Anagram-free Graph Colouring

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with 'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov. Minor changes based on referee feedbac

The Effect of Unresolved Contaminant Stars on the Cross-Matching of Photometric Catalogues

A fundamental process in astrophysics is the matching of two photometric catalogues. It is crucial that the correct objects be paired, and that their photometry does not suffer from any spurious additional flux. We compare the positions of sources in WISE, IPHAS, 2MASS, and APASS with Gaia DR1 astrometric positions. We find that the separations are described by a combination of a Gaussian distribution, wider than naively assumed based on their quoted uncertainties, and a large wing, which some authors ascribe to proper motions. We show that this is caused by flux contamination from blended stars not treated separately. We provide linear fits between the quoted Gaussian uncertainty and the core fit to the separation distributions. We show that at least one in three of the stars in the faint half of a given catalogue will suffer from flux contamination above the 1% level when the density of catalogue objects per PSF area is above approximately 0.005. This has important implications for the creation of composite catalogues. It is important for any closest neighbour matches as there will be a given fraction of matches that are flux contaminated, while some matches will be missed due to significant astrometric perturbation by faint contaminants. In the case of probability-based matching, this contamination affects the probability density function of matches as a function of distance. This effect results in up to 50% fewer counterparts being returned as matches, assuming Gaussian astrometric uncertainties for WISE-Gaia matching in crowded Galactic plane regions, compared with a closest neighbour match.Comment: 10 pages, 6 figures; accepted for publication in the Monthly Notices of the Royal Astronomical Societ

When Thinking Leads to Doing: the Relationship Between Fantasy and Reality in Sexual Offending.

This chapter is taken from the following text: Jones, T. & Wilson, D. (2008). Thinking & Doing; Fantasy & Reality: An Analysis of Convicted Paedophiles. In Birch, P., Ireland, C., & Ireland, J. (Eds.) The Assessment, Treatment and Management of Violent and Sexual Offenders, London: Willa

Signatures of the collapse and revival of a spin Schr\"{o}dinger cat state in a continuously monitored field mode

We study the effects of continuous measurement of the field mode during the collapse and revival of spin Schr\"{o}dinger cat states in the Tavis-Cummings model of N qubits (two-level quantum systems) coupled to a field mode. We show that a compromise between relatively weak and relatively strong continuous measurement will not completely destroy the collapse and revival dynamics while still providing enough signal-to-noise resolution to identify the signatures of the process in the measurement record. This type of measurement would in principle allow the verification of the occurrence of the collapse and revival of a spin Schr\"{o}dinger cat state.Comment: 5 pages, 2 figure

The Impact of Brexit on the future of UK forensic science and technology

This article seeks to assess the prospects of UK forensic science and technology in a post-Brexit world by analysing four interlocking issues: Brexit itself, the evolution of national criminal justice organisational and funding priorities, the increasing interrelationship of science and technology in the forensic domain and the relatively disadvantaged place of forensic science and technology within the contemporary ‘scientific state’ paradigm. The results are generally pessimistic for the likely future of forensic science. This conclusion is reinforced by scepticism about the wisdom of proceeding with Brexit. The article is structured to identify the potential implications of British political decisions on its national forensic science landscape. Some aspects of the analysis are likely to have a wider resonance for international discourse about the future sustainability of forensic science and technology, however, particularly the interface between the globalisation of science and technology with justice

Criminal justice and global public goods: The Prüm Forensic Biometric Cooperation

This article places sharing forensic biometric data for international criminal justice cooperation purposes within the domain of global public goods. Such cooperation is a rational response to globalization, but faces several obstacles. These range from socio-cultural and political concerns about national legal and criminal justice autonomy to the potential impact of market fundamentalism on scientific standardization and cooperation mechanism delivery. The significance of such inhibitors will vary as societal and personal perceptions of stability change. These issues are examined by analysing the progress achieved with the EU Prüm forensic biometric data exchange model. Shocks to European stability, such as the increased scale of terrorist crimes and the UK EU referendum result will inevitably test the resilience of Prüm. Combining insights from global public goods and criminal law scholarship, however, may help to identify how reactions to such shocks, including questions about future UK participation in Prüm, might be managed

Derivative Expansion of the Exact Renormalization Group

The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found: the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.Comment: 14 pages (with figures), Plain TeX, uses psfig, 4 postscript figures appended as uuencoded compressed tar file, SHEP 93/94-16, CERN-TH.7203/94. (Added small details and minor improvements in rigour : the version to be published in Phys.Lett.B