Irrelevant topics in Physics (Part II)

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Irrelevant topicsin Physics (Part II)

• Travis Hoppe

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Three more topics

• Plus-size numbers
• Theyre just big boned, more to love really
• Venn-diagrams
• Forget everything youve forgotten about them
  from third-grade
• Football betting
• Did I mention Im from Vegas?

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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.

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Progression of expression

• First-grader
• Third-grader
• Sixth-grader
• Twelfth-grader

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What's next ?

• Addition, multiplication and exponentiation are
  simply higher orders of the same function

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Time to think big

• The idea is not to generate the largest number,
  per se, but rather the largest growing
  function...
• Many different styles Conways chained arrow,
  hyper-geometric, and of course Knuths up-arrow

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Knuths Up-Arrow notation

• Each arrow starting from exponentiation forms the
  higher operators

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Knuths Up-Arrow notationNumerical examples
Note that the operator is right-associative
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Knuths Up-Arrow notationMore arrows!

• Clearly we can grow larger numbers by simply
  adding more arrows onto the expression

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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

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Grahams number

• Grahams number is so big that even Knuths 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 2n 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?
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Grahams 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.

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

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

• Number of vertices in the graph is no more than

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Special Venn Diagrams Symmetric

• Must display n-fold symmetry.
• Can be shown that these only exist when n is
  prime

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Football betting
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• 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?

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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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