MTH 210 Introduction Dr Anthony Bonato Ryerson University

MTH 210 Introduction Dr. Anthony Bonato Ryerson University

• Course outline • Course schedule

Course agreement The goal of this course is to offer a meaningful, rigorous, and rewarding experience to every student; you will build that rich experience by devoting your strongest available effort to the class. You will be challenged and supported. Please be prepared to take an active, patient, and generous role in your own learning and that of your classmates. (c/o Federico Ardila)

How to be successful in this course 1. 2. 3. 4. 5. 6. 7. 8. Attend each lecture and lab. Complete all suggested homework. Review lecture notes and rewrite them as needed. Study weeks, not days, before tests. Three hours of study for each one hour of lecture. Ask your prof or TA if you get stuck. Approach learning as fun. You are ultimately responsible for your own learning.

Discrete Mathematics

• Discrete vs continuous mathematics • Discrete math: – graphs, integers, sequences, sums • Continuous math: – functions, real numbers, series, integrals

Graph theory and networks

Graph theory in the era of Big Data • web graph, social networks, biological networks, bitcoin networks, …

Combinatorics

• Combinatorics: – the science and art of counting • Some things are easy to count – for eg, number of elements of {a, b, c, d} • Others are not – number of non-isomorphic graphs of order 10

Number theory

• Number theory studies integers • Modular arithmetic • Equations with integer solutions: Diophantine equations • Prime numbers

Tools

• Induction • Pigeonhole principle • Combinations • Asymptotics

A curious sequence 1 11 21 1211 111221 312211 13112221

Can you guess the next one? 1 11 21 1211 111221 312211 13112221

Look and say sequences • read off each digit in succession, counting the number of time it occurs, or its frequency • for example, “ 21” is read as: – “one 2, and one 1” – resulting in the sequence 1211

Properties of look-and-say sequences • They always end in 1. • They begin with 1 or 3, except for the third sequence 21. • The sequence 22 is stable, in the sense that it never changes. • Their digits are either 1, 2, or 3.

Conway’s constant • A remarkable thing about look-and-say sequences: they grow predictably in size: – the number of digits in the (n+1)th term is about 1. 3057 times the number of digits in the nth term. • In particular, the ratio of two successive terms is a constant, which is now called λ or Conway’s constant.

John Conway • Conway describes look-and-say sequences as “…the stupidest problem you can conceivably imagine leading to the most complicated answer you can conceivably imagine. ”

Where does λ come from? • λ is an algebraic number that is the unique real root of a polynomial of degree 71:

MTH 210 Introducing graphs Dr. Anthony Bonato Ryerson University

• a graph G=(V(G), E(G))=(V, E) consists of a nonempty set of vertices or nodes V, and a set of edges E • write edges: uv for vertices u and v • say u and v are adjacent vertices edges

Relations • we can think of edges as a binary relation on vertices • edges: {AB, AC, AE, BC, CD, CE, DE}

Drawings • we can also think of graphs by their drawing • but a graph can have many drawings

Loops • edges of the form uu for a vertex u are loops

Simple graphs • edge sets in general graphs are multisets • graphs without multiple edges or loops are simple

Notation • if uv is an edge, u and v are called endpoints • the edge uv is said to be incident with u and v • multiple edges are called parallel

Digraphs • in directed graphs (digraphs) E need not be symmetric

A directed graph

• number of nodes: order, |V| • number of edges: size, |E|

Real World Graphs

The web graph • nodes: web pages • edges: links • over 1 trillion nodes, with billions of nodes added each day

Example: On-line Social Networks (OSNs) • nodes: users on some OSN • edges: friendship (or following) links • maybe directed or undirected Anthony Bonato - The web graph 36

Example: Co-author graph • nodes: mathematicians and scientists • edges: coauthorship • undirected 37

Example: Co-actor graph • nodes: actors • edges: co-stars • Hollywood graph • undirected 38

Example: protein interaction networks • nodes: proteins in a living cell • edges: biochemical interaction • undirected Introducing the Web Graph - Anthony Bonato 39

Bitcoin graph • nodes: users • edges: transactions or protocols 40

Ryerson Nuit Blanche City of Toronto Four Seasons Hotel Frommer’s small world property Greenland Tourism

Exercises