Lecture

Linearity of Expectation: First Moment Method

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MATH-467: Probabilistic methods in combinatorics

The 'probabilistic method' is a fundamental tool in combinatorics. The basic idea is as follows: to prove that an object (for example, graph) with certain properties exists, it suffices to prove that if the object is chosen at random, then it has the desir ...

Description

This lecture covers the concept of Linearity of Expectation and introduces the First Moment Method, which is used to analyze random matrices and eigenvalues. The lecture also delves into indicator random variables and Buffon's Needle problem in probability theory, discussing the calculation of probabilities and lengths of needles. Furthermore, it explores transitive tournaments and Ham paths, providing insights into the minimum number of copies of a given structure in a graph. The instructor demonstrates how to force vectors to be orthogonal and discusses assignments that ensure orthogonality in vector spaces.

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

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Graph (abstract data type)

In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as edges but also sometimes arrows or arcs.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

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