Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi--Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.Comment: 65 pages, 67 reference

Commuting varieties of rr-tuples over Lie algebras

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let \g be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of \g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties C_r(\mathfrak{gl}_2), C_r(\fraksl_2) and $C_r(\N)$ where $\N$ is the nilpotent cone of \fraksl_2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when \g=\fraksl_2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of \fraksl_3, we are able to verify the aforementioned properties for C_r(\fraku). Finally, applying our calculations on the commuting variety C_r(\overline{\calO_{\sub}}) where \overline{\calO_{\sub}} is the closure of the subregular orbit in \fraksl_3, we prove that the nilpotent commuting variety $C_r(\N)$ has singularities of codimension $\ge 2$.Comment: To appear in Journal of Pure and Applied Algebr

On classical finite and affine W-algebras

This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.Comment: 12 page

There is no "Theory of Everything" inside E8

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin's classic papers and are written for mathematicians.Comment: Final version, to appear in Communications in Mathematical Physics. The main difference, from the previous version, is the creation of a new section, containing a response to Lisi's objections to our wor

Autobiography as unconventional history: Constructing the author

The experience of historians as autobiographers has led them to reconsider the nature of historical knowledge and the function of the historian as an intermediary between the past and present. In the new theoretical context of the social sciences and historiography, we can take this proposal further and consider autobiography as a valid form of history—or, at least, as ‘unconventional history’, understood as negotiations with history that transcend or subvert traditional chronological monographs, posit the ‘subjective’ as a useful form of knowledge, and engage the constructed nature of the text. Taking this hypothesis as a starting point, this article reads historians' autobiographical texts to explore if we can/should continue to defend the classic distinction between subject and object, historian scientist and historian author. In this article I compare the work of several historian autobiographers that permit us to identify different methodologies in approaching the story of the self that also reflects different theoretical conceptions of history. I argue that historians that may be considered ‘constructionist’, such as Fernand Braudel, Annie Kriegel, George Duby, and Eric Hobsbawm, design their autobiographies in the same way they articulate their historical texts: by foregrounding objectivity and establishing critical distance between the subject—the historian who narrates the story—and the object—one's own life. Unconventional or experimental approaches, such as those espoused by Robert Rosenstone, Dominick LaCapra, or Clifford Geertz, result in more self-conscious autobiographies, which are, paradoxically, often more realistic and more revealing of the epistemological nature of life writing. ----------------- La experiencia de los historiadores como autobiógrafos les ha llevado a reconsiderar la naturaleza del conocimiento histórico y la función del historiador como un intermediario entre el pasado y el presente. En el nuevo contexto teórico de las ciencias sociales y la historiografía podemos tomar esta propuesta más allá y considerar la autobiografía como una forma válida de historia-o, al menos, de historia ‘poco convencional’-, entendida como negociaciones con la historia que trascienden o subvierten las tradicionales monografías cronológicas, plantean lo "subjetivo" como una forma útil de conocimiento y participan de la naturaleza construida del texto. Tomando esta hipótesis como punto de partida, este artículo lee los textos autobiográficos de los historiadores para explorar si se puede / debe seguir defendiendo la clásica distinción entre sujeto y objeto, historiador científico e historiador escritor. En este artículo comparo el trabajo de varios historiadores autobiógrafos que nos permiten identificar las diferentes metodologías para acercarse a la historia del yo y que también reflejan las diferentes concepciones teóricas de la historia. Sostengo que los historiadores que pueden considerarse "constructivistas", como Fernand Braudel, Annie Kriegel, George Duby y Eric Hobsbawm, diseñan sus autobiografías de la misma forma que articulan sus textos históricos: poniendo en primer plano la objetividad y estableciendo una distancia crítica entre el sujeto -el historiador que narra la historia-y el objeto- la vida de cada uno. Enfoques no convencionales o experimentales, como los expuestos por Robert Rosenstone, Dominick LaCapra, o Clifford Geertz, resultan autobiografías más autoconscientes, que son, paradójicamente, a menudo más realistas y más reveladoras de la naturaleza epistemológica de la escritura de la vida

Nanoscale synchrotron X-ray speciation of iron and calcium compounds in amyloid plaque cores from Alzheimer’s disease subjects

