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The ADE correspondences are ubiquitous in mathematics. We begin with the regular polyhedra (known to the ancient Greeks) and invite the reader on a journey of discovery.
For the benefit of students, we provide an introduction to areas of mathematics we need: vector spaces, polytopes, groups (discrete and continuous), conjugacy representations, etc.
We treat some more advanced topics: monstrous (and other) moonshine, Monster and E_8, Niemeier lattices, the triangle property, generalized line graphs, quiver representations, cluster algebras, von Neumann algebras, catastrophes, Calabi–Yau, elliptic fibrations.
We discuss some areas where the ADE classification arises: polytopes, tessellations, root systems, Coxeter groups, spectra of graphs, binary polyhedral groups, reflections, Clifford algebras, Lie groups and algebras.
The ADE diagrams, shown on the cover, constitute one of the most universal and mysterious patterns in all of mathematics. John McKay's remarkable insights unveiled a connection between the 'double covers' of the groups of regular polyhedra, known since ancient Greek times, and the exceptional Lie algebras, recognised since the late nineteenth century. The correspondence involves the ADE diagrams being interpreted in different ways: as quivers associated with the groups and as Dynkin diagrams of root systems of Lie algebras. The ADE diagrams arise in many areas of mathematics, including topics in algebraic geometry, string theory, spectral theory of graphs and cluster algebras. Accessible to students, this book explains these connections with exercises and examples throughout. An excellent introduction for students and researchers wishing to learn more about this unifying principle of mathematics, it also presents standard undergraduate material from a novel perspective.
To evaluate the risk of surgical site infection (SSI) following complicated appendectomy in individual patients receiving delayed primary closure (DPC) versus primary closure (PC) after adjustment for individual risk factors.
Design:
Secondary analysis of randomized controlled trial (RCT) with prediction model.
Setting:
Referral centers across Thailand.
Participants:
Adult patients who underwent appendectomy via a lower-right-quadrant abdominal incision due to complicated appendicitis.
Methods:
A secondary analysis of a published RCT was performed applying a counterfactual prediction model considering interventions (PC vs DPC) and other significant predictors. A multivariable logistic regression was applied, and a likelihood-ratio test was used to select significant predictors to retain in a final model. Factual versus counterfactual SSI risks for individual patients along with individual treatment effect (iTE) were estimated.
Results:
In total, 546 patients (271 PC vs 275 DPC) were included in the analysis. The individualized prediction model consisted of allocated intervention, diabetes, type of complicated appendicitis, fecal contamination, and incision length. The iTE varied between 0.4% and 7% for PC compared to DPC; ∼38.1% of patients would have ≥2.1% lower SSI risk following PC compared to DPC. The greatest risk reduction was identified in diabetes with ruptured appendicitis, fecal contamination, and incision length of 10 cm, where SSI risks were 47.1% and 54.1% for PC and DPC, respectively.
Conclusions:
In this secondary analysis, we found that most patients benefited from early PC versus DPC. Findings may be used to inform SSI prevention strategies for patients with complicated appendicitis.
The Association of Transnational Law Schools [ATLAS] is a consortium of seven law schools from four continents that launched an annual academic summer program, called the Agora, for doctoral students this past July 2008. As the name of the consortium would suggest, the program focused on transnational law. The Agora is one of several multi-school initiatives aimed at furthering the study of the globalizing legal environment. The Agora both reflects and furthers a trend in legal scholarship, and as a consequence legal education, toward a focus on a set of interrelated concerns, which include globalization, international governance, transnational law, comparative legal studies, legal transplantation and the apparent conceptual challenges that these pose. In important respects these new conceptual challenges have a long pedigree in questions about the scope of legal pedagogy and theory. The pedagogical controversy is rooted in questions about the purpose of legal education, namely, whether it is trade training and should focus on practical legal skills, or whether it should be conceived of as broader than this. Intimately connected to this pedagogical controversy is a legal-theoretical controversy about the scope of legal theory (and thus the nature of law and its investigation). Does the word “law” designate the organizational instruments of state power, or should we think of “law” as referring to a more diverse set of social-organizational systems that may have greater or less affinity and connection with state law?
The deep subsurface of other planetary bodies is of special interest for robotic and human exploration. The subsurface provides access to planetary interior processes, thus yielding insights into planetary formation and evolution. On Mars, the subsurface might harbour the most habitable conditions. In the context of human exploration, the subsurface can provide refugia for habitation from extreme surface conditions. We describe the fifth Mine Analogue Research (MINAR 5) programme at 1 km depth in the Boulby Mine, UK in collaboration with Spaceward Bound NASA and the Kalam Centre, India, to test instruments and methods for the robotic and human exploration of deep environments on the Moon and Mars. The geological context in Permian evaporites provides an analogue to evaporitic materials on other planetary bodies such as Mars. A wide range of sample acquisition instruments (NASA drills, Small Planetary Impulse Tool (SPLIT) robotic hammer, universal sampling bags), analytical instruments (Raman spectroscopy, Close-Up Imager, Minion DNA sequencing technology, methane stable isotope analysis, biomolecule and metabolic life detection instruments) and environmental monitoring equipment (passive air particle sampler, particle detectors and environmental monitoring equipment) was deployed in an integrated campaign. Investigations included studying the geochemical signatures of chloride and sulphate evaporitic minerals, testing methods for life detection and planetary protection around human-tended operations, and investigations on the radiation environment of the deep subsurface. The MINAR analogue activity occurs in an active mine, showing how the development of space exploration technology can be used to contribute to addressing immediate Earth-based challenges. During the campaign, in collaboration with European Space Agency (ESA), MINAR was used for astronaut familiarization with future exploration tools and techniques. The campaign was used to develop primary and secondary school and primary to secondary transition curriculum materials on-site during the campaign which was focused on a classroom extra vehicular activity simulation.
We numerically model the dynamics of the Enceladus plume ice grains and define our nominal plume model as having a particle size distribution n(R) ~ R−q with q = 4 and a total particulate mass rate of 16 kg s−1. This mass rate is based on average plume brightness observed by Cassini across a range of orbital positions. The model predicts sample volumes of ~1600 µg for a 1 m2 collector on a spacecraft making flybys at 20–60 km altitudes above the Enceladus surface. We develop two scenarios to predict the concentration of amino acids in the plume based on these assumed sample volumes. We specifically consider Glycine, Serine, α-Alanine, α-Aminoisobutyric acid and Isovaline. The first ‘abiotic’ model assumes that Enceladus has the composition of a comet and finds abundances between 2 × 10−6 to 0.003 µg for dissolved free amino acids and 2 × 10−5 to 0.3 µg for particulate amino acids. The second ‘biotic’ model assumes that the water of Enceladus's ocean has the same amino acid composition as the deep ocean water on Earth. We compute the expected captured mass of amino acids such as Glycine, Serine, and α-Alanine in the ‘biotic’ model to be between 1 × 10−5 to 2 × 10−5 µg for dissolved free amino acids and dissolved combined amino acids and about 0.0002 µg for particulate amino acids. Both models consider enhancements due to bubble bursting. Expected captured mass of amino acids is calculated for a 1 m2 collector on a spacecraft making flybys with a closest approach of 20 km during mean plume activity for the given nominal particle size distribution.
We describe the Schwarzian equations for the 328 completely replicable functions with integral $q$-coefficients [Ford et al., ‘More on replicable functions’, Comm. Algebra 22 (1994) no. 13, 5175–5193].