Associate Professor: George R. Fegan (Chair)
Assistant Professor: Steve Chiappari
MASTER OF SCIENCE PROGRAM
The Applied Mathematics Program is open to those students who have earned a B.S. degree in Engineering, Science, or Mathematics, provided the student has completed a program in undergraduate mathematics that parallels the program of the mathematics major at Santa Clara University. The undergraduate program at Santa Clara includes calculus and differential equations, abstract algebra, linear algebra, advanced calculus and/or real analysis, and a minimum of five upper-division courses chosen from the areas of analysis, complex variables, partial differential equations, numerical analysis, logic, probability, or statistics.
Courses for the master’s degree must result in a total of 45 units. These units may include courses from other fields with permission of the Applied Mathematics Department advisor. There is a requirement of a minimum of 12 quarter units in 300-level courses.
COURSE DESCRIPTIONS
Undergraduate
Courses 106. Differential Equations
First-order linear differential equations, systems of linear differential equations, homogeneous systems of linear differential equations with constant coefficients, the Laplace transform, the solution of differential equations by Laplace transform. Prerequisite: Math 21. (4 units)
108. Probability and Statistics
Definitions of probability, sets, sample spaces, conditional and total probability, random variables, distributions, functions of random variables, sampling, estimation of parameters, testing hypotheses. Prerequisite: Math 21. (4 units)
118. Numerical Methods
Numerical solution of algebraic and transcendental equations, finite differences, numerical differentiation and integration, and solution of ordinary differential equations. Solution of representative problems on the digital computer. Prerequisites: Ability to program in some computer language (including MATLAB commands), COEN 44 or some other course in a formal language, approved by Department Chair, Math 21, and AMTH 106. (4 units)
Graduate Courses
All 200-level applied mathematics courses are assumed to be first-year graduate courses. The minimum preparation for these courses is a working knowledge of calculus and a course in differential equations. A course in advanced calculus is desirable. The 300-level courses are graduate courses in mathematics that should be taken only by students who have completed several 200-level courses.
202. Mathematical Methods in Mechanical Engineering
Analytic solution of ordinary differential equations. Fourier series. Analytic solution of linear partial differential equations by separation of variables. Numerical solution of ordinary differential equations by iterative and direct methods. Introduction to numerical solution of partial differential equations using explicit, implicit, ADI, and relaxation-type methods. Eigenvalue problems. Introduction to complex variables. (Also listed as MECH 202.) (2 units)
210. Introduction to Probability I
Definitions, sets, conditional and total probability, binomial distribution approximations, random variables, important probability distributions, functions of random variables, moments, characteristic functions, joint probability distributions, marginal distributions, sums of random variables-convolu- tions, correlation, sequences of random variables, limit theorems. The emphasis is on discrete random variables. (2 units)
211. Continuous Probability
Continuation of AMTH 210. A study of continuous probability distributions, their probability density functions, their characteristic functions, and their parameters. These distributions include the continuous uniform, the normal, the beta, the gamma with special emphasis on the exponential, Erlang, and chi-squared. The applications of these distributions are stressed. Joint probability distributions are covered. Functions of single and multiple random variables are stressed, along with their applications. Order statistics. Correlation coefficients and their applications in prediction, limiting distributions, the central limit theorem. Properties of estimators, maximum likelihood estimators, and efficiency measures for estimators. Prerequisite: AMTH 210. (2 units)
212. Discrete and Continuous Probability
Combination of AMTH 210 and 211. (4 units)
213. Introduction to the Tools of Reliability
A review of the integration and differentiation tools for finding probability density functions (pdf ) and cumulative distribution functions (cdf ). A review of techniques for arriving at probability mass functions by means of combinatorics and moment generating functions (mgf ). The basic reliability definitions, including the reliability function and the hazard function. A study of the basic probability distributions and their parameters, which are used in reliability. These will include the discrete distributions such as the discrete uniform binomial, Poisson, and geometric as well as the Erlang with emphasis on the exponential. Students must obtain approval of their advisor. (2 units)
214. Engineering Statistics I
Frequency distributions, sampling, sampling distributions, univariate and bivariate normal distributions, analysis of variance, two- and three-factor analysis, regression and correlation, design of experiments. Prerequisite: A course in probability. (2 units)
215. Engineering Statistics II
Continuation of AMTH 214. Prerequisite: AMTH 214. (2 units)
216. Product Reliability Modeling
Statistical models for reliability. Binomial, normal, lognormal, gamma, Weibull, and exponential models. Availability and spares. Stress-strength analysis. Plotting papers, censored data, electronic systems, and software reliability. Prior exposure to statistics useful but not essential. Prerequisite: A course in probability. (2 units)
