Discrete mathematics MAA 103 What is mathematics? Six degrees of - - PowerPoint PPT Presentation

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Discrete mathematics MAA 103 What is mathematics? Six degrees of - - PowerPoint PPT Presentation

Jul 17, 2023 2.03k likes •2.79k views

Discrete mathematics MAA 103 What is mathematics? Six degrees of separation What is the common point between What is the common point between What is the common point between What is the common point between Networks of

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Discrete mathematics MAA 103

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What is mathematics?

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Six degrees of separation

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What is the common point between…

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What is the common point between…

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What is the common point between…

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Networks of interacting objects!

What is the common point between…

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Networks of interacting

  • bjects
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Networks of interacting objects

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Networks of interacting objects

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Six degrees of separation

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Six degrees of separation

Frigyes Karinthy, Chain-links, 1929:

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Six degrees of separation

"There has to be something of crucial importance," I said in the middle of debate […]:

Frigyes Karinthy, Chain-links, 1929:

Planet Earth has never been as tiny as it is now

  • relatively speaking,
  • f course
  • due to the quickening pulse of both

physical and verbal communication.

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Six degrees of separation

Frigyes Karinthy, Chain-links, 1929:

One of us suggested to select any person from the 1.5 billion inhabitants of the Earth. He bet us that, using no more than five other individuals, he could contact the selected individual using nothing except the network of personal acquaintances.

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Our friend Chosen person

Six degrees of separation

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Six degrees of separation

Frigyes Karinthy, Chain-links, 1929:

  • "An interesting idea!" – said a friend – "Let's give it

a try. How would you contact Selma Lagerlöf?"

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Selma Lagerlöf Gustav of Sweden Gustav

  • f Sweden

Kehrling Kehrling

Our Friend

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Six degrees of separation… in movies

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In 1994, Kevin Bacon commented that he had worked with everybody in Hollywood or someone who's worked with them.

Six degrees of separation… in movies

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The Bacon number of an actress or an actor is the degree

  • f separation she or he has with Kevin Bacon.

For example: what is the Bacon number of Ryan Gosling?

Six degrees of separation… in movies

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The Bacon number of Ryan Gosling is 1.

Ryan Gosling Kevin Bacon

Crazy Stupid Love

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Six degrees of separation… in movies

What is the Bacon number of Natalie Portman ?

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Natalie Portman Robert de Niro Robert de Niro Kevin Bacon

The Bacon number of Natalie Portman is 2 (because she didn’t shoot with Bacon). Heat Sleapers

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Cf oracleofbacon.org !

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Six degrees of separation… in mathematics

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Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

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Paul Erdős (1913-1996) wrote more than 1500 articles with more than 500 collaborators! The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

Six degrees of separation… in science

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What is the Erdős number of Natalie Portman ?

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

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The Erdős number of Natalie Portman is 5:

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

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  • Natalie Portman wrote an article with Abigail A. Baird ;

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős. The Erdős number of Natalie Portman is 5:

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  • Abigail A. Baird wrote an article with Michael S. Gazzaniga ;
  • Natalie Portman wrote an article with Abigail A. Baird ;

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős. The Erdős number of Natalie Portman is 5:

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  • Michael S. Gazzaniga wrote an article with Jonathan D. Victor ;
  • Natalie Portman wrote an article with Abigail A. Baird ;
  • Abigail A. Baird wrote an article with Michael S. Gazzaniga ;

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős. The Erdős number of Natalie Portman is 5:

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  • Jonathan D. Victor wrote an article with Joseph Gillis.
  • Natalie Portman wrote an article with Abigail A. Baird ;
  • Abigail A. Baird wrote an article with Michael S. Gazzaniga ;
  • Michael S. Gazzaniga wrote an article with Jonathan D. Victor ;

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős. The Erdős number of Natalie Portman is 5:

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  • Natalie Portman wrote an article with Abigail A. Baird ;
  • Abigail A. Baird wrote an article with Michael S. Gazzaniga ;
  • Michael S. Gazzaniga wrote an article with Jonathan D. Victor ;
  • Jonathan D. Victor wrote an article with Joseph Gillis.
  • Joseph Gillis wrote an article with Paul Erdős.

Six degrees of separation… in science

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős. The Erdős number of Natalie Portman is 5:

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My Erdős number is 4 :

  • I have written an article with Céline Abraham;

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

Six degrees of separation… in science

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My Erdős number is 4 :

  • I have written an article with Nicolas Curien;
  • Céline Abraham has written an article with Jean-François Le Gall;

Six degrees of separation… in mathematics

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

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My Erdős number is 4 :

  • Jean-François Le Gall has written an article with Samuel J. T

aylor;

Six degrees of separation… in mathematics

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

  • I have written an article with Nicolas Curien;
  • Nicolas Curien has written an article with Jean-François Le Gall;
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My Erdős number is 4 :

  • Samuel J. T

aylor has written an article with Paul Erdős.

