 51, 52, and 53
PHYS 31 and 32, with the associated laboratory section for PHYS 32. Students with a special interest in the application of mathematics in the social sciences or economics may substitute ECON 170 or 173 for PHYS 32. Students planning to teach in secondary schools may substitute, with approval of the department chair, PHYS 11 and 12 for PHYS 31 and 32.
Seven approved upper-division courses in mathematics or computer science, one of which must be MATH 102 and at least one of which must be MATH 103, 111, or 176.

Students planning to undertake graduate studies in pure mathematics should plan to take MATH 105, 111, 112, 113, 153, and 154. Students planning to undertake graduate studies in applied mathematics should complete the emphasis in applied mathematics and take MATH 105, 144, 153, 154, and 155.

Emphasis in Applied Mathematics

Complete the requirements for a Bachelor of Science in Mathematics with the following specifications:

MATH 102, 122, 123, 166, 176
Two courses from MATH 144, 155, 165, 178, CSCI 164, or an approved alternative upper-division mathematics (but not computer science) course

Emphasis in Financial Mathematics

Complete the requirements for a Bachelor of Science degree in Mathematics with the following specifications and additions:

MATH 102, 122, 123, 125, 144, 166
BUSN 70
ACTG 11, 12
FNCE 121, 124

Emphasis in Mathematics Education

Complete the requirements for a Bachelor of Science degree in Mathematics with the following specifications and additions:

MATH 101, 102, 111, 122, 123 (or 8), 170, 175 (or 178)
EDUC 198B

Students are strongly recommended to complete the Urban Education minor.

Major in Computer Science (Mathematics)

MATH 11, 12, 13, 14, 51, 52, 53
CSCI 10, 60, 61
PHYS 31 and 32 with the associated laboratory section for PHYS 32
COEN 20, COEN (or ELEN) 21 and 21L
CSCI 163 and one course from CSCI 161, 166, or 167
Two upper-division courses from the following list and two approved upper- division courses not on the list: MATH 144, 176, 177; CSCI 161, 162, 164, 165, 166, 167, 168, 169, 181, 182, 196. Computer science majors may not take CSCI 165 or 166 as MATH 165 or 166. (Although not required, MATH 122 is highly recommended.)
COEN 177 and one approved COEN upper-division course
One additional approved upper-division course from COEN, CSCI or MATH 144, 176 or 177

Students are encouraged to select one of the following areas of focus to guide their choices of upper-division courses:

Foundations: CSCI 161, MATH 176 and 177, COEN 173
Numerical Computation: MATH 144, CSCI 165 and 166, COEN 145
Software: CSCI 161 and 169, COEN 174, COEN 176 or 178
Graduate School Preparation: CSCI 166, MATH 176 and 177, COEN 175
Another area of focus developed in conjunction with the department

Emphasis in Cryptography and Security

Complete the requirements for a Bachelor of Science in Computer Science (Mathematics) with the following specifications:

MATH 178
CSCI 181
COEN 150 and either COEN 146 or 152
MATH 122 and CSCI 182 are highly recommended

For the major in either mathematics or computer science (mathematics), at least four of the required upper-division courses in the major must be taken at Santa Clara. A single upper-division course in the Department of Mathematics and Computer Science may not be used to satisfy requirements for two majors or minors.

REQUIREMENTS FOR THE MINORS

Minor in Mathematics

Students must fulfill the following requirements for a minor in mathematics:

MATH 11, 12, 13, 14; 52 or 53
Three approved upper-division mathematics courses with no more than one course selected from MATH 165 and 166. In place of MATH 165 or 166, a student may select an upper-division computer science course.

Minor in Computer Science

Students must fulfill the following requirements for a minor in computer science:

CSCI 10, 60 and 61
MATH 12 or 51
COEN 20 and 21
Three approved upper-division computer science courses. In place of an upper- division computer science course, a student may select from MATH 144, 176, or 177.

PREPARATION IN MATHEMATICS FOR ADMISSION TO TEACHER TRAINING CREDENTIAL PROGRAMS

The State of California requires that students seeking a credential to teach mathematics or computer science in California secondary schools must pass the California Subject Examination for Teachers (CSET), a subject area competency examination. The secondary teaching credential additionally requires the completion of an approved credential program, which can be completed as a fifth year of study and student teaching, or through an undergraduate summer program internship. Students who are contemplating secondary school teaching in mathematics or computer science should consult with the coordinator in the Department of Mathematics and Computer Science as early as possible.