217. Design of Scientific Experiments
Statistical techniques applied to scientific investigations. Use of reference distributions, randomization, blocking, replication, analysis of variance, Latin squares, factorial experiments, and examination of residuals. Prior exposure to statistics useful but not essential. Prerequisite: A course in probability. (2 units)
218. Process Troubleshooting and Control
Statistical methods applied to control and troubleshoot processes. Various control charts and operating characteristic curves. Analysis of means applied to both variables and attribute data. Narrow-limit gauging, sampling, disassembly and re-assembly, outliers and outgoing product quality rating. Prior exposure to statistics useful but not essential. Prerequisite: A course in probability. (2 units)
219. Analysis of Scientific Experiments
Continuation of AMTH 217. Emphasis on analysis of the scientific experiments. The theory of design of experiments so that maximal information can be derived. Prerequisites: AMTH 211 and 217. (2 units)
220. Numerical Analysis I
Solution of algebraic and transcendental equations, finite differences, interpolation, numerical differentiation and integration, solution of ordinary differential equations, matrix methods with applications to linear equations, curve fittings, programming of representative problems. (2 units)
221. Numerical Analysis II
Continuation of AMTH 220. Prerequisite: AMTH 220. (2 units)
225. Vector Analysis I
Algebra of vectors. Differentiation of vectors. Partial differentiation and associated concepts. Integration of vectors. Applications. Basic concepts of tensor analysis. (2 units)
226. Vector Analysis II
Continuation of AMTH 225. Prerequisite: AMTH 225. (2 units)
230. Applied Mathematics I
Orthogonal functions. Fourier series. Solution of ordinary differential equations by series. Legendre polynomials. Laplace transforms, basic transforms, applications. Gamma and beta functions. Bessel functions. (2 units)
231. Applied Mathematics
II Continuation of AMTH 230. Prerequisite: AMTH 230. (2 units)
235. Complex Variables I
Algebra of complex numbers, calculus of complex variables, analytic functions, harmonic functions, power series, residue theorems, application of residue theory to definite integrals, conformal mappings. (2 units)
236. Complex Variables II
Continuation of AMTH 235. Prerequisite: AMTH 235. (2 units)
240. Discrete Mathematics for Computer Science
Relations and operation on sets, orderings, combinatories, recursion, logic, method of proof, and algebraic structures. (2 units)
241. Modern Algebra I
Introduction to postulational systems; study of integral domains, fields, and rings; special emphasis on group theory and applications. Polya theory and coding theory; Boolean algebra and its relation to switching functions; general theory of lattices. (2 units)
242. Modern Algebra II
Continuation of AMTH 241. Special topics in the application of group theory and lattice theory. Prerequisite: AMTH 241. (2 units)
245. Linear Algebra I
Vector spaces, transformations, matrices, characteristic value problems, canonical forms, and quadratic forms. (2 units)
246. Linear Algebra II
Continuation of AMTH 245. Prerequisite: AMTH 245. (2 units)
247.Linear Algebra I and II
Combination of AMTH 245 and 246. (4 units)
250. System Reliability Theory
Qualitative and quantitative system reliability analysis (fault tree analysis, reliability block diagrams, system structure analysis, exact system reliability for systems with nonrepairable components). Component importance. System with repairable components and/or standby components (Markov and semi-Markov models) modeling of dependent failures. Applications of reliability theory to systems of interest (e.g., disk arrays with redundancy). (2 units)
256. Applied Graph Theory I
Elementary treatment of graph theory. The basic definitions of graph theory are covered; the fundamental theorems are explored. Subgraphs, complements, graph isomorphisms, and some elementary algorithms make up the content. Note: This course cannot be taken for credit by applied math majors; all other majors should see their advisor before registering for this course. Prerequisite: Mathematical maturity. (2 units)
257. Applied Graph Theory II
This course is an extension of AMTH 256. Networks, Hamiltonian and planar graphs are covered in detail. Edge colorings and Ramsey numbers may also be covered. Prerequisite: AMTH 256. (2 units)
258. Applied Graph Theory I and II
This course covers the material in AMTH 256 and AMTH 257 in one quarter. Prerequisite: Mathematical maturity. (4 units)
280. Combinatorial Mathematics I
Permutations, combinations, partitions, enumeration methods, generating functions, evaluation of discrete sums, and introduction to graph theory. Recurrence relations, solution of difference equations, the principle of inclusion and exclusion, Polya theory of counting. Applications. Knowledge of matrices suggested. (2 units)
281. Combinatorial Mathematics II
Continuation of AMTH 280. Prerequisite: AMTH 280. (2 units)
285. Compression
Different compression techniques are examined. Measures of efficiency are compared. The mathematical basis for the methods will be analyzed. The course should be of specific interest to computer engineers. Prerequisite: Some knowledge of Number Theory. (2 units)
299. Special Problems
By special arrangement. (1–2 units)
305. Advanced Numerical Analysis I
Numerical solution of partial differential equations, finite difference methods. Monte Carlo techniques, relaxation methods, programming of representative problems. Prerequisites: AMTH 220 and 221 and the ability to program in some computer language. (2 units)
