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    Exact solutions to the Erdős-Rothschild problem

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    Let k := (k1,...,k2) be a sequence of natural numbers. For a graph G, let F (G;k) denote the number of colourings of the edges of G with colours 1,...,s such that, for every c ∈ {1,...,s}, the edges of colour c contain no clique of order kc. Write F (n; k) to denote the maximum of F (G;k) over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for n → ∞: (i) A sufficient condition on k which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case k = (3,..., 3) of length 7, the unique extremal graph is the complete balanced 8-partite graph, with colourings coming from Hadamard matrices of order 8. (iii) In the case k = (k+ 1, k), for which the sufficient condition in (i) does not hold, for 3 ≤ k ≤ 10, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible

    Periodic Bedrock Ridges at Oxia Planum and Chryse Planitia, Mars: Evidence for widespread aeolian erosion of an ancient surface by regional paleowinds

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    Periodic Bedrock Ridges (PBRs) are repeating, symmetrical, wind-transverse, bedrock-abraded linear ridges that occur on Mars as parallel sets. Here, we extend our previous survey of PBRs at Oxia Planum – the landing site of ESA's ExoMars Rosalind Franklin rover – to include three additional sites along the margins of the circum-Chryse basin to understand patterns in PBR orientation and occurrence. We analyzed PBR crestline orientation at each study site and found them to be consistent across this large region, but their orientations do not align with global circulation model winds, suggesting contemporary winds are not responsible for PBR development. Furthermore, we used observations of landscape-level stratigraphic relationships at Oxia Planum to constrain the formation age of PBRs to be late Noachian to early Amazonian. Their consistent orientations and age suggest that the Chryse-margin PBRs formed concurrently and represent modification of an ancient palaeosurface. In addition, we find a tendency for PBRs to preferentially occur in regions where Fe/Mg-rich phyllosilicate minerals were detected in hyperspectral remote sensing data. We conclude that regional scale aeolian processes formed the circum-Chryse PBRs, and that exposed bedrock with Fe/Mg-rich phyllosilicate detections were either more susceptible to PBR formation, or better preserve PBRs than other regional lithologies

    Clinical Foundations: A brief history of relational practice

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    This chapter traces a brief history of relational practice in the clinical context of distress and recovery. Starting with psychotherapeutic theory and the centrality of the therapeutic relationship, we explore how relational thinking has, and has not, been taken up by contemporary adult mental health practice and policy. The chapter argues that contemporary conceptions of the person as individual and independent have resulted in services that, on the whole, no longer recognise the essential role of interdependency, nor fully embrace the opportunity that relationships offer for interventions that can make a lasting difference. We look at alternative and innovative examples of relational interventions, including therapeutic communities, Open Dialogue, and peer-led interventions like Hearing Voices groups. We then conclude by considering how well relational thinking has been integrated into mainstream mental health settings and where practice may go from here

    Introduction: Why relationships matter for mental health

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    In this brief introductory chapter, we set out our hopes for the book and provide an overview of the chapter structure

    Conclusions and recommendations

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    This conclusion presents some closing thoughts of the key concepts discussed in the preceding chapters of this book. The book emphasizes the importance of structurally including design in all of its guises, methodologies and perspectives as standard practice in emergency management. It points out, in order to enable design for emergency management, a broader culture of design needs to become established within government agencies. The book offers suggestions of partners who can be engaged in co-designing, such as for-profit entities in the retail sector. It also offers another creative solution to this conundrum by showcasing the value of collaborating with higher education institutes. The book examines how to mobilize people by showcasing the interplay of various communication aspects, that is text, imagery, and an auditory component, which work in harmony to foreground urgency in a warning, and are intended to assist the audience in sense making and subsequent action taking

    Is there H<sub>2</sub>O stacking disordered ice I in the Solar System?

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    Water ice exists in large quantities across the Solar System, and it is involved in a wide range of atmospheric and geological processes. Here we focus on the question if stacking disordered ice I (ice Isd) is present in the Solar System. The conditions required to form ice Isd are described and we argue that previous descriptions of ‘cubic ice’ (ice Ic) in the literature may in fact have been concerned with ice Isd. In contrast to the stable hexagonal ice I (ice Ih) and ice Ic, ice Isd is a highly complex material that encompasses a wide range of possible stacking regimes and structures. The most fundamental quantity to describe a given ice Isd sample is its cubicity which reflects the fraction of cubic stacking. Following an introduction into the characterisation techniques used to identify and characterise ice Isd, we discuss the various environments in the Solar System where ice Isd may exist and the relevance its existence may have. This includes the atmospheres of the inner planets, various icy moons as well as comets and other icy objects in the far reaches of the Solar System. The details of the stacking disorder may contain information about the formation and thermal history of ice Isd samples. This offers the exciting prospect of using ice Isd as a marker material for atmospheric and geological processes. The crystallographic space group of ice Isd allows polar structures which could be an important factor for the accretion of ice particles in space. We conclude that ice Isd should exist at several locations in the Solar System and in potentially large quantities. The definitive identification of ice Isd in a natural environment is a next major milestone in our understanding of the importance of water ice across the Solar System