LOWER-DIVISION COURSES: MATHEMATICS

4. The Nature of Mathematics
For liberal arts students. Topics chosen from the theory of numbers, combinatorics, geometry, and other suitable areas. Material will generally be presented in a historical setting that allows students to participate in the discovery and development of important mathematical ideas and enhances their appreciation of the beauty of mathematics in the real world. Emphasis on problem solving and doing mathematics. Formerly MATH 41. (4 units)

6. Finite Mathematics for Social Science
Introduction to finite mathematics with applications to the social sciences. Sets, logic, combinatorial problems, probability, vectors, and matrices. (4 units)

7. Calculus for Social Science
Introduction to differential and integral calculus with applications to the social sciences. Ordinarily, only one of MATH 7, 11, or 30 may be taken for credit. (4 units)

8. Introduction to Statistics
Elementary topics in statistics chosen from descriptive statistics, probability, random variables and distributions, sampling, estimation, hypothesis testing, regression, and correlation. (4 units)

9. Precalculus
College algebra and trigonometry for students intending to take calculus. Does not fulfill the University Core Curriculum requirement in mathematics. (4 units)

11. Calculus and Analytic Geometry I
Differentiation and applications, introduction to integration. Ordinarily, only one of MATH 7, 11, or 30 may be taken for credit. Prerequisite: Four years of high school mathematics (including trigonometry) or satisfactory grade in MATH 9. If MATH 9 is taken, a grade of C- or higher is strongly recommended before taking MATH 11. (4 units)

12. Calculus and Analytic Geometry II
Continuation of 11. Methods and applications of integration, transcendental functions. Only one of MATH 12 or 31 may be taken for credit. Prerequisite: MATH 11 or equivalent. A grade of C- or higher in MATH 11 is strongly recommended before taking MATH 12. (4 units)

13. Calculus and Analytic Geometry III
Infinite series, vectors, vector functions, quadric surfaces. Prerequisite: MATH 12 or equivalent. A grade of C- or higher in MATH 12 is strongly recommended before taking MATH 13. (4 units)

14. Calculus and Analytic Geometry IV
Curvilinear coordinate systems, partial derivatives, multiple integrals, vector calculus. Prerequisite: MATH 13 or equivalent. A grade of C- or higher in MATH 13 is strongly recommended before taking MATH 14. Formerly MATH 21. (4 units)

22. Differential Equations
Explicit solution techniques for first order differential equations and higher order linear differential equations. Use of numerical, series, and Laplace transform methods. Applications. Only one of MATH 22 and AMTH 106 may be taken for credit. Prerequisite: MATH 14. (4 units)

30. Calculus for Business I
Differentiation and its applications to business, including marginal cost and profit, maximization of revenue, profit, utility, and cost minimization. Natural logarithms and exponential functions and their applications, including compound interest and elasticity of demand. Study of the theory of the derivative normally included in MATH 11, except trigonometric functions not included here. Ordinarily, only one of MATH 7, 11, or 30 may be taken for credit. Note: MATH 30 is not a suitable prerequisite for MATH 12. Prerequisite: Three years of high school mathematics (excluding trigonometry) or MATH 9. If MATH 9 is taken, a grade of C- or higher is strongly recommended before taking MATH 30. (4 units)

31. Calculus for Business II
Integration and its applications to business, including consumer surplus and present value of future income. Functions of several variables and their derivatives; Lagrange multipliers and constrained optimization. Emphasis throughout the sequence on mathematical modeling, the formulation of practical problems in mathematical terms. Only one of MATH 12 or 31 may be taken for credit. Prerequisite: MATH 30 or equivalent. A grade of C- or higher in MATH 30 is strongly recommended before taking MATH 31. (4 units)

44. Mathematics for Elementary Teachers I
Problem solving and logical thinking approach to whole numbers: their nature, counting, place value, computational operations, properties, and patterns. Intuitive two-dimensional geometry and measurement, especially metric. Arrupe Center participation required. (4 units) NCX

45. Mathematics for Elementary Teachers II
Problem solving and logical thinking approach to fractional numbers, integers, rational numbers, and real numbers: their nature, computational operations, properties, and patterns. Intuitive three-dimensional geometry and measurement, especially metric. Functions, relations, and graphs. Prerequisite: MATH 44. (4 units) NCX