306. Advanced Numerical Analysis II
Matrix computations, eigenvalues of finite matrices, application of matrix methods to the solution of systems of linear equations, programming of representative problems. Prerequisites: AMTH 220 and 221 or equivalent. (2 units)
308. Theory of Wavelets
Construction of Daubechies’ wavelets and the application of wavelets to image compression and numerical analysis. Multiresolution analysis and the properties of the scaling function, dilation equation, and wavelet filter coefficients. Pyramid algorithms and their application to image compression. (2 units)
309. Modeling with Discrete Dynamical Systems
The logistic population model; perioddoubling, Feigenbaum diagrams, chaos, symbolic dynamics, Sharkovskii’s theorem. Numerical modeling: Newton’s method, nonconvergent behavior, rational maps on the complex plane, Julia and Fatou sets, the Mandelbrot set. (2 units)
310. Linear Statistical Models
Basic and multifactor analysis of variance (ANOVA); single-factor analysis; two-factor studies; multifactor studies; analysis of randomized block design, Latin Square design, and factorial design. Prerequisite: AMTH 211. (2 units)
312. Nonparametric Statistics
Assumptions of parametric statistics. Nonparametric and distribution-free approaches. Single-sample procedures. Methods for independent or related multiple samples. Tests for independence, homogeneity, or goodness-of-fit. Rank correlation and other measures of association. Prerequisite: AMTH 211. (2 units)
313. Time Series Analysis
Review of forecasting methods. Concepts in time series analysis; stationarity, auto-corre- lation, Box-Jenkins. Moving average and autoregressive processes. Mixed processes. Models for seasonal time series. Prerequisite: AMTH 211. (2 units)
315. Matrix Theory I
Properties and operations, vector spaces and linear transforms, characteristic root; vectors, inversion of matrices, applications. Prerequisite: AMTH 245. (2 units)
316. Matrix Theory II
Continuation of AMTH 315. Prerequisite: AMTH 315. (2 units)
330. Advanced Applied Mathematics I
Functional spaces, vector and matrices, systems of orthogonal functions, linear spaces, manifolds, linear operators, spectral theory, functions of operators and matrices. Green’s functions, delta functions, differential operators, eigenfunction representation of operators, perturbation methods, operators for partial differential equations. (2 units)
331. Advanced Applied Mathematics II
Continuation of AMTH 330. Prerequisite: AMTH 330. (2 units)
332. Difference Equations
Definition of difference equations, standard techniques for solution of difference equations, numerical approximations to solutions of difference equations; applications: pendulum problem, predator-prey problems. Bessel and Legendre applications, phasers as applied to chaos theory. Prerequisite: AMTH 211. (2 units)
340. Linear Programming I
Basic assumptions and limitations, problem formulation, algebraic and geometric representation. Simplex algorithm and duality. (2 units)
341. Linear Programming II
Continuation of AMTH 340. Network problems, transportation problems, production problems. Prerequisite: AMTH 340. (2 units)
342. Linear Programming
A review of the basic assumptions and limitations of linear programming, problem formulation, algebraic and geometric representation. Simplex algorithm and duality. Interior point methods for example the polynomial time method of Karmarkar. Applications then are presented based on the theory and established methodology. These include network problems, transportation problems, production problems. Prerequisite: A programming language, MATLAB is acceptable. (4 units).
344. Linear Regression
The elementary straight-line “least squares fit”; the fitting of data to linear models. Emphasis on the matrix approach to linear regressions. Multiple regression; various strategies for introducing coefficients. Examination of residuals for linearity. Introduction to nonlinear regression. Prerequisite: AMTH 211. (2 units)
346. Graph Theory I
Introduction to graph theory; Euler paths and their applications; Hamiltonian circuits; trees, circuits, and cut-sets; short- est-path problems; planarity and duality; matching theory; directed graphs. Prerequisites: AMTH 246, AMTH 315, and AMTH 316 or approval of instructor. (2 units)
347. Graph Theory II
Continuation of AMTH 346. Prerequisite: AMTH 346. (2 units)
355. Process Simulation
An introduction to the modeling of systems and processes using computer simulations. The general problem of pseudo-random number generation; methods for transforming uniform random into discrete or continuous random variables; the inverse transform method; von Neumann’s rejec- tion/acceptance method. The simulation of correlated random variables, including the K-L transform and the problems in the case of correlated binary variates. The problem of abstraction level in models. The discrete event approach. Statistical analysis of results: the effect of initial conditions and the techniques for variance reduction. Tests for statistical evaluation of the model. (2 units)
358. Fourier Transforms
Definition and basic properties. Energy and power spectra. Applications of transforms of one variable to linear systems, random functions, communications. Transforms of two variables and applications to optics. (2 units)
360. Advanced Topics in Fourier Analysis
Continuation of AMTH 358. Focus on Fourier Analysis in higher dimensions, other extensions of the classical theory, and applications of Fourier Analysis in Mathematics and Signal Processing. (2 units).
