Narrow Results

The following sections are included:

• Introduction
• Polya’s Recurrence Theorem
• Burnside’s Problem
• Conclusion
• Bibliography

We look at structure results in nearring theory, especially with regard to Jacobson style primitivity. We look at some applications in state automata and cellular automata, where we find interesting structures. We then look at generalising planarity and nearfields, some of the best structured nearrings, and find that Jacobson style primitivity plays an important role.

The paper explores some of our recent work and presents several ways forward that we think might be of value. While we hope to answer some of the questions that we raise here, it is likely that other researchers will find simpler ways forward that we are unable to see. We look forward to some surprises!

In this note we generalise the Phillips theorem [1] on the subordination of Feller processes by Lévy subordinators to the class of additive subordinators (i.e. subordinators with independent but possibly nonstationary increments). In the case where the original Feller process is Lévy we also express the time-dependent characteristics of the subordinated process in terms of the characteristics of the Lévy process and the additive subordinator.