1. Miniconference on the Mathematics of Computation
MTH 210
Introduction
Dr. Anthony Bonato
Ryerson University

2. Course outline
Course schedule

3. 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)

4. How to be successful in this course
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.

5. Discrete Mathematics

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

7. Graph theory and networks

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

9. Combinatorics

10. Some things are easy to count
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

11. Number theory

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

13. Tools

14. Induction
Pigeonhole principle
Combinations
Asymptotics

16. A curious sequence

17. Can you guess the next one?

18. 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

19. 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.

20. 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 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.

21. 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.”

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

24. Miniconference on the Mathematics of Computation
MTH 210
Introducing graphs
Dr. Anthony Bonato
Ryerson University

25. 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

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

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

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

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

30. 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

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

32. A directed graph

33. number of nodes: order, |V|
number of edges: size, |E|

34. Real World Graphs

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

36. 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

37. Example: Co-author graph
nodes: mathematicians and scientists
edges: co-authorship
undirected

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

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

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

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

42. Exercises