51. Discrete Mathematics
Relations and operations on sets, orderings, elementary combinatorial analysis, recursion, algebraic structures, logic, and methods of proof. Also listed as COEN 19. (4 units)

52. Introduction to Abstract Algebra
Groups, homomorphisms, isomorphisms, quotient groups, fields, integral domains; applications to number theory. Prerequisite: MATH 51 or permission of the instructor. (4 units)

53. Linear Algebra
Vector spaces, linear transformations, algebra of matrices, eigenvalues and eigenvectors, and inner products. Prerequisite: MATH 13. (4 units)

90. Lower-Division Seminars
Basic techniques of problem solving. Topics in algebra, geometry, and analysis. (1–4 units)

UPPER-DIVISION COURSES: MATHEMATICS

Note: Although CSCI 10 is not explicitly listed as a formal prerequisite, some upper-division courses suggested for computer science (mathematics) majors may presuppose the ability to write computer programs in some language. A number of upper-division courses do not have specific prerequisites. Students planning to enroll should be aware, however, that all upper-division courses in mathematics require some level of maturity in mathematics. Those without a reasonable background in lower-division courses are advised to check with instructors before enrolling.

100. Writing in the Mathematical Sciences
An introduction to writing and research in mathematics. Techniques in formulating research problems, standard proof methods, and proof writing. Practice in mathematical exposition for a variety of audiences. Strongly recommended for mathematics and computer science majors beginning their upper-division coursework. MATH 100 may not be taken to fulfill any mathematics or computer science upper-division requirements for students majoring or minoring in mathematics or computer science. (5 units)

101. A Survey of Geometry
Topics from projective, advanced Euclidean, and non-Euclidean geometries. Symmetry. Offered in alternate years. (5 units)

102. Advanced Calculus
Vector calculus, functions of several variables, elliptic integrals, line integrals, Stokes’s theorem, and the divergence theorem. Prerequisites: MATH 14 and 53. (5 units)

103. Linear Algebra II
Abstract vector spaces, dimensionality, linear transformations, isomorphisms, matrix algebra, Eigenspaces and diagonalization, Cayley-Hamilton Theorem, canonical forms, unitary and Hermitian operators, applications. Prerequisite: MATH 53. (5 units)

105. Theory of Functions of a Complex Variable
Analytic functions. Cauchy integral theorems, power series, conformal mapping. Riemann surfaces. Offered in alternate years. (5 units)

111. Abstract Algebra I
Topics from the theory of groups. Offered in alternate years. Prerequisites: MATH 52 and 53. (5 units)

112. Abstract Algebra II
Rings and ideals, algebraic extensions of fields, and the Galois theory. Offered in alternate years. Prerequisite: MATH 111. (5 units)

113. Topology
Topological spaces and continuous functions. Separability and compactness. Introduction to covering spaces or combinatorial topology. Offered in alternate years. Prerequisite: MATH 52, 53, or 102. (5 units)

122. Probability and Statistics I
Sample spaces; conditional probability; independence; random variables; discrete and continuous probability distributions; expectation; moment-generating functions; weak law of large numbers; central limit theorem. Prerequisite: MATH 14. (5 units)

123. Probability and Statistics II
Estimation and hypothesis testing. Maximum likelihood estimation, likelihood ratio tests, and sampling from the normal distribution. Applications. Prerequisites: MATH 53 or permission of instructor and MATH 122. (5 units)

125. Mathematical Finance
Models for the movement of stock and bond prices using Brownian motion and Poisson processes. Introduction to Ito calculus and stochastic differential equations. Discrete lattice models. Pricing models for equity and bond options via Black-Scholes and its variants. Optimal discrete and continuous time portfolio rebalancing. Solution techniques will include Monte Carlo and finite difference methods. Prerequisite: MATH 122 or AMTH 108. MATH 53 recommended but not required. (5 units)

133. Logic and Foundations
Deductive theories. Theories and models. Consistency, completeness, decidability. Theory of models. Cardinality of models. Some related topics of metamathematics and foundations. Open to upper-division science and mathematics students and to philosophy majors having sufficient logical background. Offered on demand. (5 units)

134. Set Theory
Naive set theory. Cardinal and ordinal arithmetic. Axiom of choice and continuum hypothesis. Axiomatic set theory. Offered on demand. (5 units)

