Mathematics for Computer Science

Video Lectures

Displaying all 25 video lectures.
Lecture 1
Introduction and Proofs
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Introduction and Proofs
Introduction to mathematical proofs using axioms and propositions. Covers basics of truth tables and implications, as well as some famous hypotheses and conjectures.
Lecture 2
Mathematical Induction
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Mathematical Induction
An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.
Lecture 3
Strong Induction
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Strong Induction
Covers strong induction as a tool for proofs. Introduction to invariants with different games, including the n–block game and grid puzzles.
Lecture 4
Number Theory I
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Number Theory I
Explores the basics of number theory with state machines, linear combinations, and algorithms for computation with integers.

Speaker: Marten van Dijk
Lecture 5
Number Theory II
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Number Theory II
Delves deeper into number theory, covering the basics of encryption and decryption using modular arithmetic.

Speaker: Marten van Dijk
Lecture 6
Graph Theory and Coloring
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Graph Theory and Coloring
An introduction to graph theory basics and intuition with applications to scheduling, coloring, and even sexual promiscuity.
Lecture 7
Matching Problems
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Matching Problems
Introduces the concept of matching. Discusses the mating algorithm, its fairness, and relation to practical applications.
Lecture 8
Graph Theory II: Minimum Spanning Trees
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Graph Theory II: Minimum Spanning Trees
Explores the various measures of connectivity of graphs and how these can be used to categorize and analyze graphs.
Lecture 9
Communication Networks
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Communication Networks
Covers the application of graph theory to communication networks, surveying their configuration, topology, and optimization.
Lecture 10
Graph Theory III
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Graph Theory III
Builds 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
Relations, Partial Orders, and Scheduling
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Relations, Partial Orders, and Scheduling
Covers 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
Sums
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Sums
An 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
Sums and Asymptotics
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Sums and Asymptotics
Analysis of sums, formulation of asymptotic bounds using various techniques, and introduction to asymptotic notation.
Lecture 14
Divide and Conquer Recurrences
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Divide and Conquer Recurrences
Introduces 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
Linear Recurrences
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Linear Recurrences
Covers the mechanics of solving general linear recurrences as well as applications to the graduate student job problem and Fibonacci modeling of populations.
Lecture 16
Counting Rules I
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Counting Rules I
Introduces and defines relationships between sets and covers how they are used to reason about counting.
Lecture 17
Counting Rules II
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Counting Rules II
Covers computing cardinality of sets with inclusion–exclusion, the bookkeeper rule, the subset rule, and poker hands with applications to probability and counting.
Lecture 18
Probability Introduction
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Probability Introduction
Gives an overview of probability, including basic definitions, the Monty Hall problem, and strange dice games.
Lecture 19
Conditional Probability
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Conditional Probability
Covers 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
Independence: Independent and Dependent Events
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Independence: Independent and Dependent Events
Differentiates between independent and dependent events as it pertains to probability, covering applications like coin flips, the distribution of birthdays, hashing, and cryptography.
Lecture 21
Random Variables
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Random Variables
Introduces partitioning of the probabilistic sample space using random variables. Distribution functions, notably, the binomial distribution, are discussed.
Lecture 22
Expectation I
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Expectation I
Covers expected value as it relates to random variables, discussing coin games, network latency, and the hat check problem.
Lecture 23
Expectation II
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Expectation II
Continues 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
Large Deviations
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Large Deviations
Covers 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
Random Walks
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Random Walks
Discusses random walks and their non–intuitive effect on systems, such as gambling at roulette and gambler's ruin.