Formal nonassociative number theory.

Number Theory and Formal Languages

I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in nite characteristic , automaticreal numbers, xed points of homomorphisms, automaticity, and k-regular sequences.

Gauge Theory on Nonassociative Spaces

We show how to do gauge theory on the octonions and other nonassociative algebras such as ‘fuzzy R’ models proposed in string theory. We use the theory of quasialgebras obtained by cochain twist introduced previously. The gauge theory in this case is twisting-equivalent to usual gauge theory on the underlying classical space. We give a general U(1)-Yang-Mills example for any quasi-algebra and a...

Magnetic Charge in a Nonassociative Field Theory

The violation of the Jacobi identity by the presence of magnetic charge is accomodated by using an explicitly nonassociative theory of octonionic fields. It is found that the dynamics of this theory is simplified if the Lagrangian contains only dyonic charges, but certain problems in the constrained quantisation remain. The extension of these concepts to string theory may however resolve these ...

Numeration Systems: A Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence relat...

Nonassociative Algebras