144. Partial Differential Equations
Linear partial differential equations with applications in physics and engineering, including wave (hyperbolic), heat (parabolic), and Laplace (elliptic) equations. Solutions on bounded and unbounded domains using Fourier series and Fourier transforms. Introduction to nonlinear partial differential equations. Offered in alternate years. Prerequisite: MATH 14. Recommended: MATH 22 or AMTH 106. (5 units)

153. Intermediate Analysis I
Rigorous investigation of the real number system. Concepts of limit, continuity, differentiability of functions of one real variable, uniform convergence, and theorems of differential and integral calculus. Offered in alternate years. Prerequisite: MATH 102. (5 units)

154. Intermediate Analysis II
Continuation of MATH 153. Offered in alternate years. Prerequisite: MATH 153. (5 units)

155. Ordinary Differential Equations
Solutions to systems of linear differential equations. Behavior of nonlinear autonomous two-dimensional systems. Uniqueness and existence of solutions. Offered in alternate years. (5 units)

165. Linear Programming
Algebraic background. Transportation problem. General simplex methods. Linear programming and theory of games. Numerical methods. Offered in alternate years. Also listed as CSCI 165. (5 units)

166. Numerical Analysis
Numerical algorithms and techniques for solving mathematical problems. Linear systems, integration, approximation of functions, solution of nonlinear equations. Analysis of errors involved in the various methods. Direct methods and iterative methods. Prerequisites: (1) The ability to program in some scientific language, (2) MATH 53 or permission of the instructor. Also listed as CSCI 166. (5 units)

170. Development of Mathematics
A selection of mathematical concepts with their historical context. Offered in alternate years. Prerequisite: Upper-division standing in a science major. (5 units)

172. Problem Solving
Use of induction, analogy, and other techniques in solving mathematical problems. Offered in alternate years. (5 units)

174. Differential Geometry
Introduction to curves and surfaces. Frenet-Serret formulas, Gauss’ Theorema Egregium, Gauss-Bonnet theorem. Offered in alternate years. Prerequisite: MATH 53. (5 units)

175. Theory of Numbers
Fundamental theorems on divisibility, primes, congruences. Number theoretic functions. Diophantine equations. Quadratic residues. Partitions. Offered in alternate years. Prerequisite: MATH 52. (5 units)

176. Combinatorics
Permutations and combinations, generating functions, recursion relations, inclusion-exclusion, Pólya counting theorem, and a selection of topics from combinatorial geometry, graph enumeration, and algebraic combinatorics. (5 units)

177. Graph Theory
Selected topics from planarity, connectedness, trees (enumeration), digraphs, graph algorithms, and networks. Offered in alternate years. (5 units)

178. Cryptography
History, classical cryptosystems, stream ciphers, AES, RSA, discrete log over finite fields and elliptic curves, stream ciphers, and signatures. (5 units)

190. Upper-Division Seminars
Advanced topics in algebra, geometry, or analysis. Research projects. May be repeated for credit. (1–5 units)

197. Advanced Topics
Areas of mathematics not ordinarily covered in regularly scheduled courses, often areas of current interest. May be repeated for credit. (5 units)

198. Internship/Practicum
Guided study related to off-campus practical work experience in mathematics or statistics. Enrollment restricted to majors or minors of the department. Prerequisite: Approval of a faculty sponsor. (1–5 units)

199. Independent Study
Reading and investigation for superior students under the direction of a staff member. This can be used only to extend, not to duplicate, the content of other courses. May be repeated for credit. (1–5 units)

LOWER-DIVISION COURSES: COMPUTER SCIENCE

3. Introduction to Computing and Applications
An overview course providing background on how computers process information and interact with the world; topics presented with a historical perspective; computer-related issues studied within the context of broader, more abstract concepts; the ethical and social responsibility associated with technology. (4 units)

10. Introduction to Computer Science
Introduction to computer science and programming: overview of hardware and software organization; structured programming techniques using C++; elementary algorithms and data structures; abstract data types; the ethical and societal dimensions of computers and technology. Primarily (but not exclusively) for majors in computer science, mathematics, and physical sciences. CSCI 10 may not be taken for credit if the student has received credit for a course in C++ or Java. Prerequisite: MATH 11 (may be taken concurrently). (4 units)

60. Object-oriented Programming
Object-oriented programming techniques using C++: abstract data types and objects; encapsulation; inheritance; polymorphism; the Standard Template Library; the five phases of software development (specification, design, implementation, analysis, and testing). Prerequisites: CSCI 10 or an equivalent introductory course in a scientific language. (4 units)

