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Southern Iceland is one of the main outlets of the Icelandic ice sheet and is subject to seismicity of both tectonic and volcanic origins along the South Iceland Seismic Zone (SISZ). A sedimentary complex spanning Marine Isotopic Stage 6 (MIS 6) to the present includes evidence of both activities. It includes a continuous sedimentary record since the Eemian interglacial period, controlled by a rapid deglaciation, followed by two marine glacioisostasy-forced transgressions, separated by a regression phase connected to an intra-MIS 5e glacial advance. This record has been constrained by tephrostratigraphy and dating. Analysis of this record has provided better insights into the interconnectedness of hydrology and volcanic and tectonic activity during deglaciations and glaciations. Low-intensity earthquakes recurrently affected the water-laid sedimentation during the early stages of unloading, accompanying rifting events, dyke injection, and fault reactivations. During full interglacial periods, earthquakes were significantly less frequent but of higher magnitude along the SISZ, due to stress accumulation, favored by low groundwater levels and more limited magma production. Occurrence of volcanism and seismicity in Iceland is commonly related to rifting events. Subglacial volcanic events seem moreover to have been related to stress unlocking related to limited or full unloading/deglaciation events. Major eruptions were mostly located at the melting margin of the ice sheet.
Open source design of hardware products is an emerging phenomenon that takes more and more importance today's in the society. However, open source (hardware) design implies a tremendous change in both design practices and philosophy because it is partly related to the movements of creative commons and the sharing economy. From this perspective one could think that participation is crucial in the success of open source design projects. In this paper, we analyse 9 case studies in the light of 3 hypotheses. If many studies highlight the potential of the crowd as a resource for design tasks, our study shows that for open source design communities the participation is not massive. In this study, we used an activity-based approach to build our model. As open source design processes are fairly unstructured and based on voluntary participation, it is impossible to adopt a classical task-based model. With the help of this model, we were able evaluate the overall size of the active community, the participation rate with regards to the activities. This study paves the way to deeper and extensive studies on how to support communities engaged in open source design of hardware products.
Senegal is experiencing a rising obesity epidemic, due to the nutrition transition occurring in most African countries, and driven by sedentary behaviour and high-calorie dietary intake. In addition, the anthropological local drivers of the social valorization of processed high-calorie food and large body sizes could expose the population to obesity risk. This study aimed to determine the impact of these biocultural factors on the nutritional status of Senegalese adults. A mixed methods approach was used, including qualitative and quantitative studies. Between 2011 and 2013, fourteen focus group discussions (n=84) and a cross-sectional quantitative survey (n=313 women; n=284 men) of adults in three different socio-ecological areas of Senegal (rural: n=204; suburban: n=206; urban: n=187) were conducted. Dietary intake (Dietary Diversity Scores), physical activity (International Physical Activity Questionnaire), body weight norms (Body Size Scale), weight and health statuses (anthropometric measures and blood pressure) were measured. Middle-aged and older Senegalese women were found to value overweight/obesity more than younger Senegalese in all regions. In addition, young urban/suburban adults had a tendency for daily snacking whilst urban/suburban adults tended to be less physically active and had higher anthropometric means. A binary logistic regression model showed that being female, older, living in urban/suburban areas and valuing larger body size were independently associated with being overweight/obese, but not high-calorie diet. Univariate analyses showed that lower physical activity and higher socioeconomic status were associated with being overweight/obese. Finally, overweight/obesity, which is low in men, is associated with hypertension in the total sample. The nutrition transition is currently underway in Senegal’s urban/suburban areas, with older women being more affected. Since several specific biocultural factors jointly contribute to this phenomenon, the study’s findings suggest the need for local public health interventions that target women and which account for the anthropological specificities of the Senegalese population.
