Part I I I

1. Irrelevant Topics in Physics Part III

2. What is Irrelevant? • Questions the tools we use in physics so indiscriminately • Relax assumptions • Enjoy physics outside the confines of our ‘research’.

3. Today's Topics • Cardinality of Infinity • Cantor’s crazy continuum counting • Negative Probabilities • An homage to lateral thinking • A Solution to Graph Isomorphism • What train-hopping hobos can teach us

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

5. Absolute Examples • Simple for finite sets

6. 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

7. 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...)

8. 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

9. 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

10. 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...

11. 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:

12. 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, ...).

13. 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:

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

15. Feynman’s Roulette

16. Negative Probabilities • Possible that the direct states of the system are not observable, that is:

17. 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.

18. 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

19. 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.

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

21. 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.

22. 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

23. Hoppe’s 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.

24. Adjacency Matrix A {0,1} symmetric matrix with 1 if nodes i,j are joined by an edge

25. 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:

26. 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(x^3) 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.

27. 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:

28. ex. Symmetric Nodes

29. 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.

30. 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, Rubik’s Cube solutions, etc... Solid state lattice structures Ability to successfully navigate train schedules