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MATH 61DM: Modern Mathematics: Discrete Methods

This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with Math 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.This series provides the necessary mathematical background for majors in Computer Science, Economics, Mathematics, Mathematical and Computational Science, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering majors requiring Math 50¿s beyond Math 51 are recommended to take Math 60CM.
Terms: Aut | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Vondrak, J. (PI)

MATH 62DM: Modern Mathematics: Discrete Methods

This is the second part of a theoretical (proof-based) sequence with a focus on discrete mathematics. The central objects discussed in this course are finite fields. These are beautiful structures in themselves, and very useful in large areas of modern mathematics, and beyond. Our goal will be to construct these, understand their structure, and along the way discuss unexpected applications in combinatorics and number theory. Highlights of the course include a complete proof of a polynomial time algorithm for primality testing, Sidon sets and finite projective planes, and an understanding of a lovely magic trick due to Persi Diaconis. Prerequisite: Math 61DM or 61CM.
Terms: Win | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Soundararajan, K. (PI)

MATH 63DM: Modern Mathematics: Discrete Methods

Third part of a proof-based sequence in discrete mathematics, though independent of the second part (62DM). The first half of the quarter gives a brisk-paced coverage of probability and random processes with an intensive use of generating functions and a rich variety of applications. The second half treats entropy, Bayesian inference, Markov chains, game theory, probabilistic methods in solving non-probabilistic problems. We use continuous calculus, e.g. in handling the Gaussian, but anything needed will be reviewed in a self-contained manner. Prerequisite: Math 61DM or 61CM
Terms: Spr | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Fox, J. (PI)
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