In this chapter we treat the basic theory of central simple algebras from a modern viewpoint. The main point we would like to emphasize is that, as a consequence of Wedderburn's theorem, we may characterize central simple algebras as those finite-dimensional algebras which become isomorphic to some full matrix ring over a finite extension of the base field. We then show that this extension can in fact be chosen to be a Galois extension, which enables us to exploit a powerful theory in our further investigations, that of Galois descent. Using descent we can give elegant treatments of such classical topics as the construction of reduced norms or the Skolem–Noether theorem. The main invariant concerning central simple algebras is the Brauer group, which classifies all finite-dimensional central division algebras over a field. Using Galois descent, we shall identify it with a certain first cohomology set equipped with an abelian group structure.
The foundations of the theory of central simple algebras go back to the great algebraists of the dawn of the twentieth century; we merely mention here the names of Wedderburn, Dickson and Emmy Noether. The Brauer group appears in the pioneering paper of the young Richard Brauer [1]. Though Galois descent had been implicitly used by algebraists in the early years of the twentieth century and Châtelet had considered special cases in connection with Diophantine equations, it was AndréWeil who first gave a systematic treatment with applications to algebraic geometry in mind (Weil [2]). The theory in the form presented below was developed by Jean-Pierre Serre, and finally found a tantalizing generalization in the general descent theory of Grothendieck ([1], [2]).
Wedderburn's theorem
Let k be a field. We assume throughout that all k-algebras under consideration are finite dimensional over k. A k-algebra A is called simple if it has no (two-sided) ideal other than 0 and A. Recall moreover from the previous chapter that A is central if its centre equals k.
Here are the basic examples of central simple algebras.
This book provides a comprehensive and up-to-date introduction to the theory of central simple algebras over arbitrary fields, emphasizing methods of Galois cohomology and (mostly elementary) algebraic geometry. The central result is the Merkurjev–Suslin theorem. As we see it today, this fundamental theorem is at the same time the culmination of the theory of Brauer groups of fields initiated by Brauer, Noether, Hasse and Albert in the 1930s, and a starting point of motivic cohomology theory, a domain which is at the forefront of current research in algebraic geometry and K-theory – suffice it here to mention the recent spectacular results of Voevodsky, Suslin, Rost and others. As a gentle ascent towards the Merkurjev–Suslin theorem, we cover the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. These chapters also contain a number of noteworthy additional topics. The last chapter of the book rounds off the theory by presenting the results in positive characteristic. For an overview of the contents of each chapter we refer to their introductory sections.
Prerequisites. The book should be accessible to a graduate student or a nonspecialist reader with a solid training in algebra including Galois theory and basic commutative algebra, but no homological algebra. Some familiarity with algebraic geometry is also helpful. Most of the text can be read with a basic knowledge corresponding to, say, the first volume of Shafarevich's text. To help the novice, we summarize in an appendix the results from algebraic geometry we need. The first three sections of Chapter 8 require some familiarity with schemes, and in the proof of one technical statement we are forced to use techniques from Quillen K-theory. However, these may be skipped in a first reading by those willing to accept some ‘black boxes’.
Acknowledgments
Our first words of thanks go to Jean-Louis Colliot-Thélène and Jean-Pierre Serre, from whom we learned much of what we know about the subject and who, to our great joy, have also been the most assiduous readers of the manuscript, and suggested many improvements. Numerous other colleagues helped us with their advice during the preparation of the text, or spotted inaccuracies in previous versions.
Residue maps constitute a fundamental technical tool for the study of the cohomological symbol. Their definition is not particularly enlightening at a first glance, but the reader will see that they emerge naturally during the computation of Brauer groups of function fields or complete discretely valued fields. When one determines these, a natural idea is to pass to a field extension having trivial Brauer group. There are three famous classes of such fields: finite fields, function fields of curves over an algebraically closed field and complete discretely valued fields with algebraically closed residue field. Once we know that the Brauer groups of these fields vanish, we are able to compute the Brauer groups of function fields over an arbitrary perfect field. The central result here is Faddeev's exact sequence for the Brauer group of a rational function field. We give two important applications of this theory: one to the class field theory of curves over finite fields, the other to constructing counterexamples to the rationality of the field of invariants of a finite group acting on some linear space. Following this ample motivation, we finally attack residue maps with finite coefficients, thereby preparing the ground for the next two chapters.
