# Chapter 8: 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 46 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

In addition to its freestanding master’s degree program, the Department of Applied Mathematics offers a concentration in mathematical finance within its master’s degree program. Specific course requirements change from time to time. For further information, please consult with the chair of the department.

## Course Descriptions

### Undergraduate Courses

#### AMTH 106. Differential Equations

Explicit solution techniques for first order differential equations and higher order linear differential equations. Use of numerical and Laplace transform methods. Only one of MATH 22 and AMTH 106 may be taken for credit. Prerequisite: MATH 13.

(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 112. Risk Analysis in Civil Engineering

Set theory and probability, random variables, conditional and total probability, functions of random variables, probabilistic models for engineering analysis, statistical inference, hypothesis testing. Prerequisites: MATH 14 and at least junior standing.

(4 units)

#### AMTH 118. Numerical Methods

Numerical solution of algebraic and transcendental equations, 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 one of the following: COEN 11, 44 or MECH 45 or CSCI 10.

(4 units)

#### AMTH 120. Engineering Mathematics

Review of ordinary differential equations (ODEs) and Laplace transform, vector calculus, linear algebra, orthogonal functions and Fourier Series, partial differential equations (PDEs), and introduction to numerical solutions of ODEs. Cross-listed with MECH 120. Prerequisite: AMTH 106.

(4 units)

#### AMTH 194. Peer Educator in Applied Mathematics

Peer educators in applied mathematics work closely with a faculty member to help students understand course material, think more deeply about course material, benefit from collaborative learning, feel less anxious about testing situations, and help students enjoy learning. Prerequisite: Instructor Approval.

(2 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. Prerequisite: AMTH 106 or equivalent.

(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 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 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 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. Differential Equations with Variable Coefficients

Solution of ordinary differential equations with variable coefficients using power series and the method of Frobenius. Solution of Legendre differential equation. Orthogonality of Legendre polynomials, Sturm-Liouville differential equation. Eigenvalues and Eigenfunctions. Generalized Fourier series and Legendre Fourier series.

(2 units)

#### AMTH 231. Special Functions and Laplace Transforms

Review of the method of Frobenius in solving differential equations with variable coefficients. Gamma and beta functions. Solution of Bessel’s differential equation, properties and orthogonality of Bessel functions. Bessel Fourier series. Laplace transform, basic transforms, and applications. Prerequisite: AMTH 230.

(2 units)

#### AMTH 232. Biostatistics

Statistical principles used in bioengineering; distribution-based analyses and Bayesian methods applied to biomedical device and disease testing; methods for categorical data, comparing groups (analysis of variance), and analyzing associations (linear and logistic regression). Special emphases on computational approaches used in model optimization, test-method validation, sensitivity analysis (ROC curves), and survival analysis. Also listed as BIOE 232. Prerequisite: AMTH 108, BIOE 120, or equivalent.

(2 units)

#### AMTH 232L. Biostatistics Laboratory

Laboratory for AMTH 232. Also listed as BIOE 232L. Co-requisite: AMTH 232.

(1 unit)

#### 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 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 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 297. Directed Research

By arrangement. Prerequisite: Permission of the chair of applied mathematics. May be repeated for credit with permission of the chair of applied mathematics.

(1–8 units)

#### AMTH 299. Special Problems

By 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 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 dyadically 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 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 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 351. Quantum Computing

Introduction to quantum computing, with emphasis on computational and algorithmic aspects. Prerequisite: AMTH 246 or 247.

(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. Prerequisites: Calculus sequence, elementary differential equations, fundamentals of linear algebra, and familiarity with MATLAB (preferably) or other high-level programming language.

(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 instructor approval.

(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 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. Undergraduate mathematical background in calculus, probability, and matrices or instructor approval. Calculus background should be up to and including multivariable calculus. Probability background should include knowledge of mean, variance, binomial and normal random variables, the covariance of random variables and the central limit theorem. Matrices background need only cover matrix and vector multiplication and the transpose and inverse of a matrix. Some background in computer programming is recommended as well. Also listed as FNCE 3489 and as MATH 125 and FNCE 116.

(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 non-gradient 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 379. Advanced Design and Analysis of Algorithms

Amortized and probabilistic analysis of algorithms and data structures: disjoint sets, hashing, search trees, suffix arrays and trees. Randomized, parallel, and approximation algorithms. Also listed as COEN 379. Prerequisite: AMTH 377/COEN 279.

(4 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 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)