61. Data Structures
Specification, implementations, and analysis of basic data structures (stacks, queues, graphs, hash tables, binary trees) and their applications in sorting and searching algorithms. Prerequisite: CSCI 60. CSCI 61 and COEN 12 cannot both be taken for credit. (4 units)

90. Lower-Division Seminars
Basic techniques of problem solving. Topics in computer science. (1–4 units)

UPPER-DIVISION COURSES: COMPUTER SCIENCE

Note: Although CSCI 10 is not explicitly listed as a formal prerequisite, some upper-division courses suggested for computer science (mathematics) majors may presuppose the ability to write computer programs in some language. A number of upper-division courses do not have specific prerequisites. Students planning to enroll should be aware, however, that all upper-division courses in computer science require some level of maturity in computer science and mathematics. Those without a reasonable background in lower-division courses are advised to check with instructors before enrolling.

161. Theory of Automata and Languages I
Classification of automata, formal languages, and grammars. Chomsky hierarchy. Representation of automata and grammars, BNF. Deterministic and nondeterministic finite state automata. Regular expressions and languages. Push-down automata. Context-free languages. Context-sensitive grammars and linear bounded automata. Recursively enumerable languages. Turing machines; normal forms; undecidability. Offered in alternate years. Prerequisites: MATH 52 and CSCI 61 or equivalent. (5 units)

162. Theory of Automata and Languages II
Continuation of CSCI 161. Offered in alternate years. Prerequisite: CSCI 161. (5 units)

163. Theory of Algorithms
Introduction to techniques of design and analysis of algorithms: asymptotic notations and running times of recursive algorithms; design strategies: brute-force, divide and conquer, decrease and conquer, transform and conquer, dynamic programming, greedy technique. Intractability: P and NP, approximation algorithms. Also listed as COEN 179. Prerequisites: MATH 51 or 52, or equivalent, and CSCI 61 or equivalent. (5 units)

164. Computer Simulation
Techniques for generation of probability distributions. Computer models of queueing in inventory and scheduling. Simulation of economic systems. Monte Carlo methods for physical systems. Offered in alternate years. Prerequisite: The ability to program in some scientific language. MATH 122 recommended but not required. (5 units) NCX

166. Numerical Analysis
Numerical algorithms and techniques for solving mathematical problems. Linear systems, integration, approximation of functions, solution of nonlinear equations. Analysis of errors involved in the various methods. Direct methods and iterative methods. Also listed as MATH 166. Prerequisites: (1) The ability to program in some scientific language, (2) MATH 53 or permission of the instructor. (5 units)

167. Switching Theory and Boolean Algebra
Switching algebra and Boolean algebra. Minimization via Karnaugh maps and Quine-McCluskey, state compatibility, and equivalence. Machine minimization. Faults. State identification, finite memory, definiteness, information losslessness. Offered on demand. (5 units)

168. Computer Graphics
Systematic and comprehensive overview of interactive computer graphics, such as mathematical techniques for picture transformations and curve and surface approximations. Prerequisite: The ability to program in some scientific language. MATH 53 recommended but not required. (5 units)

169. Programming Languages
Comparative study of major classes of programming languages. Introduction to theoretical definitions of languages and run-time concerns, with emphasis on good points and deficiencies of various languages and on using the appropriate language for a given task. Programs written in several languages (e.g., LISP, FORTRAN-2003, C, C++, MPI). Offered in alternate years. (5 units)

181. Applied Cryptography
Key management, hash functions, stream ciphers, web of trust, time stamping, secret sharing, quantum cryptography, running time analysis, cryptanalytic techniques. Prerequisite: MATH 178. (5 units)

182. Digital Steganography
History and applications; Techniques: substitution, transform domain, distortion, statistical, cover; Evaluation: benchmarking, statistical analysis; Attacks: distortion, counterfeiting, detection; Theory: perfect and computational security. (5 units)

190. Upper-Division Seminars
Advanced topics in computer science. Research projects. May be repeated for credit. (1–5 units)

197. Advanced Topics
Areas of computer science not ordinarily covered in regularly scheduled courses, often areas of current interest. May be repeated for credit. (5 units)

198. Internship/Practicum
Guided study related to off-campus practical work experience in computer science. Enrollment restricted to majors or minors of the department. Prerequisite: Approval of a faculty sponsor. (1–5 units)

Santa Clara University

Undergraduate Bulletins - Department-of-Mathematics-and-Computer-Science