Chapter 7: Department of Applied Mathematics

Associate Professor:听Aaron Melman (Department Chair)
Assistant Professor:听Francisco Villarroya Alvarez
Lecturers:听Magda Metwally, Robert Kleinhenz

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 expectation for admission based on 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鈥檚 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 AMTH听肠辞耻谤蝉别蝉.

Course Descriptions

Undergraduate Courses

Please see the undergraduate bulletin for undergraduate course descriptions. www.scu.edu/bulletin/undergraduate-bulletin/

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 applied mathematics courses are graduate courses in mathematics that should be taken only by students who have completed several 200-level courses.

AMTH听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. Also listed as MECH 200. (2 units)

AMTH听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.听Also, listed as MECH 201. Prerequisite: AMTH/MECH 200. (2 units)

AMTH听202. Advanced Engineering Mathematics I & II

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听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听211. Probability II

Continuation of AMTH听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听210. (2 units)

AMTH听212. Probability I and II

Combination of AMTH听210 and 211. (4 units)

AMTH听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听215. Engineering Statistics II

Continuation of AMTH听214.听笔谤别谤别辩耻颈蝉颈迟别: AMTH听214. (2 units)

AMTH听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 is useful but not essential. Prerequisite: Solid background in discrete and continuous probability. (2 units)

AMTH听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听221. Numerical Analysis II

Continuation of AMTH听220. Prerequisite: AMTH听220.听(2 units)

AMTH听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听226. Vector Analysis II

Continuation of AMTH听225. Prerequisite: AMTH听225.听(2 units)

AMTH听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听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鈥檚 differential equation, properties, and orthogonality of Bessel functions. Bessel Fourier series. Laplace transform, basic transforms, and applications. Prerequisite: AMTH听230.听(2 units)

AMTH听232. Biostatistics

This course will cover the statistical principles used in Bioengineering encompassing distribution-based analyses and Bayesian methods applied to biomedical device and disease testing including methods for categorical data, comparing groups (analysis of variance), and analyzing associations (linear and logistic regression). Special emphasis will be placed on computational approaches used in model optimization, test-method validation, sensitivity analysis (ROC curve), and survival analysis. Also listed as BIOE 232 Prerequisites: AMTH 108, BIOE 120, or equivalent. (2 units)

AMTH听232L. Biostatistics Laboratory

Laboratory for AMTH听232. Also listed as BIOE 232L. Co-requisite: AMTH听232.听(1 unit)

AMTH听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听236. Complex Variables II

Continuation of AMTH听235. Prerequisite: AMTH听235. (2 units)

AMTH听240. Discrete Mathematics for Computer Science

Relations and operation on sets, orderings, combinatorics, recursion, logic, method of proof, and algebraic structures. (2 units)

AMTH听245. Linear Algebra I

Vector spaces, transformations, matrices, characteristic value problems, canonical forms, and quadratic forms. (2 units)

AMTH听246. Linear Algebra II

Continuation of AMTH听245. Prerequisite: AMTH听245.听(2 units)

AMTH听247. Linear Algebra I and II

Combination of AMTH听245 and 246. (4 units)

AMTH 250. Fundamental Mathematics for Artificial Intelligence

The class consists of three main components: linear algebra, probability and statistics, and optimization. LINEAR ALGEBRA: systems of equations and matrices, vector spaces, bases, linear transformations, image and kernel of a linear transformation, orthogonality, projections, reflections, least squares, Gram-Schmidt orthogonalization, QR decomposition, determinants, eigenvalues and eigenvectors, matrix symmetries, spectral theorem, positive definite matrices, singular value decomposition. PROBABILITY AND STATISTICS: Axioms of probability, random variables, discrete and continuous distributions, measure of centrality, variance, conditional probability, Bayes' law, central limit theorem, parameter estimation, hypothesis testing, confidence intervals. NONLINEAR OPTIMIZATION: Optimality conditions, unconstrained optimization methods, Steepest Descent, Newton, Gauss-Newton, linear and nonlinear least squares, regularization, brief excursion into unconstrained optimization methods, convergence analysis, classical problems. The class is accompanied by programming in Python. (4 units)

