AMTH
Courses
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)
# Units: 2
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. Overlaps with MECH 201. Prerequisite: AMTH/MECH 200 (2 units)
# Units: 2
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)
# Units: 4
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, correlations, sequences of random variables, limit theorems. The emphasis is on discrete random variables.(2 units)
# Units: 2
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)
# Units: 2
Combination of AMTH 210 and 211. (4 units)
# Units: 4
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# Units: 2
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)
# Units: 2
Continuation of AMTH 214. Prerequisite: AMTH 214 (2 units)
# Units: 2
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# Units: 2
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 couse in probability.
# Units: 2
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# Units: 2
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)
# Units: 2
Continuation of AMTH 220. Prerequisite: AMTH 220. (2 units)
# Units: 2
Combination of Amth 217 and Amth 219. Prerequisite: Amth 211 or 212 (4 units)
# Units: 4
Algebra of vectors. Differentiation of vectors. Partial differentiation and associated concepts. Integration of vectors. Applications. Basic concepts of tensor analysis.
# Units: 2
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# Units: 2
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)
# Units: 2
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 Bessell functions. Bessell Fourier series. Laplace transform, basic transforms and applications Prerequisite: AMTH 230. (2 units)
# Units: 2
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 emphases 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)
# Units: 2
Laboratory for AMTH 232. Also listed as BIOE 232L. Co-requisite: AMTH 232. (1 unit)
# Units: 1
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)
# Units: 2
Continuation of AMTH 235. Prerequisite: AMTH 235 ( 2 units)
# Units: 2
Relations and operation on sets, orderings, combinatorics, recursion, logic, method of proof, and algebraic structures. (2 units)
# Units: 2
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# Units: 2
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# Units: 2
Vector spaces, transformations, matrices, characteristic value problems, canonical forms, and quadratic forms.(2 units)
# Units: 2
Continuation of AMTH 245. Prerquisite: AMTH 245. (2 units)
# Units: 2
Combination of AMTH 245 and 246.(4 units)
# Units: 4
ntroduction to Bayesian statistics. Computational statistics via the R programming language. Estimation. Prediction. Approximate Bayesian computation. Hypothesis testing. Evidence. Simulation. Hierarchical models. Prerequisites: Calculus sequence, introductory mathematical statistics, fundamentals of linear algebra, and familiarity with MATLAB, Python, or R, or instructor approval.
# Units: 4
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)
# Units: 2
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# Units: 2
This course covers the material in AMTH 256 and AMTH 257 in one quarter. Prerequisite: Mathematical maturity.
# Units: 4
Selection of topics chosen from the following list: Fractals, Chaos theory. Wavelets. Graph algorithms, Neural networks 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. Prerequisite: Familiarity with calculus, linear algebra, and computer programming.
# Units: 2
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# Units: 2
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# Units: 2
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.
# Units: 4
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# Units: 4
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# Units: 2
By arrangement. Prerequisites: Permission of the chair of applied mathematics. May be repeated for credit with permission of the chair of applied mathematics. (1-8 units)
# Units: 1 - 8
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# Units: 1 - 8
By special arrangement.
# Units: 1 - 2
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.
# Units: 2
Matix 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.
# Units: 2
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. Prerequisites: Familiarity with MATLAB or other high-level language, Fourier analysis and linear algebra. (2 units)
# Units: 2
The logistic population model; period-doubling, Reigenbaum 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 Madelbrot set.
# Units: 2
Assumptions of parametric statistics. Non-parametric 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.
# Units: 2
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# Units: 2
Properties and operations, vector spaces and linear transforms, characteristic root; vectors, inversion of matrices, application. Prerequites: AMTH 245.
# Units: 2
Continuation of AMTH 315. Prerequisite: AMTH 315.
# Units: 2
The objective of this course is to give an overview of very recent developments in theory and application of wavelets.
# Units: 2
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# Units: 2
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# Units: 2
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.
# Units: 2
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# Units: 2
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# Units: 2
Basic assumptions and limitations, problem formulation, algebraic and geometric representation. Simplex algorithm and duality.
# Units: 2
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# Units: 2
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. Application 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.
# Units: 4
The elementary straight-line "least squares fit"; the fitting of data to linear models. Emphasis on the matrix approach to linear regressions. Mulitple regression; various strategies for introducing coefficents. Examination of residuals for linearity. Introduction to nonlinear regression. Prerequisite: AMTH 211.
# Units: 2
Introduction to graph theory; Euler paths and their applications; Hamiltonian circuits; tress, circuits, and cut-sets; shortest-path problems planarity and duality; matching theory; directed graph. Prerequisites; AMTH 246, 315, and 316, or approval of instructor.
# Units: 2
Continuation of AMTH 346. Prerequisite: AMTH 346.
# Units: 2
Introduction to quantum computing, with emphasis on computational and algorithmic aspects. Prerequisites: AMTH 246 or 247. (2 units)
# Units: 2
Definitions 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.
# Units: 2
Continuation of AMTH 358. Focus on Fourier Analysis in higher dimensions, other extension of the classical theory, and application of Fourier Analysis in mathematics and signal processing. Prerequisites: AMTH 358 or instructor approval. (2 units)
# Units: 2
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)
# Units: 2
Continuation of AMTH 362 (dependent of sufficient demand). Prerequisite: AMTH 211.
# Units: 2
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.
# Units: 2
Measures, integration, Fourier analysis, probability theory, and related notions.
# Units: 2
Continuation of AMTH 365
# Units: 2
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 with FNCE 116, MATH 125 and FNCE 3489. (4 units)
# Units: 4
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 246 or 247. (2 units)
# Units: 2
Optimization problems in multidimensional spaces involving equality constraints and inequality constraints by gradient and nongradient methods. Special topics. Prerequisites: AMTH 370 (2 units)
# Units: 2
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# Units: 2 - 3
Relation between particular solutions, general solutions and boundary values. Existence and uniqueness theorems. Wave equation and Cauchy's problem. Heat equation. (2 units)
# Units: 2
Continuation of AMTH 374. Prerequisite: AMTH 374 (2 units)
# Units: 2
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)
# Units: 2
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. (4 units)
# Units: 4
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 377 OR CSEN 279. (4 units).
# Units: 4
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# Units: 2
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# Units: 2
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 cipher inclusing 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.
# Units: 2
Continuation of AMTH 385.
# Units: 2
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)
# Units: 4
Topics may include advanced cryptology and cryptoanalysis. May be repeated for credit if topics differ. Prerequisite: AMTH 387
# Units: 2
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 ve repeated for credit with permission of the thesis advisor and chair of Applied Mathematics.
# Units: 2
By arrangement. Limited to master's students in applied mathematics. (1-9 units)
# Units: 1 - 9
By arrangement. Prerequisite: Instructor consent.
# Units: 1 - 4
Special course for alums only (50% discount)
# Units: 1