# Mathematics for Computer Science

## Video Lectures

Displaying all 25 video lectures.
 Lecture 1 Play Video Introduction and ProofsIntroduction to mathematical proofs using axioms and propositions. Covers basics of truth tables and implications, as well as some famous hypotheses and conjectures. Lecture 2 Play Video Mathematical InductionAn introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction. Lecture 3 Play Video Strong InductionCovers strong induction as a tool for proofs. Introduction to invariants with different games, including the n–block game and grid puzzles. Lecture 4 Play Video Number Theory IExplores the basics of number theory with state machines, linear combinations, and algorithms for computation with integers. Speaker: Marten van Dijk Lecture 5 Play Video Number Theory IIDelves deeper into number theory, covering the basics of encryption and decryption using modular arithmetic. Speaker: Marten van Dijk Lecture 6 Play Video Graph Theory and ColoringAn introduction to graph theory basics and intuition with applications to scheduling, coloring, and even sexual promiscuity. Lecture 7 Play Video Matching ProblemsIntroduces the concept of matching. Discusses the mating algorithm, its fairness, and relation to practical applications. Lecture 8 Play Video Graph Theory II: Minimum Spanning TreesExplores the various measures of connectivity of graphs and how these can be used to categorize and analyze graphs. Lecture 9 Play Video Communication NetworksCovers the application of graph theory to communication networks, surveying their configuration, topology, and optimization. Lecture 10 Play Video Graph Theory IIIBuilds upon previous lectures to cover additional graph classifications and criteria, including tournament graphs and directed acyclic graphs. Also covers Euler Tours, Hamiltonian paths, and adjacency matrices. Lecture 11 Play Video Relations, Partial Orders, and SchedulingCovers definitions and examples of basic relations, equivalence classes, Hasse diagrams and topological sorts, as well as other topics. Speaker: Marten van Dijk The last 30 minutes of this video are not available. Lecture 12 Play Video SumsAn introduction to sums through examination of real–world problems like annuities. Covers finding closed form solutions and bounds with the perturbation, derivative, and integral methods. Lecture 13 Play Video Sums and AsymptoticsAnalysis of sums, formulation of asymptotic bounds using various techniques, and introduction to asymptotic notation. Lecture 14 Play Video Divide and Conquer RecurrencesIntroduces the concept of recursion applied to various recurrence problems, such as the Towers of Hanoi and the Merge Sort algorithm, as well as their asymptotic analysis using the Akra–Bazzi method. Lecture 15 Play Video Linear RecurrencesCovers the mechanics of solving general linear recurrences as well as applications to the graduate student job problem and Fibonacci modeling of populations. Lecture 16 Play Video Counting Rules IIntroduces and defines relationships between sets and covers how they are used to reason about counting. Lecture 17 Play Video Counting Rules IICovers computing cardinality of sets with inclusion–exclusion, the bookkeeper rule, the subset rule, and poker hands with applications to probability and counting. Lecture 18 Play Video Probability IntroductionGives an overview of probability, including basic definitions, the Monty Hall problem, and strange dice games. Lecture 19 Play Video Conditional ProbabilityCovers conditional probability and its applications to examples including medical testing, gambling, and court cases. Speaker: Tom Leighton Instructor's Note: The actual details of the Berkeley sex discrimination case may have been different than what was stated in the lecture, so it is best to consider the description given in lecture as fictional but illustrative of the mathematical point being made. Lecture 20 Play Video Independence: Independent and Dependent EventsDifferentiates between independent and dependent events as it pertains to probability, covering applications like coin flips, the distribution of birthdays, hashing, and cryptography. Lecture 21 Play Video Random VariablesIntroduces partitioning of the probabilistic sample space using random variables. Distribution functions, notably, the binomial distribution, are discussed. Lecture 22 Play Video Expectation ICovers expected value as it relates to random variables, discussing coin games, network latency, and the hat check problem. Lecture 23 Play Video Expectation IIContinues exploring expectation with a discussion of likelihood in cases of card games, bit transmission errors, and algorithms, and concludes with definitions of variance and standard deviation for random variables. Lecture 24 Play Video Large DeviationsCovers large deviation. Like expectation, it gives three other notions in solving bounds and many frequently experienced problems in computer science, such as determining the probability a random variable will deviate from its expectation. Lecture 25 Play Video Random WalksDiscusses random walks and their non–intuitive effect on systems, such as gambling at roulette and gambler's ruin.