Residue maps for the Brauer group first appeared in the work of the German school on class field theory; the names of Artin, Hasse and F. K. Schmidt are the most important to be mentioned here. It was apparently Witt who first noticed the significance of residue maps over arbitrary discretely valued fields. Residue maps with finite coefficients came into the foreground in the 1960s in the context of étale cohomology; another source for their emergence in Galois cohomology is work by Arason [1] on quadratic forms.
Cohomological dimension
Before embarking on the study of fields with vanishing Brauer group it is convenient to discuss the relevant cohomological background: this is the theory of cohomological dimension for profinite groups, introduced by Tate.
Recall that for an abelian group B and a prime number p, the notation B﹛p﹜ stands for the p-primary torsion subgroup of B, i.e. the subgroup of elements of p-power order.
We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol.
The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others.
Profinite groups and Galois groups
It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension ks|k, where ks is a separable closure of k.
As a prelude to the book, we present here our main objects of study in the simplest case, that of quaternion algebras. Many concepts that will be ubiquitous in what follows, such as division algebras, splitting fields or norms appear here in a concrete and elementary context. Another important notion we shall introduce is that of the conic associated with a quaternion algebra; these are the simplest examples of Severi–Brauer varieties, objects to which a whole chapter will be devoted later. In the second part of the chapter two classic theorems from the 1930s are proven: a theorem of Witt asserting that the associated conic determines a quaternion algebra up to isomorphism, and a theorem of Albert that gives a criterion for the tensor product of two quaternion algebras to be a division algebra. The discussion following Albert's theorem will lead us to the statement of one of the main theorems proven later in the book, that of Merkurjev concerning division algebras of period 2.
The basic theory of quaternion algebras goes back to the nineteenth century. The original references for the main theorems of the last two sections are Witt [1] and Albert [1], [5], respectively.
Basic properties
In this book we shall study finite-dimensional algebras over a field. Here by an algebra over a field k we mean a k-vector space equipped with a not necessarily commutative but associative k-linear multiplication. All k-algebras will be tacitly assumed to have a unit element.
Historically the first example of a finite-dimensional noncommutative algebra over a field was discovered by W. R. Hamilton during a walk with his wife (presumably doomed to silence) on 16 October 1843. It is the algebra of quaternions, a 4-dimensional algebra with basis 1, i, j, k over the field R of real numbers, the multiplication being determined by the rules
This is in fact a division algebra over R, which means that each nonzero element x has a two-sided multiplicative inverse, i.e. an element y with xy = yx = 1. Hamilton proved this as follows.
In Chapter 1 we associated with each quaternion algebra a conic with the property that the conic has a k-point if and only if the algebra splits over k. We now generalize this correspondence to arbitrary dimension: with each central simple algebra A of degree n over an arbitrary field k we associate a projective k-variety X of dimension n − 1 which has a k-point if and only if A splits. Both objects will correspond to a class in H1(G,PGLn(K)), where K is a Galois splitting field for A with group G. The varieties X arising in this way are called Severi–Brauer varieties; they are characterized by the property that they become isomorphic to some projective space over the algebraic closure. This interpretation will enable us to give another, geometric construction of the Brauer group. Another central result of this chapter is a theorem of Amitsur which states that for a Severi–Brauer variety X with function field k(X) the kernel of the natural map Br (k) → Br (k(X)) is a cyclic group generated by the class of X. This seemingly technical statement (which generalizes Witt's theorem proven in Chapter 1) has very fruitful algebraic applications. At the end of the chapter we shall present one such application, due to Saltman, which shows that all central simple algebras of fixed degree n over a field k containing the n-th roots of unity can be made cyclic via base change to some large field extension of k.
