Santa Clara University

Department of Applied Mathematics

Senior Lecturer: Stephen A. Chiappari (Chair)
Renewable Term Lecturer: Aaron Melman

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 that 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, and 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. A minimum of 12 quarter units must be in 300-level courses.

Concentration in Mathematical Finance within the Master of Science in Applied Mathematics

The Department of Applied Mathematics offers a concentration in mathematical finance within its master’s degree program. For further information, please consult with the chair of the department.

Required Components for Concentration

• ACTG 300 Financial Accounting (3 units)
• ACTG 303 Corporate Financial Reporting or 319 Financial Statement Analysis (3 units)
• AMTH 210 and 211 (or 212) Discrete and Continuous Probability (4 units)
• AMTH 220 and 221 Numerical Analysis (4 units)
• AMTH 245 and 246 (or 247) Linear Algebra (4 units)
• AMTH 313 Time Series Analysis (2 units)
• AMTH 344 Linear Regression (2 units)
• AMTH 362 Stochastic Processes I (2 units)
• AMTH 374 Partial Differential Equations I (2 units)
• ECON 401 Economics for Business Decisions (3 units)
• FNCE 451 Financial Management (3 units)
• FNCE 455 Investments (3 units)
• FNCE 474 Risk Management with Derivative Securities (3 units)
• FNCE 696 (cross-listed as AMTH 367 and MATH 125 and FNCE 116)
• Mathematical Finance (3 units)
• Any additional units required to meet the engineering graduate core

Recommended, but not Required, Components for Concentration

• AMTH 358 Fourier Transforms (2 units)
• AMTH 363 Stochastic Processes II (2 units)
• AMTH 397 Master’s thesis or project (3 units)
• FNCE 484 Financial Engineering (3 units)
• FNCE 712 Monte Carlo Simulation (1 unit)
• FNCE 710 Default Modeling (1 unit)

COURSE DESCRIPTIONS

Undergraduate Courses
AMTH 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 14. (4 units)

AMTH 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 14. (4 units)

AMTH 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: AMTH 106 or MATH 22, and COEN 44 or 45. (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.

AMTH 200. Advanced Engineering Mathematics I
Method of solution of the first, second, and higher order differential equations (ODEs). Integral transforms including Laplace transforms, Fourier series and Fourier transforms. Cross listed with MECH 200. Prerequisite: AMTH 106 or equivalent. (2 units)

AMTH 201. Advanced Engineering Mathematics II
Method of solution of partial differential equations (PDEs) including separation of variables, Fourier series and Laplace transforms. Introduction to calculus of variations. Selected topics from vector analysis and linear algebra. Cross listed with MECH 201. Prerequisite: AMTH/MECH 200. (2 units)

AMTH 202. Advanced Engineering Mathematics
Method of solution of first, second, and higher order ordinary differential equations, Laplace transforms, Fourier series, and Fourier transforms. Method of solution of partial differential equations, including separation of variables, Fourier series, and Laplace transforms. Selected topics in linear algebra, vector analysis, and calculus of variations. Also listed as MECH 202. (4 units)

AMTH 210. 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, convolutions, correlation, sequences of random variables, limit theorems. The emphasis is on discrete random variables. (2 units)

AMTH 211. Probability II
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)

AMTH 212. Probability I and II
Combination of AMTH 210 and 211. (4 units)

AMTH 213. 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 discrete distributions, such as the discrete uniform binomial, Poisson, and geometric, as well as the Erlang with emphasis on the exponential. Prerequisite: Advisor’s approval. (2 units)

AMTH 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: Solid background in discrete and continuous probability. (2 units)

AMTH 215. Engineering Statistics II
Continuation of AMTH 214. Prerequisite: AMTH 214. (2 units)

AMTH 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: Solid background in discrete and continuous probability. (2 units)

AMTH 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: Solid background in discrete and continuous probability. (2 units)

