Resolving G-torsors by abelian base extensions

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H^1(K, S) -> H^1(K, G) is surjective for every field extension K/k. We give several applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. In particular, we prove that for every z in H^1(K, G) there exists an abelian field extension L/K such that z_L \in H^1(L, G) is represented by a G-torsor over a projective variety. From this we deduce that z_L has trivial point obstruction. We also show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension =< 1. The last assertion, if true, would prove conjectures of Bogomolov and Koenigsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E_8.Comment: New material added on the no-name lemma in Section 4 and on Hilbert's 13th problem in Section 9. A mistake in the proof of Proposition 2.3 is correcte

Three-point Lie algebras and Grothendieck's dessins d'enfants

We define and classify the analogues of the affine Kac-Moody Lie algebras for the ring corresponding to the complex projective line minus three points. The classification is given in terms of Grothendieck's dessins d'enfants. We also study the question of conjugacy of Cartan subalgebras for these algebras.Comment: 16 page

A cohomological proof of Peterson-Kac's theorem of conjugacy of Cartan subalgebras of affine Kac-Moody Lie algebras

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings.Fil: Chernousov, V.. University of Alberta; CanadáFil: Gille, P.. Centre National de la Recherche Scientifique. Ecole Normale Supérieure; FranciaFil: Pianzola, Arturo. University of Alberta; Canadá. Universidad Centro de Altos Estudios en Ciencias Exactas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Yahorau, U.. University of Alberta; Canad

DETERMINATION OF DEFORMABILITY PARAMETERS OF CONCRETE SAMPLES BY THE FORMULAS OF FRACTURE MECHANICS

To determine the deformability parameters of concrete samples by the formulas of fracture mechanics, equilibrium tests were carried out at the stage of local deformation of the sample, which showed the correspondence of the change in external forces to the internal forces of the material resistance with the corresponding static development of the main crack. For the same purpose, the samples are tested for bending with an initial notch and the "load-deflection" diagram is recorded. In this work, we presented a test scheme for a specimen with a notch (crack) and constructed a diagram of the deformation of a specimen under bending "load-deflection". Based on it, it is possible to predict the destruction of the material, that is, to determine the value of the load at which the limit value of deflection or the displacement of the outer edges of the notch (opening the throat of the crack on the lower surface of the specimen) can be taken as the moment of loss of the resource of the material. Also, we examined the deformation of a concrete sample during three-point bending and presented a diagram of the deformation of a concrete sample within the plastic zone. Dependencies were derived for determining the ultimate relative strains under tension and bending. Based on the results obtained, the state diagrams of the stretched concrete and the deformation scheme of the normal section of the concrete sample were constructed. As a result, the conclusion and convergence of the results

Conjugacy theorems for loop reductive group schemes and Lie algebras

The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildingsComment: Publi\'e dans Bulletin of Mathematical Sciences 4 (2014), 281-32

Isotropy of unitary involutions

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.Comment: final version, to appear in Acta Mat

ПЭТ-КТ в диагностике и дифференциальной диагностике адренокортикального рака.