Severi–Brauer varieties were introduced in the pioneering paper of Châtelet [1], under the name ‘variétés de Brauer’. Practically all results in the first half of the present chapter stem from this work. The term ‘Severi–Brauer variety’ was coined by Beniamino Segre in his note [1], who expressed his discontent that Châtelet had ignored previous work by Severi in the area. Indeed, in the paper of Severi [1] Severi–Brauer varieties are studied in a classical geometric context, and what is known today as Châtelet's theorem is proven in some cases. As an amusing feature, we may mention that Severi calls the varieties in question ‘varietà di Segre’, but beware, this does not refer to Beniamino but to his second uncle Corrado Segre.
In the preceding chapters, when working with Galois cohomology groups or K-groups modulo some prime, a standing assumption was that the groups under study were torsion groups prime to the characteristic of the base field. We now remove this restriction. In the first part of the chapter the central result is Teichmüller's theorem, according to which the p-primary torsion subgroup in the Brauer group of a field of characteristic p > 0 is generated by classes of cyclic algebras – a characteristic p ancestor of the Merkurjev–Suslin theorem. We shall give two proofs of this statement: a more classical one due to Hochschild which uses central simple algebras, and a totally different one based on a presentation of the p-torsion in Br (k) via logarithmic differential forms. The key tool here is a famous theorem of Jacobson–Cartier characterizing logarithmic forms. The latter approach leads us to the second main topic of the chapter, namely the study of the differential symbol. This is a p-analogue of the Galois symbol which relates the Milnor K-groups modulo p to a certain group defined using differential forms. We shall prove the Bloch–Gabber–Kato theorem establishing its bijectivity, and obtain as an application the absence of p-torsion in Milnor K-groups of fields of characteristic p, a statement due to Izhboldin.
Teichmüller's result first appeared in the ill-famed journal Deutsche Mathematik (Teichmüller [1]); see also Jacobson [3] for an account of the original proof. The role of derivations and differentials in the theory of central simple algebras was noticed well before the Second World War; today the most important work seems to be that of Jacobson [1]. This line of thought was further pursued in papers by Hochschild [1], [2], and, above all, in the thesis of Cartier [1], which opened the way to a wide range of geometric developments. The original references for the differential symbol are the papers of Kato [2] and Bloch–Kato [1]; they have applied the theory to questions concerning higher-dimensional local fields and p-adic Hodge theory.
The theorems of Teichmüller and Albert
In what follows k will denote a field of characteristic p > 0, and ks will be a fixed separable closure of k.
The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.
Between 2010 and 2012, 3 outbreaks of nosocomial infections in German neonatal intensive care units (NICUs) attracted considerable public interest. Headlines on national television channels and in newspapers had important consequences for the involved institutions and a negative impact on the relationship between families and staff in many German NICUs.
OBJECTIVE
To determine whether NICU outbreaks reported in the media influenced provider behavior in the community of neonatal care and led to more third-line antibiotic prescribing.
DESIGN
Observational cohort study.
METHODS
To investigate secular trends, we evaluated data for very-low-birth-weight infants (VLBWIs, birth weight <1,500 g) enrolled in the German Neonatal Network (GNN) between 2009 and 2014 (N=10,253). For outbreak effects, we specifically analyzed data for VLBWIs discharged 6 months before (n=2,428) and 6 months after outbreaks (n=2,508).
RESULTS
The exposure of all VLBWIs to third-line antibiotics increased after outbreaks (19.4% before vs 22.5% after; P=.007). This trend particularly affected male infants (4.6% increase; P=.005) and infants with a birth weight between 1,000 and 1,499 g (3.5% increase; P=.001)
In a logistic regression analysis, month of discharge as linear variable of time was associated with increased exposure to third-line antibiotics (odds ratio [OR], 1.01; 95% confidence interval [CI], 1.009–1.014; P<.001), and discharge within the 6-month period after outbreak reports independently contributed to this long-term trend (OR, 1.14; 95% CI, 1.017–1.270; P=.024).
CONCLUSIONS
Media reports directly affect medical practice, eg, overuse of third-line antibiotics. Future communication and management strategies must be based on objective dialogues between the scientific community and investigative journalists.