    Realistic Ambitions: Technology Transfer for Biologics Platform Technologies

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    This chapter argues that the recent international support for expanding vaccine production in African countries offers local policymakers and industrialists an opportunity that should be seized, to transition local manufacturing capabilities to produce biologics. Biologics offer a broader portfolio of cancer therapies. Biologics such as monoclonal antibodies represent an incremental innovation for vaccine manufacturers, with lower learning and transition costs than those that would be faced by manufacturers of chemical drugs seeking to move into biologics. However, biologics production in Africa is not only a technological project, it is also political and economic, feeding into geo-politics debates

    Evaluation and comparison of spatial cluster detection methods for improved decision making of disease surveillance: a case study of national dengue surveillance in Thailand

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    Background: Dengue is a mosquito-borne disease that causes over 300 million infections worldwide each year with no specific treatment available. Effective surveillance systems are needed for outbreak detection and resource allocation. Spatial cluster detection methods are commonly used, but no general guidance exists on the most appropriate method for dengue surveillance. Therefore, a comprehensive study is needed to assess different methods and provide guidance for dengue surveillance programs. Methods: To evaluate the effectiveness of different cluster detection methods for dengue surveillance, we selected and assessed commonly used methods: Getis Ord Gi∗, Local Moran, SaTScan, and Bayesian modeling. We conducted a simulation study to compare their performance in detecting clusters, and applied all methods to a case study of dengue surveillance in Thailand in 2019 to further evaluate their practical utility. Results: In the simulation study, Getis Ord Gi∗ and Local Moran had similar performance, with most misdetections occurring at cluster boundaries and isolated hotspots. SaTScan showed better precision but was less effective at detecting inner outliers, although it performed well on large outbreaks. Bayesian convolution modeling had the highest overall precision in the simulation study. In the dengue case study in Thailand, Getis Ord Gi∗ and Local Moran missed most disease clusters, while SaTScan was mostly able to detect a large cluster. Bayesian disease mapping seemed to be the most effective, with adaptive detection of irregularly shaped disease anomalies. Conclusions: Bayesian modeling showed to be the most effective method, demonstrating the best accuracy in adaptively identifying irregularly shaped disease anomalies. In contrast, SaTScan excelled in detecting large outbreaks and regular forms. This study provides empirical evidence for the selection of appropriate tools for dengue surveillance in Thailand, with potential applicability to other disease control programs in similar settings

    Flux-conserving directed percolation

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    We discuss a model for directed percolation in which the flux of material along each bond is a dynamical variable. The model includes a physically significant limiting case where the total flux of material is conserved. We show that the distribution of fluxes is asymptotic to a power law at small fluxes. We give an implicit equation for the exponent, in terms of probabilities characterising site occupations. In one dimension the site occupations are exactly independent, and the model is exactly solvable. In two dimensions, the independent-occupation assumption gives a good approximation. We explore the relationship between this model and traditional models for directed percolation

    A hopeful future for mobile language learning

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    Mobile language learning has long inspired teachers and researchers to innovate, improve their classroom practice and enhance learning for their students, but the Covid-19 pandemic drew attention to more far-reaching educational challenges that could be addressed through the adoption of mobile approaches to learning. From a position of optimism and hope, the chapter sets out some of those educational challenges and the ways in which mobile learning offers multiple ways of responding to them. Language education plays a central role, as it aims to empower individuals and communities, to promote intercultural understanding and to connect people by equipping them to overcome linguistic barriers in communication. At the same time, teachers and learners are living through difficult times and any innovations should be introduced appropriately. The chapter outlines reasons to be hopeful, in the context of profound changes that are taking place in digital and online learning. It offers observations regarding interrelationships between languages and mobile technologies and their relationships with informal learning, where learners are leading the way. Out of these reflections arise some pointers towards possible and desirable futures for mobile language learning around the world


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