Kenneth G. Monks

The autoconjugacy of the 3x+1 function

The 3x+1 map T is defined on the 2-adic integers by T(x)=x/2 for even x and T(x)=(3x+1)/2 for odd x and the 3x+1 conjecture states that the T-orbit of any positive integer contains 1. We define and study properties of the unique nontrivial autoconjugacy ? of T. This autoconjugacy sends x to the unique 2-adic integer whose parity vector is the ones complement of the parity vector of x. We prove that if ? maps rational numbers to rational numbers then there are no divergent T-orbits of positive integers. The map ? is then used to restate the 3x+1 conjecture in a parity neutral form. We derive a necessary and sufficient condition for a cycle to be self-conjugate and show that self-conjugate cycles contain only positive elements. It is then shown that the only self-conjugate cycle of integers is {1,2}. Finally, we prove that for any rational 2-adic integer x, lim??n(x)n+lim?n(?(x))/n=1 where ?n(x) is the number of ones in the first n digits of the parity vector of x, and we use this along with generalizations of known restrictions on lim??n(x)/n to prove most of the results in the paper.

Change of basis, monomial relations, andPts bases for the Steenrod algebra

Abstract The relationship between several common bases for the mod 2 Steenrod algebra is explored and a family of bases consisting of monomials in distinct P t s s is developed. A recursive change of basis formula is produced to convert between the Milnor basis and each of the bases for which the change of basis matrix in every grading is upper triangular. In particular, it is shown that the basis of admissible monomials, the P t s bases, and two bases due to Arnon, are all bases having this property, and the corresponding change of basis formula is produced for each of them. Some monomial relations for the mod 2 Steenrod algebra are then obtained by exploring the change of basis transformations.

A Web-Based Toolkit for Mathematical Word Processing Applications with Semantics

Lurch is an open-source word processor that can check the steps in students’ mathematical proofs. Users write in a natural language, but mark portions of a document as meaningful, so the software can distinguish content for human readers from content it should analyze.