p-adic path set fractals and arithmetic

This paper considers a class C(Z_p) of closed sets of the p-adic integers obtained by graph-directed constructions analogous to those of Mauldin and Williams over the real numbers. These sets are characterized as collections of those p-adic integers whose p-adic expansions are describeed by paths in the graph of a finite automaton issuing from a distinguished initial vertex. This paper shows that this class of sets is closed under the arithmetic operations of addition and multiplication by p-integral rational numbers. In addition the Minkowski sum (under p-adic addition) of two set in the class is shown to also belong to this class. These results represent purely p-adic phenomena in that analogous closure properties do not hold over the real numbers. We also show the existence of computable formulas for the Hausdorff dimensions of such sets.Comment: v1 24 pages; v2 added to title, 28 pages; v3, 30 pages, added concluding section, v.4, incorporate changes requested by reviewe

Stabilizing Heegaard Splittings of High-Distance Knots

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to these Heegaard surfaces.Comment: 19 pages, 8 figure

Algebraic constructive quantum field theory: Integrable models and deformation techniques

Several related operator-algebraic constructions for quantum field theory models on Minkowski spacetime are reviewed. The common theme of these constructions is that of a Borchers triple, capturing the structure of observables localized in a Rindler wedge. After reviewing the abstract setting, we discuss in this framework i) the construction of free field theories from standard pairs, ii) the inverse scattering construction of integrable QFT models on two-dimensional Minkowski space, and iii) the warped convolution deformation of QFT models in arbitrary dimension, inspired from non-commutative Minkowski space.Comment: Review article, 57 pages, 3 figure

Convolution, Separation and Concurrency

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page