362. Stochastic Processes I
Types of stochastic processes, stationarity, ergodicity, differentiation and integration of stochastic processes, correlation and power spectral density functions, linear systems, band limit processes, estimation, nonstationary processes, normal processes, Markov processes. Prerequisite: AMTH 210, AMTH 211, and knowledge of multivariate distributions. (2 units)
363. Stochastic Processes II
Continuation of AMTH 362 (dependent on sufficient demand). Prerequisite: AMTH 211. (2 units)
365. Real Analysis I
Measures, integration, Fourier analysis, probability theory, and related notions. (2 units)
366. Real Analysis II
Continuation of AMTH 365. Prerequisite: AMTH 365. (2 units)
370. Optimization Techniques I
Optimization techniques with emphasis on experimental methods. One-dimensional search methods. Multi-dimensional unconstrained searches: random walk, steepest descent, conjugate gradient, variablemetric. Prerequisites: Ability to program in FORTRAN or similar language and AMTH 246 or equivalent.(2 units)
371. Optimization Techniques II
Optimization problems in multidimensional spaces involving equality constraints and inequality constraints by gradient and nongradient methods. Special topics. Prerequisite: AMTH 370.(2 units)
374. Partial Differential Equations I
Relation between particular solutions, general solutions, and boundary values. Existence and uniqueness theorems. Wave equation and Cauchy’s problem. Heat equation. Prerequisites: AMTH 230 and 231. (2 units)
375. Partial Differential Equations II
Continuation of AMTH 374. Prerequisite: AMTH 374. (2 units)
377. Design and Analysis of Algorithms
This course is equivalent to AMTH 378 and AMTH 379. Depending upon available resources, AMTH 377 will be offered as a 4-unit, 1-quarter class covering the same material as AMTH 378 and 379. Prerequisite: COEN 9. Advisor approval required. (4 units)
378. Design and Analysis of Algorithms I
Designed to meet both the specific needs of the software design program and the general needs of computer engineering. Material to be covered: methods of specifying algorithms, methods for proving the correctness of algorithms, methods for development from specifications. Measures of Complexity: time complexity, storage complexity, NP-Completeness, techniques that lead to efficient algorithms and maintainable code. Prerequisite: COEN 9. Advisor approval required. (2 units)
379. Design and Analysis of Algorithms II
Continuation of AMTH 378. Prerequisites: AMTH 378 and COEN 9. (2 units)
380. Queueing Systems I
Systems of flow. Notation and structure for basic queueing systems. Discrete-time and continuous-time Markov chains. Birthdeath processes. Birth-death queueing systems in equilibrium: discouraged arrivals, responsive servers, m-server case, finite storage, finite customer population, and combinations of these cases. Markovian queues in equilibrium: bulk arrival system, bulk service system, Erlangian distribution, networks of Markovian queues. Introduction to advanced queueing systems. Prerequisites: AMTH 210, 211. (2 units)
381. Queueing Systems II
Direct applications of queueing theory in the areas of operating systems design, processor performance evaluation, and design of computer networks. Prerequisite: AMTH 380. (2 units)
385. Cryptology I
Cryptology covers both cryptography: writing information with the objective of keeping it secret; and cryptanalysis: the science of attacking ciphers or secret messages. The course will examine both symmetric encryption systems including the Data Encryption Standard and asymmetric ciphers (public key ciphers) including the RSA system and the ElGamal system. The course will survey random number generators and connect the theory of groups and finite fields to the methods of cryptography. The course should be of special interest to computer engineers. Prerequisites: Some knowledge of number theory and abstract algebra. (2 units).
386. Cryptology II
Continuation of AMTH 385. Prerequisite: AMTH 385. (2 units)
387. Cryptology
Mathematical foundations for information security (number theory, finite fields, discrete logarithms, information theory, elliptic curves). Cryptography. Encryption systems (classical, DES, Rijndael, RSA). Cryptanalytic techniques. Simple protocols. Techniques for data security (digital signatures, hash algorithms, secret sharing, zeroknowledge techniques). Prerequisite: Mathematical maturity at least at the level of upperdivision engineering students. (4 units).
