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Irrelevant topics in Physics (Part II)

Travis Hoppe. Irrelevant topics in Physics (Part II). Three more topics…. Plus-size numbers They’re just big boned, more to love really Venn-diagrams Forget everything you’ve forgotten about them from third-grade Football betting Did I mention I’m from Vegas?. Really really big numbers ….

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Irrelevant topics in Physics (Part II)

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  1. Travis Hoppe Irrelevant topicsin Physics (Part II)

  2. Three more topics… • Plus-size numbers • They’re just big boned, more to love really • Venn-diagrams • Forget everything you’ve forgotten about them from third-grade • Football betting • Did I mention I’m from Vegas?

  3. Really really big numbers … • First a contest! • Write the largest number you can think of on the note card. • Use standard mathematical functions or define your own. • The number must be verifiably finite and computable. • The number must be completely defined on the card.

  4. Progression of expression • First-grader: • Third-grader: • Sixth-grader: • Twelfth-grader:

  5. What's next … ? • Addition, multiplication and exponentiation are simply higher orders of the same function:

  6. Time to think big… The idea is not to generate the largest number, per se, but rather the largest growing function... Many different styles: Conway’s chained arrow, hyper-geometric, and of course Knuth’s up-arrow

  7. Knuth’s Up-Arrow notation • Each arrow starting from exponentiation forms the higher operators:

  8. Knuth’s Up-Arrow notation:Numerical examples … Note that the operator is right-associative:

  9. Knuth’s Up-Arrow notation:More arrows! • Clearly we can grow larger numbers by simply adding more arrows onto the expression:

  10. Why are big numbers so awesome? • We have primitive brains – • For small numbers we can only think spatially, four cows, three hens etc … • Abstract numerical systems allow us understand larger quantities • If you build it …. • Large numbers systems were invented because of their necessity. For example …

  11. Graham’s number • Graham’s number is so big that even Knuth’s up arrow notation is insufficient to contain it. It is the best known upper-bound to the problem: Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2^n vertices. Then color each of the edges of this graph using only the colors red and black. What is the smallest value of n for which every possible such coloring must necessarily contain a single-colored complete sub-graph with 4 vertices which lie in a plane?

  12. Graham’s number, G • This is an upper bound to the problem. It has been proven that the lower bound solution is at least 11. The authors state that there is, “some room for improvement”.

  13. Venn Diagrams! (originally invented by Euler) You, at the conclusion of this talk

  14. Formal definition of a Venn Diagram • Let C = { C1, C2, ..., Cn } be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the region formed by the intersection of X1, X2, ..., Xn is nonempty, where each Xi is either int(Ci ) (the interior of Ci ) or is ext(Ci ) (the exterior of Ci ). • If, in addition, each such region is connected and there are only finitely many points of intersection between curves, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram. • In other words – every subset built from a collection of n objects has to be represented only once ….

  15. Venn or not Venn? Not Venn as A U B is not represented, this is still known as an Euler diagram

  16. Converting diagrams to graphs Constructing graphs allows different Venn diagrams of the same order to be compared. Diagrams are isomorphic if their graphs are Isomorphic.

  17. “Special” Venn Diagrams Minimum • Number of vertices in the graph is no more than:

  18. “Special” Venn Diagrams Symmetric • Must display “n-fold” symmetry. • Can be shown that these only exist when n is prime

  19. Football betting • Which squares are better and by how much? • Should some squares cost more? • Are you more likely to be a win or lose? • Is the post-season different from regular season?

  20. Football for squares • Data collection – wrote script to dump all scores from 1994-2007 season (no box scores were found pre-1994). • Partitioned data into regular season and post (wildcard, playoffs and Super Bowl) games. • Took into account home-field advantage; 07 is different from 70, with the first score the home team (irrelevant in games where neither team has home field advantage). • Each score (x,y) was given an expected value:

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