Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron \Cal M of $W$ and an acceptable (corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors orthogonal to faces of \Cal M (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by $M$. We show that \goth g has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if \goth g has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $\rho$ are called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on results and ideas. 31 pages, no figures. AMSTe

The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras

We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: A_{1,0} = 2 0 -1 0 2 -2 -1 -2 2 and A_{1,I} = 2 -2 -1 -2 2 -1 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form \chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of \chi_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te

Integrative taxonomic, ecological and genotyping study of charophyte populations from the egyptian western-desert oases and sinai peninsula

Present-day information available on the charophyte macroalgae in Egypt, including their phylogenetic affinities, remains largely incomplete. In this study, nine charophyte populations were collected from different aquatic biotopes across the Egyptian Western-Desert Oases and Sinai Peninsula. All populations were investigated using an integrative polyphasic approach including phylogenetic analyses inferred from the chloroplast-encoded gene (rbcL) and the internal transcribed spacer (ITS1) regions, in parallel with morphotaxonomic assignment, ultrastructure of the oospore walls, and autecology. The specimens identified belonged to the genera Chara, Nitella, and Tolypella, with predominance of the first genus to which five species were assigned though they presented some interesting aberrant taxonomic features: C. aspera, C. contraria, C. globata, C. tomentosa, and C. vulgaris. Based on our integrative study, the globally rare species C. globata was reported for the second time for the whole African continent. The genus Nitella was only represented by N. flagellifera, and based on the available literature, it is a new record for North Africa. Noteworthy, an interesting Tolypella sp., morphologically very similar to T. glomerata, was collected and characterized and finally designated with the working name ‘Tolypella sp. PBA–1704 from a desert, freshwater wetland’, mainly based on its concatenated rbcL+ITS1 phylogenetic position. This study not only improved our understanding on the diversity, biogeography and autecological preferences of charophytes in Egypt, but it also broadened our knowledge on this vulnerable algal group in North Africa, emphasizing the need of more in-depth research work in the future, particularly in the less–impacted desert habitats

The mean meridional circulation and midlatitude ozone buildup

International audienceThe wintertime ozone buildup over the Northern Hemisphere (NH) midlatitudes and its connection with the mean meridional circulation in the stratosphere are examined statistically on a monthly basis from October to March (1980?2002). The ozone buildup begins locally in October with positive total ozone tendencies over the North Pacific, which spread eastward and westward in November and finally cover all midlatitudes in December. The local onset of the buildup in October is not evident in zonal mean ozone tendency, which is close to zero. From November to March, zonal mean total ozone tendency (50°?60° N) shows a strong correlation (|r|=0.7) with several zonal mean parameters associated to the mean meridional circulation, namely: eddy heat flux, temperature tendency, the vertical residual velocity and the residual streamfunction. At the same time, on the latitude-altitude cross section, correlation patterns between ozone tendency and widely used eddy heat flux are not uniform during winter. The strongest correlations are located equatorward (almost throughout the stratosphere) or poleward (only in the lower stratosphere) of the edge of the polar vortex. Such distribution may depend on the existence of the midlatitude and polar waveguides which defined refraction of upward propagating waves from the troposphere either to the midlatitude stratosphere or to the polar stratosphere. As a consequence of the nonuniform correlation patterns, heat flux averaged over the common region 45°?75° N, 100 hPa is not always an optimum proxy for statistical models describing total ozone variability in midlatitudes. Other parameters approximating the strength of the mean meridional circulation have more uniform and stable correlation patterns with ozone tendency during winter. We show that the NH midlatitude ozone buildup has a stable statistical relationship with the mean meridional circulation in all months from October to March and half of the interannual variability in monthly ozone tendencies can be explained by applying different proxies of the mean meridional circulation

Neurophysiophenomenology – predicting emotional arousal from brain arousal in a virtual reality roller coaster

Arousal is a core affect constituted of both bodily and subjective states that prepares an agent to respond to events of the natural environment. While the peripheral physiological components of arousal have been examined also under naturalistic conditions, its neural correlates were suggested mainly on the basis of simplifed experimental designs.   We used virtual reality (VR) to present a highly immersive and contextually rich scenario of roller coaster rides to evoke naturalistic states of emotional arousal. Simultaneously, we recorded EEG to validate the suggested neural correlates of arousal in alpha frequency oscillations (8-12Hz) over temporo-parietal cortical areas. To fnd the complex link between these alpha components and the participants’ continuous subjective reports of arousal, we employed a set of complementary analytical methods coming from machine learning and deep learning