Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks

We initiate the study of distributed protocols over directed anonymous networks that are not necessarily strongly connected. In such networks, nodes are aware only of their incoming and outgoing edges, have no unique identity, and have no knowledge of the network topology or even bounds on its parameters, like the number of nodes or the network diameter. Anonymous networks are of interest in various settings such as wireless ad-hoc networks and peer to peer networks. Our goal is to create distributed protocols that reduce the uncertainty by distributing the knowledge of the network topology to all the nodes. We consider two basic protocols: broadcasting and unique label assignment. These two protocols enable a complete mapping of the network and can serve as key building blocks in more advanced protocols. We develop distributed asynchronous protocols as well as derive lower bounds on their communication complexity, total bandwidth complexity, and node label complexity. The resulting lower bounds are sometimes surprisingly high, exhibiting the complexity of topology extraction in directed anonymous networks

Multipliers of embedded discs

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of $\ell^2$ which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of $\ell^2$ which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety $V$ in $\mathbb B_2$ such that the multiplier algebra is not all of $H^\infty(V)$. We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.Comment: 34 pages; the earlier version relied on a result of Davidson and Pitts that the fibre of the maximal ideal space of the multiplier algebra over a point in the open ball consists only of point evaluation. This result fails for $d = \infty$, and has necessitated some changes; to appear in Complex Analysis and Operator Theor

Non-linear Cyclic Codes that Attain the Gilbert-Varshamov Bound

We prove that there exist non-linear binary cyclic codes that attain the Gilbert-Varshamov bound