Compactifying Exchange Graphs I: Annuli and Tubes

Connectivity of Soft Random Geometric Graphs Over Annuli

Nodes are randomly distributed within an annulus (and then a shell) to form a point pattern of communication terminals which are linked stochastically according to the Rayleigh fading of radio-frequency data signals. We then present analytic formulas for the connection probability of these spatially embedded graphs, describing the connectivity behaviour as a dense-network limit is approached. T...

Half-space Theorem, Embedded Minimal Annuli and Minimal Graphs in the Heisenberg Group

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil3 is entire. Also, the sister surface of an entire minimal graph in Nil3 is an entire constant mean curv...

Compactifying Moduli of Hyperelliptic Curves

We construct a new compactification of the moduli space Hg of smooth hyperelliptic curves of genus g. We compare our compactification with other well-known remarkable compactifications of Hg.

Genetic Order and Compactifying Garbage Collectors

Retinal arteriolar annuli.

Arterial annuli occur at the junction of side-arm branches of retinal arteries in a number of species, and we have studied them in the owl monkey, cat, dog, rat, and pig. They were initially described as being PAS-positive, but they can also be demonstrated with the Masson's, elastic-Van Gieson's, and Gridley's quadruple stains. In most instances the annuli were hypercellular, and some of the c...