arXiv subjects
Publications and source records attributed to L. F. Johnson.
A fast simple O(\log n) iteration algorithm for individual Lucas numbers is given. This is faster than using Fibonacci based methods because of the structure of Lucas numbers. Using a sqrt 5 conversion factor on Lucus numbers gives a faster Fibonacci algorithm. In addition, a fast simple recursive algorithm for individual Lucas numbers is given that is O(log n).
Two very fast and simple O(lg n) algorithms for individual Fibonacci numbers are given and compared to competing algorithms. A simple O(lg n) recursion is derived that can also be applied to Lucas. A formula is given to estimate the largest n, where F_n does not overflow the implementation's data type. The danger of timing runs on input that is too large for the computer representation leads to false research results.
Statements of Shannon's Noiseless Coding Theorem by various authors, including the original, are reviewed and clarified. Traditional statements of the theorem are often unclear as to when it applies. A new notation is introduced and the domain of application is clarified. An examination of the bounds of the Theorem leads to a new symmetric restatement. It is shown that the extended upper bound is an acheivable upper bound, giving symmetry to the theorem.The relation of information entropy to the physical entropy of Gibbs and Boltmann is illustrated. Consequently, the study of Shannon Entropy is strongly related to physics and there is a physical theory of information. This paper is the beginning of of an attempt to clarify these relationships.