The objective of the study was to develop criteria for the early diagnosis and differential diagnosis of adrenocortical cancer (ACC) during positron emission tomography combined with computed tomography (PET / CT) with 18F-fluorodeoxyglucose (18F-FDG).Materials and methods. Diagnostic tests were performed on a General Electric the Discovery PET / CT 610. After scanning 60–90 minutes after the administration of the radiopharmaceutical, PET-grams were formed showing the distribution of the radiopharmaceutical throughout the body in both physiological and pathological areas. Using software, automatic "fusion" of images obtained on PET and CT scanners was carried out. Initially, a radiologist performed an analysis of structural pathological changes in organs, followed by an assessment of the level of accumulation and determination of CT densitometry of pathological adrenal formations. Subsequently, metabolic activity was determined by the level of SUV (standart uptake value).Results. A retrospective analysis of PET / CT data with 18F-FDG was performed in 50 patients with adrenal tumor formations. Based on the data studied, all patients were divided into 2 groups: 1st group (n = 21) with a subsequent diagnosis of ACC, 2nd group (n = 29) with benign adrenal neoplasms. The age of the patients was 68 (32–76) years and did not statistically differ in both groups (p ˃0.05). After check up tumor parameters the following results were obtained: the average tumor size in patients of the 1st group was 4.5 Ѓ} 0.75 cm, of the 2nd group – 4.9 Ѓ} 1.1 cm, no significant difference was found (p ˃0.05). The diagnostic model obtained by constructing and subsequent analysis of the ROC curve of the SUV level in patients with malignant and benign neoplasms of the adrenal glands showed a high quality model with a sesitivity of 90 %, specificity of 95 % (area under the ROC curve (AUC) 0.93 at p &lt;0.0001 (z = 27.37)) and a cut-off point of 3. When studying CT densitometric parameters of a tumor in patients of the 1st and 2nd groups, a direct correlation relationship between these parameters and the diagnosis of malignant lesions was also obtained (Rs = 0.67; coefficient τ Kendall 0.64; p = 0.001). Interpretation of the results of the ROC analysis of the tumor density level (CT density in Hounsfield units (HU)) in patients of the 1st and 2nd groups showed 80 % sensitivity and 90 % specificity (AUC 0.89; p &lt;0.0001) with clipping point equal to 10 HU.Conclusion. Obtaining integral information on the SUV level together with the measurement of CT density during PET / CT with 18F-FDG is one of the most modern diagnostic methods that allow both diagnostics and differential diagnosis of ACC in the early stages of the pathological process. The above factors currently indicate the high diagnostic value of PET / CT with 18F-FDG and the priority nature of this study in cases of suspected ACC in patients with adrenal neoplasms of 1–5 cm in size.Цель исследования – разработка критериев ранней и дифференциальной диагностики адренокортикального рака (АКР) при проведении позитронной эмиссионной томографии, совмещенной с компьютерной томографией (ПЭТ / КТ), с 18F-фтордезоксиглюкозой (18F-ФДГ).Материалы и методы. Диагностические исследования выполнены на аппарате General Electric the Discovery PET / CT 610. После проведения сканирования и введения радиофармацевтического препарата (РФП) через 60–90 мин сформированы изображения ПЭТ, показывающие распределение РФП во всем теле как в физиологических, так и в патологических зонах. С помощью программного обеспечения проводилось автоматическое «слияние» снимков, полученных на ПЭТи КТ-сканерах. Первоначально врачом-рентгенологом выполнялся анализ структурных патологических изменений в органах с последующей оценкой уровня накопления РФП и определением КТ-денситометрии образований надпочечников. Метаболическую активность опухолей надпочечников определяли по уровню SUV (стандартизированный показатель накопления РФП).Результаты. Проведен ретроспективный анализ данных ПЭТ / КТ с 18F-ФДГ у 50 пациентов с опухолевыми образованиями надпочечников. На основании изученных данных пациенты были разделены на 2 группы: 1‑я группа (n = 21) с диагностированным в последующем АКР, 2‑я группа (n = 29) с доброкачественными новообразованиями надпочечников. Возраст пациентов составил 68 (32–76) лет и статистически не различался в обеих группах (р &gt;0,05). При изучении параметров образования получены следующие результаты: средний размер опухоли у пациентов 1‑й группы составил 4,5 Ѓ} 0,75 см, 2‑й группы – 4,9 Ѓ} 1,1 см, при этом достоверной разницы между группами не выявлено (р &gt;0,05). Диагностическая модель, полученная при построении и последующем анализе ROC-кривой уровня SUV у пациентов со злокачественными и доброкачественными новообразованиями надпочечников, показала высокое качество с чувствительностью метода 90 %, специфичностью 95 % (площадь под ROC-кривой (AUC) 0,93 при p &lt;0,0001 (z = 27,37)) и точкой отсечения, равной 3. При изучении КТ-денситометрических показателей опухоли надпочечников у пациентов 1‑й и 2‑й групп получена прямая корреляционная связь данных параметров с диагностикой злокачественного поражения (Rs = 0,67; коэффициент τ Кендалла 0,64; р = 0,001). Интерпретация результатов ROC-анализа уровня плотности опухоли (КТ-плотность в единицах Хаунсфилда (HU)) у пациентов 1‑й и 2‑й групп показала 80 % чувствительность и 90 % специфичность (AUC 0,89; p &lt;0,0001) с точкой отсечения, равной 10 HU.Заключение. Получение интегральной информации по уровню SUV вместе с измерением КТ-плотности опухоли при проведении ПЭТ / КТ с 18F-ФДГ является одним из наиболее современных диагностических методов, позволяющих проводить дифференциальную диагностику патологического процесса в надпочечниках и на ранних этапах обследования выявлять АКР. Перечисленные выше факторы в настоящее время указывают на высокую диагностическую ценность ПЭТ / КТ с 18F-ФДГ и приоритетный характер выполнения данного исследования при подозрении на АКР у пациентов с новообразованиями надпочечников размером 1–5 см