AMTH 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 reassembly, outliers and outgoing product quality rating. Prior exposure to statistics useful but not essential. Prerequisite: Solid background in discrete and continuous probability. (2 units)

AMTH 219. Analysis of Scientific Experiments
Continuation of AMTH 217. Emphasis on the analysis of scientific experiments. The theory of design of experiments so that maximal information can be derived. Prerequisites: AMTH 211 or 212, and 217. (2 units)

AMTH 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)

AMTH 221. Numerical Analysis II
Continuation of AMTH 220. Prerequisite: AMTH 220. (2 units)

AMTH 222. Design and Analysis of Scientific Experiments
Combination of AMTH 217 and AMTH 219. Prerequisite: AMTH 211 or 212. (4 units)

AMTH 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)

AMTH 226. Vector Analysis II
Continuation of AMTH 225. Prerequisite: AMTH 225. (2 units)

AMTH 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)

AMTH 231. Applied Mathematics II
Continuation of AMTH 230. Prerequisite: AMTH 230. (2 units)

AMTH 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)

AMTH 236. Complex Variables II
Continuation of AMTH 235. Prerequisite: AMTH 235. (2 units)

AMTH 240. Discrete Mathematics for Computer Science
Relations and operation on sets, orderings, combinatorics, recursion, logic, method of proof, and algebraic structures. (2 units)

AMTH 241. Modern Algebra I
Introduction to postulation 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)

AMTH 242. Modern Algebra II
Continuation of AMTH 241. Special topics in the application of group theory and lattice theory. Prerequisite: AMTH 241. (2 units)

AMTH 245. Linear Algebra I
Vector spaces, transformations, matrices, characteristic value problems, canonical forms, and quadratic forms. (2 units)

AMTH 246. Linear Algebra II
Continuation of AMTH 245. Prerequisite: AMTH 245. (2 units)

AMTH 247. Linear Algebra I and II
Combination of AMTH 245 and 246. (4 units)

AMTH 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 non repairable 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). Prerequisite: AMTH 215. (2 units)

AMTH 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. Prerequisite: Mathematical maturity. (2 units)

AMTH 257. Applied Graph Theory II
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)

AMTH 258. Applied Graph Theory I and II
Combination of AMTH 256 and AMTH 257. Prerequisite: Mathematical maturity. (4 units)

AMTH 260. Jewels of Modern Mathematics and Their Wide Applications
Selected topics chosen from the following list: Fractals. Chaos theory. Wavelets. Graph algorithms. Neural network and artificial intelligence. Variational principles and finite element methods. Finite state machines and Turing machines. Finite calculus. Convex optimization. Functions as vectors: Hilbert space. Applications. (2 units)

AMTH 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)

AMTH 281. Combinatorial Mathematics II
Continuation of AMTH 280. Prerequisite: AMTH 280. (2 units)

AMTH 285. Compression
Different compression techniques are examined. Measures of efficiency are compared. The mathematical basis for the methods. Of specific interest to computer engineers. Prerequisite: Some knowledge of number theory. (2 units)

AMTH 299. Special Problems
By special arrangement. (1–2 units)

AMTH 308. Theory of Wavelets
Construction of Daubechies’ wavelets and the application of wavelets to image compression and numerical analysis. Multi resolution analysis and the properties of the scaling function, dilation equation, and wavelet filter coefficients. Pyramid algorithms and their application to image compression. Prerequisites: Familiarity with MATLAB or other high-level language, Fourier analysis, and linear algebra. (2 units)

AMTH 309. Modeling with Discrete Dynamical Systems
The logistic population model; period doubling, Tannenbaum diagrams, chaos, symbolic dynamics, Sharkovskii’s theorem. Numerical modeling: Newton’s method, non convergent behavior, rational maps on the complex plane, Julia and Fatou sets, the Mandelbrot set. (2 units)

