Robust ambiguity for contact tracing

A known drawback of `decentralised' contact tracing architectures is that users who have been in contact with an infected person are able to precisely identify the relevant contact, and thereby perhaps identify the infected person. In their proposal, the PACT team discuss a simple DH-based protocol to mitigate this problem, but dismiss it because it is vulnerable to a malicious user who may deviate from the specified behaviour. This note presents a modified protocol which achieves robustness against a fully malicious user, and establishes some simple security properties.Comment: 5 page

A computer assisted proof of universality for cubic critical maps of the circle with Golden Mean rotation number

In order to explain the universal metric properties associated with the breakdown of invariant tori in dissipative dynamical systems, Ostlund, Rand, Sethna and Siggia together with Feigenbaum, Kadanoff and Shenker have developed a renormalisation group analysis for pairs of analytic functions that glue together to make a map of the circle. Using a method of Lanford's, we have obtained a proof of the existence and hyperbolicity of a non-trivial fixed point of the renormalisation transformation for rotation number equal to the golden mean (√5 - 1/2). The proof uses numerical estimates obtained rigorously with the aid of a computer. These computer calculations were based on a method of Eckmann, Koch and Wittwer

A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics

From the output produced by a memoryless deletion channel from a uniformly random input of known length $n$, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length $m$, and it is natural to ask for which outputs this is extremized. This question was posed in a previous work, where it was conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant strings $\texttt{000}\ldots$ and $\texttt{111}\ldots$ and the alternating strings $\texttt{0101}\ldots$ and $\texttt{1010}\ldots$ respectively. In the present work we confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics. We show how the analytic-combinatorial methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and $n\rightarrow\infty$, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure

Widths of regular and context-free languages

Given a partially-ordered finite alphabet $\Sigma$ and a language $L\subseteq \Sigma^*$, how large can an antichain in $L$ be (where $L$ is given the lexicographic ordering)? More precisely, since $L$ will in general be infinite, we should ask about the rate of growth of maximum antichains consisting of words of length $n$. This fundamental property of partial orders is known as the width, and in a companion work we show that the problem of computing the information leakage permitted by a deterministic interactive system modeled as a finite-state transducer can be reduced to the problem of computing the width of a certain regular language. In this paper, we show that if $L$ is regular then there is a dichotomy between polynomial and exponential antichain growth. We give a polynomial-time algorithm to distinguish the two cases, and to compute the order of polynomial growth, with the language specified as an NFA. For context-free languages we show that there is a similar dichotomy, but now the problem of distinguishing the two cases is undecidable. Finally, we generalise the lexicographic order to tree languages, and show that for regular tree languages there is a trichotomy between polynomial, exponential and doubly exponential antichain growth.Comment: 22 page

Analysis of Clumps in Molecular Cloud Models: Mass Spectrum, Shapes, Alignment and Rotation

Observations reveal concentrations of molecular line emission on the sky, called ``clumps,'' in dense, star-forming molecular clouds. These clumps are believed to be the eventual sites of star formation. We study the three-dimensional analogs of clumps using a set of self-consistent, time-dependent numerical models of molecular clouds. The models follow the decay of initially supersonic turbulence in an isothermal, self-gravitating, magnetized fluid. We find the following. (1) Clumps are intrinsically triaxial. This explains the observed deficit of clumps with a projected axis ratio near unity, and the apparent prolateness of clumps. (2) Simulated clump axes are not strongly aligned with the mean magnetic field within clumps, nor with the large-scale mean fields. This is in agreement with observations. (3) The clump mass spectrum has a high-mass slope that is consistent with the Salpeter value. There is a low-mass break in the slope at \sim 0.5 \msun, although this may depend on model parameters including numerical resolution. (4) The typical specific spin angular momentum of clumps is $4 \times 10^{22} {\rm cm^2 s^{-1}}$. This is larger than the median specific angular momentum of binary stars. Scaling arguments suggest that higher resolution simulations may soon be able to resolve the scales at which the angular momentum of binary stars is determined.Comment: 14 pages, 13 figures, to appear in 2003 July 20 Ap

A Survey of Requirements for COVID-19 Mitigation Strategies. Part I: Newspaper Clips

The COVID-19 pandemic has influenced virtually all aspects of our lives. Across the world, countries have applied various mitigation strategies for the epidemic, based on social, political, and technological instruments. We postulate that one should {identify the relevant requirements} before committing to a particular mitigation strategy. One way to achieve it is through an overview of what is considered relevant by the general public, and referred to in the media. To this end, we have collected a number of news clips that mention the possible goals and requirements for a mitigation strategy. The snippets are sorted thematically into several categories, such as health-related goals, social and political impact, civil rights, ethical requirements, and so on. In a forthcoming companion paper, we will present a digest of the requirements, derived from the news clips, and a preliminary take on their formal specification

A Two-Fluid Method for Ambipolar Diffusion

We present a semi-implicit method for isothermal two-fluid ion-neutral ambipolar drift that is second-order accurate in space and time. The method has been implemented in the RIEMANN code for astrophysical fluid dynamics. We present four test problems that show the method works and correctly tracks the propagation of MHD waves and the structure of two-fluid C-shocks. The accurate propagation of MHD waves in the two-fluid approximation is shown to be a stringent test of the algorithm. We demonstrate that highly accurate methods are required in order to properly capture the MHD wave behaviour in the presence of ion-neutral friction.Comment: 29 pages, 16 figures, accepted to MNRA

Cloud angular momentum and effective viscosity in global SPH simulations with feedback

We examine simulations of isolated galaxies to analyse the effects of localized feedback on the formation and evolution of molecular clouds. Feedback contributes to turbulence and the destruction of clouds, leading to a population of clouds that is younger, less massive, and with more retrograde rotation. We investigate the evolution of clouds as they interact with each other and the diffuse interstellar medium, and determine that the role of cloud interactions differs strongly with the presence of feedback: in models without feedback, scattering events dramatically increase the retrograde fraction, but in models with feedback, mergers between clouds may slightly increase the prograde fraction. We also produce an estimate of the viscous time-scale due to cloud–cloud collisions, which increases with increasing strength of feedback (tν ∼ 20 Gyr versus tν ∼ 10 Gyr), but is still much smaller than previous estimates (tν ∼ 1000 Gyr); although collisions become more frequent with feedback, less energy is lost in each collision than in the models without feedback