arXiv subjects
Publications and source records attributed to Samuel L. Marateck.
This note describes a way of obtaining e that differs from the standard one. It could be used as an alternate way of showing how the value of e is obtained. No attempt is made to show the existence of the limit in the definition of e that appears in the final equation.
The Yang-Mills field strength incorporating a non-Abelian feature is one of the cornerstones of the standard model. Although Yang-Mills gauge theories have been around for over fifty years, surprisingly the derivation of the Yang-Mills field strength using classical gauge theory does not appear anywhere in the literature. In their 1954 paper, Yang and Mills had to invent a non-Abelian field strength to satisfy certain criteria. In Section 5 we use Yang's gauge transformation in a heuristic derivation of the Yang-Mills field strength. The preceding sections cover material relating to the derivation. Section 3 shows where Pauli in the article cited by Yang and Mills gives an expression for the electro-magnetic field strength in terms of a commutator. For some reason, Yang and Mills did not use this approach.
Warnsdorffs rule for a knights tour is a heuristic, i.e., it is a rule that does not produce the desired result all the time. It is a classic example of a greedy method in that it is based on a series of locally optimal choices. This note describes an analysis that determines how good the heuristic is on an 8 X 8 chessboard. The order of appearance in a permutation of the eight possible moves a knight can make determines the path the knight takes. A computer analysis is done of the 8! permutations of the order of a knights moves in Warnsdorffs rule on an 8 X 8 chessboard for tours starting on each of the 64 squares. Whenever a tie occurs for moves to vertices that have the lowest degree, the first of these vertices encountered in the programming loop is chosen. The number of permutations of the 8! total that yield non-Hamiltonian paths is tallied. This will be the same value if we consistently choose the last of these vertices encountered.
It is noted that a given pairing of the phase factor and gauge transformation to retain gauge symmetry is not unique. In their seminal paper, when Yang and Mills (YM) discuss the phase factor - gauge transformation relationship, they cite Pauli's review paper. It is interesting that although Pauli in that paper presents the electromagnetic field strength in terms of a commutator, for whatever reason YM did not extrapolate the commutator's use to obtain the Yang-Mills field strength -- they obtained it by trial and error. Presented is a derivation of this field strength using the commutator approach detailing how certain terms cancel each other. Finally, the Yang-Mills field transformation is derived in a slightly different way than is traditionally done.