# MATH-Mathematics

## MATH 10A Basic Calculus

Algebra and trigonometry review, differentiation, applications of differentiation, integration, applications of integration, series, vectors and matrices, differential equations. Designed for students who desire to learn calculus with less detail than in the corresponding 11A-B sequence. Students cannot receive credit for both a 10 course and the corresponding 11 course.

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## MATH 10B Basic Calculus

Algebra and trigonometry review, differentiation, applications of differentiation, integration, applications of integration, series, vectors and matrices, differential equations. Designed for students who desire to learn calculus with less detail than in the corresponding 11A-B sequence. Students cannot receive credit for both a 10 course and the corresponding 11 course

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## MATH 12C Applied Linear Algebra

Systems of linear equations, matrices, determinants, euclidean spaces, eigenvalues, and eigenvectors. Extensive computational work on computers in a campus Macintosh lab. Integrates both the computational and the theoretical aspects of linear algebra. No prior computer experience is needed. Concurrent enrollment in course 12L is required. One quarter of college mathematics is recommended.

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## MATH 13 Vector Calculus

The derivative as a linear transformation. Directional derivatives, gradients, divergence, and curl. Maxima and minima. Integrals on paths and surfaces. Green, Stokes, and Gauss theorems.

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## MATH 28 Introduction to Computational Number Theory

Prime numbers, congruences, Euclid's algorithm. The theorems of Fermat and Lagrange. The reduction of arithmetic calculations to the case of prime-power modulus. The use of quadratic residues. Computational aspects of primality testing, and factorization techniques. Students cannot receive credit for this course and course 18. High school algebra is recommended; knowledge of a computer language is useful.

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## MATH 28L Laboratory for Computational Number Theory

Laboratory sequence illustrating topics covered in course 28. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 28 is required.

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## MATH 50 Introduction to Fractal Geometry I: Iterated Function Systems and L-Systems

Definition and measurement of fractal dimension, self-similarity and self-affinity, iterated function and L-systems, random fractals, diffusion limited aggregation models, multi-fractals, Julia and Mandelbrot sets, rewriting systems, image compression, chaos and fractals. Concurrent enrollment in course 50L is required.

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## MATH 50L Fractal Geometry I Laboratory

Laboratory sequence illustrating topics covered in course 150. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 150 is required.

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## MATH 80A What is Mathematics?

A broad overview of the subject, intended primarily for liberal arts students. What do mathematicians do, and why do they do it? Examines the art of proving theorems, from both the philosophical and aesthetic points of view, using examples such as non-Euclidean geometrics, prime numbers, abstract groups, and uncountable sets. Emphasis on appreciating the beauty and variety of mathematical ideas. Includes a survey of important results and unsolved problems which motivate mathematical research.

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## MATH 80B Nature of Mathematics

A survey course, aimed at non-science students, providing an understanding of some fundamental concepts of higher mathematics. Is mathematics a natural science? Do mathematicians discover or invent? Discussion of such questions in connection with such topics as numbers (whole, prime, negative, irrational, round, imaginary, binary, transfinite, infinitesimal, etc.), logic, symmetry, various types of geometry, and some recent developments. Basic high school algebra and geometry is sufficient background. Students cannot receive credit for this course and course 80A.

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## MATH 124A Topology and Manifolds

Intrinsic properties of spaces such as tori, Möbius strips, and Klein bottles and methods for classifying them. A: Basic point set topology, covering spaces, homotopy, simplicial complexes, genus, Fundamental Theorem of Surfaces. B: Manifolds, simplicial homology, differential forms, and DeRham cohomology.

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## MATH 124B Topology and Manifolds

Intrinsic properties of spaces such as tori, Möbius strips, and Klein bottles and methods for classifying them. A: Basic point set topology, covering spaces, homotopy, simplicial complexes, genus, Fundamental Theorem of Surfaces. B: Manifolds, simplicial homology, differential forms, and DeRham cohomology.

