Normalizers of maximal tori and real forms of Lie groups

For a complex reductive Lie group $G$ Tits defined an extension $W_G^T$ of the corresponding Weyl group $W_G$. The extended group is supplied with an embedding into the normalizer $N_G(H)$ of the maximal torus $H\subset G$ such that $W_G^T$ together with $H$ generate $N_G(H)$. We give an interpretation of the Tits classical construction in terms of the maximal split real form $G(\mathbb{R})\subset G(\mathbb{C})$, leading to a simple topological description of $W^T_G$. We also propose a different extension $W_G^U$ of the Weyl group $W_G$ associated with the compact real form $U\subset G(\mathbb{C})$. This results into a presentation of the normalizer of maximal torus of the group extension $U\ltimes {\rm Gal}(\mathbb{C}/\mathbb{R})$ by the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. We also describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.Comment: 17 page

Особенности современной семьи: модель взаимоотношений в России и мире

Modern society is currently undergoing the stage of transition. Such a change has an impact on all social institutions, including the family and family-marital relations. People are becoming increasingly liberated and independent. This affects marital relations, which are currently being built according to new paradigms associated with greater responsibility for oneself and less for the partner. All these are new phenomena of our social reality, requiring a new understanding and development of new social practice. To validly disclose the features of the modern model of family relations, we will build our considerations in line with evolutionary, functional, empirical and interactionist approaches, based on the assertion that the family is, first of all, a small social group, where each partner has their own, often opposing, interests, and which at the same time acts as an integral social system.Современное общество в настоящее время переживает переходный этап. Такое изменение оказывает влияние на все социальные институты, включая семью и семейно-брачные отношения. Люди становятся все более раскрепощенными и независимыми. Это сказывается на супружеских отношениях, которые в настоящее время строятся в соответствии с новыми парадигмами, связанными с большей ответственностью за себя и меньшей степени за партнера. Все это - новые явления нашей социальной реальности, требующие нового понимания и развития новой социальной практики. Чтобы достоверно раскрыть особенности современной модели семейных отношений, мы строим наши рассуждения в русле эволюционного, функционального, эмпирического и интеракционистского подходов, основанных на утверждении, что семья — это прежде всего небольшая социальная группа, где каждый партнер имеет свои, часто противоположные, интересы и которая в то же время выступает как целостная социальная система

Comment on ``Damping of energetic gluons and quarks in high-temperature QCD''

Burgess and Marini have recently pointed out that the leading contribution to the damping rate of energetic gluons and quarks in the QCD plasma, given by $\gamma=c g^2\ln(1/g)T$, can be obtained by simple arguments obviating the need of a fully resummed perturbation theory as developed by Braaten and Pisarski. Their calculation confirmed previous results of Braaten and Pisarski, but contradicted those proposed by Lebedev and Smilga. While agreeing with the general considerations made by Burgess and Marini, I correct their actual calculation of the damping rates, which is based on a wrong expression for the static limit of the resummed gluon propagator. The effect of this, however, turns out to be cancelled fortuitously by another mistake, so as to leave all of their conclusions unchanged. I also verify the gauge independence of the results, which in the corrected calculation arises in a less obvious manner.Comment: 5 page

Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group G with a maximal torus H⊂G, Tits defined an extension WTG of the corresponding Weyl group WG. The extended group is supplied with an embedding into the normalizer NG(H) such that WTG together with H generate NG(H). In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form G(R)⊂G, which leads to a simple topological description of WTG. We also consider a variation of the Tits construction associated with compact real form U of G. In this case we define an extension WUG of the Weyl group WG, naturally embedded into the group extension U˜:=U⋊Γ of the compact real form U by the Galois group Γ=Gal(C/R). Generators of WUG are squared to identity as in the Weyl group WG. However, the non-trivial action of Γ by outer automorphisms requires WUG to be a non-trivial extension of WG. This gives a specific presentation of the maximal torus normalizer of the group extension U˜. Finally, we describe explicitly the adjoint action of WTG and WUG on the Lie algebra of G

On q-deformed gl(l+1)-Whittaker function II

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For \ell=1, this provides an interpretation of non-specialized q-deformed gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Gamma-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.Comment: Extended version submitted in Comm. Math. Phys., 24 page

The application process of the Ornstein-Ulenbek to the formation of cavitation bubbles

In this paper we propose the process of formation of cavitation bubbles in the pilot valve, considered as the steady and homogeneous Markov process

LINE-1 retrotransposon methylation in chorionic villi of first trimester miscarriages with aneuploidy

Purpose High frequency of aneuploidy in meiosis and cleavage stage coincides with waves of epigenetic genome reprogramming that may indicate a possible association between epigenetic mechanisms and aneuploidy occurrence. This study aimed to assess the methylation level of the long interspersed repeat element 1 (LINE-1) retrotransposon in chorionic villi of first trimester miscarriages with a normal karyotype and aneuploidy. Methods The methylation level was assessed at 19 LINE-1 promoter CpG sites in chorionic villi of 141 miscarriages with trisomy of chromosomes 2, 6, 8-10, 13-15, 16, 18, 20-22, and monosomy X using massive parallel sequencing. Results The LINE-1 methylation level was elevated statistically significant in chorionic villi of miscarriages with both trisomy (45.2 +/- 4.3%) and monosomy X (46.9 +/- 4.2%) compared with that in induced abortions (40.0 +/- 2.4%) (p < 0.00001). The LINE-1 methylation levels were specific for miscarriages with different aneuploidies and significantly increased in miscarriages with trisomies 8, 14, and 18 and monosomy X (p < 0.05). The LINE-1 methylation level increased with gestational age both for group of miscarriages regardless of karyotype (R = 0.21, p = 0.012) and specifically for miscarriages with trisomy 16 (R = 0.48, p = 0.007). LINE-1 methylation decreased with maternal age in miscarriages with a normal karyotype (R = - 0.31, p = 0.029) and with trisomy 21 (R = - 0.64, p = 0.024) and increased with paternal age for miscarriages with trisomy 16 (R = 0.38, p = 0.048) and monosomy X (R = 0.73, p = 0.003). Conclusion Our results indicate that the pathogenic effects of aneuploidy in human embryogenesis can be supplemented with significant epigenetic changes in the repetitive sequences

Letter of interest for a neutrino beam from Protvino to KM3NeT/ORCA

The Protvino accelerator facility located in the Moscow region, Russia, is in a good position to offer a rich experimental research program in the field of neutrino physics. Of particular interest is the possibility to direct a neutrino beam from Protvino towards the KM3NeT/ORCA detector, which is currently under construction in the Mediterranean Sea 40 km offshore Toulon, France. This proposal is known as P2O. Thanks to its baseline of 2595 km, this experiment would yield an unparalleled sensitivity to matter effects in the Earth, allowing for the determination of the neutrino mass ordering with a high level of certainty after only a few years of running at a modest beam intensity of ≈ 90 kW. With a prolonged exposure (≈1500 kWyear), a 2σ sensitivity to the leptonic CP-violating Dirac phase can be achieved. A second stage of the experiment, comprising a further intensity upgrade of the accelerator complex and a densified version of the ORCA detector (Super-ORCA), would allow for up to a 6σ sensitivity to CP violation and a 10º−17º resolution on the CP phase after 10 years of running with a 450 kW beam, competitive with other planned experiments. The initial composition and energy spectrum of the neutrino beam would need to be monitored by a near detector, to be constructed several hundred meters downstream from the proton beam target. The same neutrino beam and near detector set-up would also allow for neutrino-nucleus cross section measurements to be performed. A short-baseline sterile neutrino search experiment would also be possible