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We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.
Serre. In the mean time the fundamental contributions of Grothendieck and Deligne are explained: rationality and the functional equation for L-functions of l-adic sheaves in characteristic p, with essentially complete proofs; the theory of weights; and some theorems of Deligne on the Riemann hypothesis (the last of Weil’s conjectures), this time without proof. Notably, one will find an exposition of functional equations of the L-functions developed by Grothendieck, and a more precise statement and fairly complete parts of the proof of Grothendieck and Deligne’s theorem on the rationality and the functional equation of Hasse–Weil L-functions in characteristic > 0, confirming a conjecture of Serre in this case.
This chapter is dedicated to the Weil conjectures. They are all proven, except the hardest of them: the “Riemann hypothesis”. An overview of Dwork’s p-adic proof of the rationality of zeta functions of varieties over a finite field is given (obtained before the development of Grothendieck’s cohomological methods!).
This last chapter is dedicated to motives and their zeta functions. We consider only an elementary case: that of pure motives of Grothendieck associated to smooth projective varieties over a finite field. One can go much further using the triangulated categories of motives introduced by Voevodsky and developed by Ivorra, Ayoub, and Cisinski–D’eglise, but this would go beyond the scope of the book. Nonetheless, it is explained how this viewpoint considerably clarifies how Weil cohomologies are used to prove rationality and the functional equation, and this theory is applied to prove a somewhat forgotten theorem of Weil: Artin’s conjecture for non-abelian L-functions in positive characteristic.
This chapter returns to more elementary mathematics, introducing Dirichlet, Hecke, and Artin L-functions. A proof of Dirichlet’s theorem on arithmetic progressions is given, by the method expounded by Serre; it would however be a shame to omit Dirichlet’s original method, which gave additional information and anticipated the analytic class number formulae. The two main generalisations of Dirichlet’s L-functions are then introduced: those of Hecke and Artin. Hecke’s main theorem is stated without proof: existence of an analytic continuation and a functional equation, and it is then explained how Artin and Brauer derived the same results for non-abelian L-functions.
This chapter introduces zeta functions of Z-schemes of finite type. It is essentially dedicated to the proof of the Riemann hypothesis for curves over a finite field. An idea of Weil’s proof of the Castelnuovo–Severi inequality is included, and the easy case of curves of genus 1 (due to Hasse) is given. For the general case, the proofs of Mattuck–Tate and Grothendieck, which rely on an a priori weaker inequality, are given; the two inequalities are compared and it is shown that we can recover the first one using the second and the additivity of the numerically trivial divisors.
The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story.
Longitudinal studies of the relationship between cognition and functioning in bipolar disorder are scarce, although cognition is thought to be a key determinant of functioning. The causal structure between cognition and psychosocial functioning in bipolar disorder is unknown.
We sought to examine the direction of causality between cognitive performance and functional outcome over 2 years in a large cohort of euthymic patients with bipolar disorder.
The sample consisted of 272 adults diagnosed with bipolar disorder who were euthymic at baseline, 12 and 24 months. All participants were recruited via the FondaMental Advanced Centers of Expertise in Bipolar Disorders. We used a battery of tests, assessing six domains of cognition at baseline and 24 months. Residual depressive symptoms and psychosocial functioning were measured at baseline and 12 and 24 months. The possible causal structure between cognition and psychosocial functioning was investigated with cross-lagged panel models with residual depressive symptoms as a covariate.
The analyses support a causal model in which cognition moderately predicts and is causally primary to functional outcome 1 year later, whereas psychosocial functioning does not predict later cognitive performance. Subthreshold depressive symptoms concurrently affected functioning at each time of measure.
Our results are compatible with an upward causal effect of cognition on functional outcome in euthymic patients with bipolar disorder. Neuropsychological assessment may help specify individual prognoses. Further studies are warranted to confirm this causal link and evaluate cognitive remediation, before or simultaneously with functional remediation, as an intervention to improve functional outcome.
The electricity sector is in the midst of a structural change driven by new technologies. In Brazil, the electricity sector regulation has mechanisms to foster innovation, including investments in R&D. Recently, the regulatory agency and the industry have been calling for approaches to increase the rate at which R&D departments generate solutions that end up being adopted. As a result, novel approaches to R&D project management have entered the agenda. In this context, the objective of this paper is to characterise Agile Product Development and its application in a highly regulated sector. The paper presents a systematic literature review with the debates about Agile and new product development. Then, a case study exploring an early adoption of the Agile approach in R&D project management in the Brazilian electricity sector is presented. Results include the identification of the Agile features most frequently mentioned in the literature. Moreover, the case study explores the Agile features that were more easily absorbed in early adoption, such as iterative patterns, and discusses implementation challenges in team structure, feedback loops, and communication.
Cognitive deficits are a well-established feature of bipolar disorders (BD), even during periods of euthymia, but risk factors associated with cognitive deficits in euthymic BD are still poorly understood. We aimed to validate classification criteria for the identification of clinically significant cognitive impairment, based on psychometric properties, to estimate the prevalence of neuropsychological deficits in euthymic BD, and identify risk factors for cognitive deficits using a multivariate approach.
We investigated neuropsychological performance in 476 euthymic patients with BD recruited via the French network of BD expert centres. We used a battery of tests, assessing five domains of cognition. Five criteria for the identification of neuropsychological impairment were tested based on their convergent and concurrent validity. Uni- and multivariate logistic regressions between cognitive impairment and several clinical and demographic variables were performed to identify risk factors for neuropsychological impairment in BD.
One cut-off had satisfactory psychometric properties and yielded a prevalence of 12.4% for cognitive deficits in euthymic BD. Antipsychotics use were associated with the presence of a cognitive deficit.
This is the first study to validate a criterion for clinically significant cognitive impairment in BD. We report a lower prevalence of cognitive impairment than previous studies, which may have overestimated its prevalence. Patients with euthymic BD and cognitive impairment may benefit from cognitive remediation.
The relationship between residual depressive symptoms, cognition and functioning in patients with euthymic bipolar disorder is a subject of debate.
To assess whether cognition mediates the association between residual depressive symptoms and functioning in patients with bipolar disorder who were euthymic.
We included 241 adults with euthymic bipolar disorder in a multicentre cross-sectional study. We used a battery of tests to assess six cognition domains. A path analysis was then used to perform a mediation analysis of the relationship between residual depressive symptoms, cognitive components and functioning.
Only verbal and working memory were significantly associated with better functioning. Residual depressive symptoms were associated with poorer functioning. No significant relationship was found between residual depressive symptoms and any cognitive component.
Cognition and residual depressive symptoms appear to be two independent sources of variation in the functioning of people with euthymic bipolar disorder.