Grunge Template

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Irrelevant Topics in Physics
Part III
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What is Irrelevant?

• Questions the tools we use in physics so
  indiscriminately
• Relax assumptions
• Enjoy physics outside the confines of our
  research.

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Today's Topics

• Cardinality of Infinity
• Cantors crazy continuum counting
• Negative Probabilities
• An homage to lateral thinking
• A Solution to Graph Isomorphism
• What train-hopping hobos can teach us

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Absolute beginnings

• Reconsider the absolute value x
• For physicists -gt distance
• More than one type of norm L1,L2,L3,L_infinty ...
• For Set Theorists -gt counting, known as the
  cardinality of the set

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Absolute Examples

• Simple for finite sets

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Absolute Infinity

• What about infinite sets?
• Only comparisons can be made now
• Sets are equal, iff they can be put in a
  one-to-one correspondence with each other

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Absolute Natural

• Even numbers have the same cardinality as the
  natural numbers
• Holds for all infinite partitions of the natural
  numbers (odds, divisible by 13, primes, etc...)

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Absolute Rational

• Rational numbers also have the same cardinality
  as the natural numbers!
• Omit the repeats to get
• the sequence
• These sets are known as countable

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Absolute (un)Real

• Do all infinite sets have the same cardinality?
  NO! There are more reals then rationals!
• Any subset of the real numbers ie. 0,1 can be
  put in correspondence with any other subset, or
  even the entire line.
• The ? above is known as the continuum
  hypothesis, which can neither be proved or
  disproved when assuming the axiom of choice

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Absolute (un)Real

• Proof by contradiction, assume a mapping exists,
  for example take

Each real on the right is infinite, and the
length of this list is also infinite. Take a
diagonal of the list .4297..... Add one to each
of the numbers (mod 10) .5308....
This new number is NOT on the list above, as it
differs from the first digit for the first
number, second digit for the second number, etc...
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Absolute Crazy

• The cardinality of the square is equal to a line
  segment.
• Higher order cardinalities exists by taking
  multisets, ie. The set of all sets

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Absolute Irrelevant

• Does this have an impact on physics?
• Sets of higher order then the real numbers have
  never found use in physics.
• However, the language of quantum mechanics uses
  discrete (quanta of energy, spin, ...) and
  continuous variables (position, momenta, ...).

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Negative Probabilities

• Relax the assumption that each probability must
  be positive, however still enforce that the sum
  of all events must be unity.
• Consider a concrete example of a roulette wheel
  with two conditions

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Feynmans Roulette
A (.7) B (.3)
1 0.3 -0.4
2 0.6 1.2
3 0.1 0.2

• The table is known to have two states,A,B and
  separate probabilities for each ofthe numbers
  coming up.

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Feynmans Roulette
A (.7) B (.3)
1 0.3 -0.4
2 0.6 1.2
3 0.1 0.2
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Negative Probabilities

• Possible that the direct states of the system are
  not observable, that is

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Negative Probabilities

• Why not rearrange the calculation or theory so
  probabilities are positive in all intermediate
  states?
• The accountant who subtracts all paymentsbefore
  adding in the profits (intermediate sum can be
  negative).
• Nothing mathematically wrong with working with
  negative probabilities.

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Hobos on a Train

• From Wikipedia
• Hobo is a term that refers to a subculture of
  wandering homeless people, particularly those who
  make a habit of hopping freight trains

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Hobosumptions

• Assume that occasionally, when a hobo wakes up,
  he is unsure of his current location.
• As a survival instinct, he has memorized all the
  train schedules for each country.
• Wine has degraded his memory, and he only
  remembers the connections.

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Hobomaps

• It is crucial when picking up a train schedule,
  no matter what country, to determine the lay of
  the land.

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Notes from the Ivory Towers

• A mathematician would call the hobo-map an
  undirected, unlabeled simple graph, where the
  process for determining two graphs are the same
  is known as graph isomorphism.
• Computationally, graph isomorphism is curious, it
  belongs to NP but it is not known to have a
  polynomial solution (P) nor is it NP-complete.

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Invariants

• An invariant is a graph property that can
  (possibly) show two graphs different.
• Examples number of nodes, degree sequence,
  number of edges, etc...
• Graphs with different invariants are not
  isomorphic the converse is NOT true in general

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Hoppes Invariant

• I propose an invariant which I think is also
  unique, that is, no two non-isomorphic graphs
  share the same invariant (working on this part).
• Moreover, the invariant is computable in
  polynomial time.

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Adjacency Matrix

• A 0,1 symmetric matrix with 1 if nodes i,j are
  joined by an edge

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Generating functions

• This matrix has nice properties. Raised to a
  power n the element i,j tells you the number of
  trips starting at i and ending at j of length n.
• A generating function can be found that gives all
  the terms

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Big-Oh! Notation

• This step is assumed to be a polynomial time
  operation. Finding the det. of matrix can be
  doing using LU decomposition O(x3) by dividing
  and multiplying rows. When the matrix elements
  themselves are polynomials, the number of
  operations is surely increased, but (seems to be)
  bounded by polynomial time.

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Symmetric Nodes

• Once each polynomial is found, it encodes all the
  powers of A into it by taking higher terms of z.
  If two nodes share the property
• We will call them symmetric.
• The unordered set is a graph
• invariant

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ex. Symmetric Nodes
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Hobomorphism

• The question then is
• Does this set uniquely define a graph? Ie. Can
  one produce a graph simply by knowing how many
  walks of length n lead back home for all n and
  for all nodes?
• If so, the graph isomorphism problem has a
  polynomial time solution.

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Hobos and YOU

• Why graph isomorphism?
• Computationally an unsolved problem
• Chemical structure evaluation
• Symmetric groups for coupled Josephson-Junction
  systems (cf. Sam Kennerly)
• Discrete mathematics generalized knights tours,
  Rubiks Cube solutions, etc...
• Solid state lattice structures
• Ability to successfully navigate train schedules