Altered metabolism of biometals in the brain is a key feature of Alzheimer's disease, and biometal interactions with amyloid-β are linked to amyloid plaque formation. Iron-rich aggregates, including evidence for the mixed-valence iron oxide magnetite, are associated with amyloid plaques. To test the hypothesis that increased chemical reduction of iron, as observed in vitro in the presence of aggregating amyloid-β, may occur at sites of amyloid plaque formation in the human brain, the nanoscale distribution and physicochemical states of biometals, particularly iron, were characterised in isolated amyloid plaque cores from human Alzheimer's disease cases using synchrotron x-ray spectromicroscopy. In situ x-ray magnetic circular dichroism revealed the presence of magnetite: a finding supported by ptychographic observation of an iron oxide crystal with the morphology of biogenic magnetite. The exceptional sensitivity and specificity of x-ray spectromicroscopy, combining chemical and magnetic probes, allowed enhanced differentiation of the iron oxides phases present. This facilitated the discovery and speciation of ferrous-rich phases and lower oxidation state phases resembling zero-valent iron as well as magnetite. Sequestered calcium was discovered in two distinct mineral forms suggesting a dynamic process of amyloid plaque calcification in vivo. The range of iron oxidation states present and the direct observation of biogenic magnetite, provides unparalleled support for the hypothesis that chemical reduction of iron arises in conjunction with the formation of amyloid plaques. These new findings raise challenging questions about the relative impacts of amyloid-β aggregation, plaque formation, and disrupted metal homeostasis on the oxidative burden observed in Alzheimer's disease

Changing Agendas on Sleep, Treatment and Learning in Epilepsy (CASTLE) Sleep-E: A protocol for a randomised controlled trial comparing an online behavioural sleep intervention with standard care in children with Rolandic epilepsy

This is the final version. Available from BMJ Publishing Group via the DOI in this record. INTRODUCTION: Sleep and epilepsy have an established bidirectional relationship yet only one randomised controlled clinical trial has assessed the effectiveness of behavioural sleep interventions for children with epilepsy. The intervention was successful, but was delivered via face-to-face educational sessions with parents, which are costly and non-scalable to population level. The Changing Agendas on Sleep, Treatment and Learning in Epilepsy (CASTLE) Sleep-E trial addresses this problem by comparing clinical and cost-effectiveness in children with Rolandic epilepsy between standard care (SC) and SC augmented with a novel, tailored parent-led CASTLE Online Sleep Intervention (COSI) that incorporates evidence-based behavioural components. METHODS AND ANALYSES: CASTLE Sleep-E is a UK-based, multicentre, open-label, active concurrent control, randomised, parallel-group, pragmatic superiority trial. A total of 110 children with Rolandic epilepsy will be recruited in outpatient clinics and allocated 1:1 to SC or SC augmented with COSI (SC+COSI). Primary clinical outcome is parent-reported sleep problem score (Children's Sleep Habits Questionnaire). Primary health economic outcome is the incremental cost-effectiveness ratio (National Health Service and Personal Social Services perspective, Child Health Utility 9D Instrument). Parents and children (≥7 years) can opt into qualitative interviews and activities to share their experiences and perceptions of trial participation and managing sleep with Rolandic epilepsy. ETHICS AND DISSEMINATION: The CASTLE Sleep-E protocol was approved by the Health Research Authority East Midlands (HRA)-Nottingham 1 Research Ethics Committee (reference: 21/EM/0205). Trial results will be disseminated to scientific audiences, families, professional groups, managers, commissioners and policymakers. Pseudo-anonymised individual patient data will be made available after dissemination on reasonable request. TRIAL REGISTRATION NUMBER: ISRCTN13202325.National Institute for Health and Care Research (NIHR)National Health and Medical Research Council (NHMRC, Australia)Victorian Government’s Operational Infrastructure Support Progra

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

Changes in the Policy Environment For Infant and Young Child feeding in Vietnam, Bangladesh, and Ethiopia, and the Role of Targeted Advocacy

There is limited literature examining shifts in policy environments for nutrition and infant and young child feeding (IYCF) over time, and on the potential contribution of targeted advocacy to improved policy environments in low- and middle-income countries. This study tracked changes in the policy environment over a four-year period in three countries, and examined the role of targeted nutrition and IYCF advocacy strategies by a global initiative