Six degrees of separation… in mathematics

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

  • Jean-François Le Gall has written an article with Samuel J. T

aylor;

  • I have written an article with Nicolas Curien;
  • Nicolas Curien has written an article with Jean-François Le Gall;
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My Erdős number is 4 :

  • Samuel J. T

aylor has written an article with Paul Erdős.

Six degrees of separation… in mathematics

The Erdős number of a mathematicain is the degree of separation, in terms of mathematical collaboration, she or he has with Paul Erdős.

  • Jean-François Le Gall has written an article with Samuel J. T

aylor;

  • I have written an article with Nicolas Curien;
  • Nicolas Curien has written an article with Jean-François Le Gall;
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How can we explain the phenomenon of six degrees of separation?

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Modelling: graphs

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Notion of graph

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Notion of graph

A graph is a collection of vertices and of edges connecting some vertices between them.

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Another example of a graph:

  • the vertices are the people in this room
  • two persons are connected by an edge if they know each other

Notion of graph

A graph is a collection of vertices and of edges connecting some vertices between them.

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The distance between two vertices of a graph is the minimal number of edges separating them.

Distance between 2 points

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A B The distance between A and B is 2.

Distance between 2 points

The distance between two vertices of a graph is the minimal number of edges separating them.

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Diameter of a graph

The diameter of a graph is the distance between two vertices who are the farthest apart.

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6 Graphs: Diameter: 2 2

Diameter of a graph

The diameter of a graph is the distance between two vertices who are the farthest apart.

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Saying that a graph satisfies the six degrees of separation phenomenon amounts to saying that its diameter is at most 6! 6 Graphs: Diameter: 2 2

Diameter of a graph

The diameter of a graph is the distance between two vertices who are the farthest apart.

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How can we explain the phenomenon of six degrees of separation?

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A « simulator »

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Graphs and randomness

In 1999, Albert and Barabási suggested to use a simulator based on randomness and on a mechanism of popularity reinforcment.

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Graphs and randomness

In 1999, Albert and Barabási suggested to use a simulator based on randomness and on a mechanism of popularity reinforcment.

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Conclusion

In this simulator, the diameter grows (roughly) by 1 when one multiplies by 10 the number of vertices. Karinthy was partly right, partly wrong. The simulator of Albert and Barabási produces graphs that look like those of the real world.

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What is discrete mathematics?

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Mathematics can by roughly divided in two realms:

  • the continuous (real numbers)
  • the discrete (integer numbers)

Discrete mathematics

Comparison with watches:

  • analog watches: continuous
  • digital watches: discrete

Discrete mathematics:

  • provides excellent tools and models for analyzing real-world

phenomena.

  • is the entrance to mathematics.
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What is mathematics?

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Mathematics

What is success in mathematics? In mathematics, the main success is a proof. A proof is an essay in which, starting from axioms, an assertion (such as « there are infinitely many prime numbers ») is incontrovertibly and universally shown to be correct. A main goal of this course is to learn how to write proofs. Proofs train us to think and communicate clearly, and present our case logically. Actually, mathematics is proofs.

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Mathematics in the real world

In mathematics, the axioms are taken for granted. What are the axioms of the real world?

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Mathematics in the real world

In order to apply mathematics in the real world, one has to choose a model ( =«the axioms ») to make predictions. If the model has nothing to do with reality, one can still study the model, but beware the consequences! Questions concern models in the real world/our society:

  • how to define happiness?
  • how to define the quality of Education?
  • …
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Mathematics

Proofs train us to think and communicate clearly, present our case logically and keep a critical mind.

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Discrete mathematics MAA 103

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Discrete mathematics MAA 103

  • Course on the blackboard, following Discrete Mathematics by

Schneierman.

  • Course webpage (there is a link on the moodle):


http://www.cmap.polytechnique.fr/~kortchemski/dmaths/ Grading:

  • 50% homework (each exercise sheet will contains a homework

assignement, which has to be handed the next week to your T A). You ARE are allowed and encouraged to discuss the homework problems with other students. However, all written solutions must be individually submitted and must not be copied from somewhere else.

  • 25% midterm exam (probably November 12)
  • 25% final exam (end of January)
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Discrete mathematics MAA 103

Work regularly and keep up the pace!

  • Tutoring sessions (only if needed)