AMTH 310. Linear Statistical Models
Basic and multi-factor analysis of variance (ANOVA); single-factor analysis; two- factor studies; multi-factor studies; analysis of randomized block design, Latin Square design, and factorial design. Prerequisite: AMTH 211 or 212. (2 units)

AMTH 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 or 212. (2 units)

AMTH 313. Time Series Analysis
Review of forecasting methods. Concepts in time series analysis; stationarity, auto-correlation, Box-Jenkins. Moving average and auto-regressive processes. Mixed processes. Models for seasonal time series. Prerequisite: AMTH 211 or 212. (2 units)

AMTH 315. Matrix Theory I
Properties and operations, vector spaces and linear transforms, characteristic root; vectors, inversion of matrices, applications. Prerequisite: AMTH 246 or 247. (2 units)

AMTH 316. Matrix Theory II
Continuation of AMTH 315. Prerequisite: AMTH 315. (2 units)

AMTH 318. Advanced Topics in Wavelets
An overview of very recent developments in the theory and application of wavelets. Study of a new generation of wavelet-like objects, such as beamlets, which exhibit unprecedented capabilities for the compression and analysis of 3D data. The beamlet framework consists of five major components: The beamlet dictionary, a dyadic ally organized library of line segments over a range of locations, orientations, and scales. The beamlet transform, a collection of line integrals of the given 3D data along the line segments in the beamlet dictionary. The beamlet pyramid, the set of all beamlet transform coefficients arranged in a hierarchical data structure according to scale. The beamlet graph, the graph structure in which vertices correspond to voxel corners of the underlying 3D object, and the edges correspond to beamlets connecting pairs of vertices. The beamlet algorithms, to extract information from the beamlet graph consistent with the structure of the beamlet graph. Study of each component in detail. Implementation issues. Selected applications in the areas of computer graphics, pattern recognition, and data compression. Prerequisite: AMTH 308. (2 units)

AMTH 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. (2 units)

AMTH 340. Linear Programming I
Basic assumptions and limitations, problem formulation, algebraic and geometric representation. Simplex algorithm and duality. (2 units)

AMTH 341. Linear Programming II
Continuation of AMTH 340. Network problems, transportation problems, production problems. Prerequisite: AMTH 340. (2 units)

AMTH 342. Linear Programming
Combination of AMTH 340 and 341. (4 units)

AMTH 344. Linear Regression
The elementary straight-line “least squares least-squares fit;” and 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 or 212. (2 units)

AMTH 349. Structure of the Internet and the World Wide Web
The Internet and World Wide Web as random graphs. Graph metrics (degree distribution, average path length, clustering coefficient, diameter, and betweenness). Classical random graph theory. Random graphs via z-transforms. Graph spectra for Poisson and power-law degree distributions. Small-world networks. Evolution of scale-free networks, including the Barabási and Albert model. Search techniques on networks. Virus epidemics and immunization in the Internet. Error and attack tolerance of networks. Congestion in a stochastic network. Prerequisites: Familiarity with graph theory, linear algebra, and probability theory. (4 units)

AMTH 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 rejection/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. Prerequisites: AMTH 211 or 212 and 215. (2 units)

AMTH 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)

AMTH 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. Prerequisite: AMTH 358 or instructor approval. (2 units)

AMTH 362. Stochastic Processes I
Types of stochastic processes, stationarity, ergodicity, differentiation and integration of stochastic processes. Topics chosen from correlation and power spectral density functions, linear systems, band-limit processes, normal processes, Markov processes, Brownian motion, and option pricing. Prerequisite: AMTH 211 or 212 or instructor approval. (2 units)

AMTH 363. Stochastic Processes II
Continuation of AMTH 362. Prerequisite: AMTH 362 or approval of instructor. (2 units)

AMTH 364. Markov Chains
Markov property, Markov processes, discrete-time Markov chains, classes of states, recurrence processes and limiting probabilities, continuous-time Markov chains, time-reversed chains, numerical techniques. Prerequisite: AMTH 211 or 212 or 362 or ELEN 233 or 236. (2 units)