AMTH听256. Applied Graph Theory I

Elementary treatment of graph theory. The basic definitions of graph theory are covered; and the fundamental theorems are explored. Subgraphs, complements, graph isomorphisms, and some elementary algorithms make up the content. Prerequisite: Mathematical maturity.听(2 units)

AMTH听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鈥�8 units)

AMTH听299. Special Problems

By arrangement. (1鈥�2 units)

AMTH听308. Theory of Wavelets

Construction of Daubechies鈥� 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听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.听笔谤别谤别辩耻颈蝉颈迟别: AMTH听211 or 212.听(2 units)

AMTH听315. Matrix Theory I

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

AMTH听316. Matrix Theory II

Continuation of AMTH听315. Prerequisite: AMTH听315.听(2 units)

AMTH听340. Linear Programming I

Basic assumptions and limitations, problem formulation, algebraic and geometric representation. Simplex algorithm and duality. (2 units)

AMTH听344. Linear Regression

The elementary straight-line 鈥渓east squares least-squares fit;鈥� 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听211 or 212. (2 units)

AMTH听351. Quantum Computing

Introduction to quantum computing, with emphasis on computational and algorithmic aspects. Prerequisite: AMTH听246 or 247.听(2 units)

AMTH听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.听(2 units)

AMTH听360. Advanced Topics in Fourier Analysis

Continuation of AMTH听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听358 or instructor approval.听(2 units)

AMTH听362. Stochastic Processes I

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

AMTH听363. Stochastic Processes II

Continuation of AMTH听362. Prerequisite: AMTH听362 or instructor approval.听(2 units)

AMTH听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.听笔谤别谤别辩耻颈蝉颈迟别: AMTH听211 or 212 or 362 or ECEN听 233 or 236.听(2 units)

AMTH听367. Mathematical Finance

Introduction to Ito calculus and stochastic differential equations. Discrete lattice models. Models for the movement of stock and bond prices using Brownian motion and Poisson processes. Pricing models for equity and bond options via Black-Scholes and its variants. Optimal portfolio allocation. Solution techniques will include Monte Carlo and finite difference methods. Prerequisite: MATH 53 or permission of instructor and MATH 122 or AMTH 108. Also listed as FNCE 116, MATH 125, AND FNCE 3489.听(4 units)

AMTH听370. Optimization Techniques I

Convex sets and functions. Unconstrained optimality conditions. Convergence and rates of convergence. Applications. Numerical methods for unconstrained optimization (and constrained optimization as time permits). Prerequisites: Proficiency in Matlab programming and AMTH听246 or 247.听(2 units)

AMTH听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听370.听(2 units)

AMTH听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听211 or 212 or 362 or 364 or ECEN听233 or 236.听(2 units)

AMTH听374. Partial Differential Equations I

Relation between particular solutions, general solutions, and boundary values. Existence and uniqueness theorems. Wave equation and Cauchy鈥檚 problem. Heat equation. (2 units)

AMTH听375. Partial Differential Equations II

Continuation of AMTH听374. Prerequisite: AMTH听374.听(2 units)

AMTH听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听377. Design and Analysis of Algorithms

Techniques of design and analysis of algorithms: proof of correctness; running times of recursive algorithms; design strategies: brute-force, divide and conquer, dynamic programming, branch-and-bound, backtracking, and greedy technique; max flow/ matching. Intractability: lower bounds; P, NP, and NP-completeness. Also listed as CSEN 279. Prerequisite: CSEN 912C or equivalent.听(4 units)

AMTH听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 CSEN 379. Prerequisite: AMTH听377/CSEN 279.听(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.听(4 units)

AMTH听388. Advanced Topics in Cryptology

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

AMTH听397. Master鈥檚 Thesis

By arrangement. Limited to master鈥檚 students in applied mathematics. (1鈥�9 units)

AMTH听399. Independent Study