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## MATH 125 Calculus of Variations

History and development of the calculus of variations; sufficient and necessary conditions for a minimum; Euler-Lagrange equations; Hamilton Jacobi theory; the theory of geodesics; parametric variational problems.

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## MATH 131B Advanced Probability Theory

Classical probability theory. Random variables, expectation and moments, independence. The major families of distributions and their properties; characteristic functions, infinite divisibility. Limit theorems on sums of random variables, random walks, Markov processes, Martingales, ruin.

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## MATH 133 Applied Regression

Topics include bivariate and multiple regression with residual, multicolinearity, and sequential selection analysis. ANOVA with multiple comparisons, unbalanced and missing data analysis and repeated measure models. Experimental designs (completely randomized block and splitplot). Statistical software packages.

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## MATH 138 Decision Theory and Game Theory

Statistical decision theory as used for modeling and resolving problems in a variety of fields, from agriculture to artificial intelligence. Basic statistical models, decision trees, utilities and rewards, risk and loss functions, principles of assessing probabilities, Bayes decisions and Bayes estimation. Also group utilities, conflicts of interest, two person zero-sum games, optimal strategies, the minimax theorem.

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## MATH 139 Introduction to Stochastic Processes

The theory and application of stochastic processes as models for empirical phenomena (time-series), with special emphasis on the following processes: Wiener, Poisson, stationary, normal, counting, renewal, Markov, and birth-death.

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## MATH 141L Nonlinear Mathematics Laboratory

Laboratory sequence illustrating topics covered in course 141. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 141 is required.

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## MATH 142 Introduction to Mathematica

Symbolic integration and differentiation; symbolic manipulation of algebraic equations; linear algebra; graphs of functions with one or two variables; parametric surfaces in three-dimensional space; solutions of ordinary differential equations; error estimates; elementary programming.

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## MATH 142L Mathematica Laboratory

Laboratory sequence illustrating topics covered in course 142. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 142 is required.

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## MATH 143 Computational Mathematics

Solving mathematical problems in a Unix environment using the C++ programming language. In addition, software packages such as LAPACK, MACSYMA, MATLAB, and Numerical Recipes are used. Computational problems in both pure and applied mathematics are discussed. Use of other machines on the Internet, such as the Cray supercomputer in San Diego is explained. Students are expected to know how to program in some language, not necessarily C++. Concurrent enrollment in course 143L is required.

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## MATH 143L Computational Mathematics Laboratory

Laboratory sequence illustrating topics covered in course 143.One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 143 is required.

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## MATH 144 Advanced Computational Mathematics

Solving mathematical problems in a C++ UNIX environment. The C++ programming language along with mathematical software tools such as MACSYMA and MATLAB are used to attack problems in pure and applied mathematics. A thorough knowledge of C is assumed. Concurrent enrollment in course 144L is required.

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## MATH 144L Advanced Computational Mathematics Laboratory

Laboratory sequence illustrating topics covered in course 144. One three-hour session per week in microcomputer laboratory. Concurrently enrollment in course 144 is required.

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## MATH 147 Dynamical Models

A survey of strategies for building dynamical models for complex natural systems, with exemplary models taken from the literature of the biological and social sciences. Concurrent enrollment in course 147L is required.

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## MATH 147L Dynamical Models Laboratory

Laboratory sequence illustrating topics covered in course 147. One three-hour sessionper week in microcomputer laboratory. Concurrent enrollment in course 147 is required.

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## MATH 151 Introduction to Fractal Geometry II

Study of random fractals and holomorphic dynamical systems. Percolation theory, diffusion limited aggregation models, Brownian motion, Julia sets for rational functions, bifurcation sets and the Mandelbrot set.

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## MATH 151L Fractal Geometry II Laboratory (.2 course credit)

Laboratory sequence illustrating topics covered in course 151. One three-hour session per week in microcomputer laboratory.

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