AMTH 365. Real Analysis I
Measures, integration, Fourier analysis, probability theory, and related notions. (2 units)

AMTH 366. Real Analysis II
Continuation of AMTH 365. Prerequisite: AMTH 365. (2 units)

AMTH 367. Mathematical Finance
Basic principles of finance and economic investments. Random processes with white noise. Topics in control theory, optimization theory, stochastic analysis, and numerical analysis. Mathematical models in finance. Financial derivatives. Software to implement mathematical finance models. Also listed as FNCE 696 and as MATH 125. Prerequisites: Mathematical maturity at least at the level of junior mathematics majors; knowledge of mean, variance, binomial and normal random variables, and the central limit theorem; or instructor approval. (4 units)

AMTH 370. Optimization Techniques I
Optimization techniques with emphasis on experimental methods. One-dimensional search methods. Multidimensional unconstrained searches: random walk, steepest descent, conjugate gradient, variable metric. Prerequisites: Ability to program in some computer language and AMTH 246 or 247. (2 units)

AMTH 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)

AMTH 372. Semi-Markov and Decision Processes
Semi-Markov processes in discrete and continuous time, continuous-time Markov processes, processes with an infinite number of states, rewards, discounting, decision processes, dynamic programming, and applications. Prerequisite: AMTH 211 or 212 or 362 or 364 or ELEN 233 or 236. (2 units)

AMTH 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. (2 units)

AMTH 375. Partial Differential Equations II
Continuation of AMTH 374. Prerequisite: AMTH 374. (2 units)

AMTH 376. Numerical Solution of Partial Differential Equations
Numerical solution of parabolic, elliptic, and hyperbolic partial differential equations. Basic techniques of finite differences, finite volumes, finite elements, and spectral methods. Direct and iterative solvers. Prerequisites: Familiarity with numerical analysis, linear algebra, and MATLAB. (2 units)

AMTH 377. Design and Analysis of Algorithms
Advanced topics in design and analysis of algorithms: amortized and probabilistic analysis; greedy technique; dynamic programming; max flow/matching. Intractability: lower bounds; P, NP, and NP-completeness; branch-and-bound; backtracking. Current topics: primality testing and factoring; string matching. Also listed as COEN 279. Prerequisite: Familiarity with data structures. (4 units)

AMTH 380. Queueing Systems I
Systems of flow. Notation and structure for basic queueing systems. Discrete-time and continuous-time Markov chains. Birth-death 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. Prerequisite: AMTH 211 or 212. (2 units)

AMTH 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)

AMTH 383. Computational Number Theory
Computer implementation of algorithms in elementary number theory. Multiple-precision arithmetic, the Fast Fourier Transform, and error analysis. Computation relevant to aspects of public-key cryptosystems such as distribution of primes, recognition of prime numbers, factorization of large numbers, and elliptic curves. Prerequisites: Mathematical maturity (algebra, Fourier analysis, and analytic geometry), basic understanding of computer architecture, and ability to program in a high-level language. (2 units)

AMTH 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, zero-knowledge techniques). Prerequisite: Mathematical maturity at least at the level of upper-division engineering students. (4 units)

AMTH 388. Advanced Topics in Cryptology
Topics may include advanced cryptography and cryptanalysis. May be repeated for credit if topics differ. Prerequisite: AMTH 387. (2 units)

AMTH 395. Mathematics Seminar for Engineering Research
Selected topics in mathematics related to doctoral research in engineering. Prerequisites: Good standing in a doctoral program in engineering, permission of thesis advisor, and permission of chair of applied mathematics. May be repeated for credit with permission of the thesis advisor and of the chair of applied mathematics. (2 units)

AMTH 397. Master’s Thesis
By arrangement. Limited to master’s students in applied mathematics. (1–9 units)

AMTH 399. Independent Study
By arrangement. Prerequisite: Instructor approval. (1–4 units)