A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.Comment: 24 pages, 6 figures (now with minor revisions and clarifications

A Generic Storage API

We present a generic API suitable for provision of highly generic storage facilities that can be tailored to produce various individually customised storage infrastructures. The paper identifies a candidate set of minimal storage system building blocks, which are sufficiently simple to avoid encapsulating policy where it cannot be customised by applications, and composable to build highly flexible storage architectures. Four main generic components are defined: the store, the namer, the caster and the interpreter. It is hypothesised that these are sufficiently general that they could act as building blocks for any information storage and retrieval system. The essential characteristics of each are defined by an interface, which may be implemented by multiple implementing classes.Comment: Submitted to ACSC 200

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."Comment: 9 pages, 4 figure

Report on the XBase Project

This project addressed the conceptual fundamentals of data storage, investigating techniques for provision of highly generic storage facilities that can be tailored to produce various individually customised storage infrastructures, compliant to the needs of particular applications. This requires the separation of mechanism and policy wherever possible. Aspirations include: actors, whether users or individual processes, should be able to bind to, update and manipulate data and programs transparently with respect to their respective locations; programs should be expressed independently of the storage and network technology involved in their execution; storage facilities should be structure-neutral so that actors can impose multiple interpretations over information, simultaneously and safely; information should not be discarded so that arbitrary historical views are supported; raw stored information should be open to all; where security restrictions on its use are required this should be achieved using cryptographic techniques. The key advances of the research were: 1) the identification of a candidate set of minimal storage system building blocks, which are sufficiently simple to avoid encapsulating policy where it cannot be customised by applications, and composable to build highly flexible storage architectures 2) insight into the nature of append-only storage components, and the issues arising from their application to common storage use-cases