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

Is sensor space analysis good enough? Spatial patterns as a tool for assessing spatial mixing of EEG/MEG rhythms

Analyzing non-invasive recordings of electroencephalography (EEG) and magnetoencephalography (MEG) directly in sensor space, using the signal from individual sensors, is a convenient and standard way of working with this type of data. However, volume conduction introduces considerable challenges for sensor space analysis. While the general idea of signal mixing due to volume conduction in EEG/MEG is recognized, the implications have not yet been clearly exemplified. Here, we illustrate how different types of activity overlap on the level of individual sensors. We show spatial mixing in the context of alpha rhythms, which are known to have generators in different areas of the brain. Using simulations with a realistic 3D head model and lead field and data analysis of a large resting-state EEG dataset, we show that electrode signals can be differentially affected by spatial mixing by computing a sensor complexity measure. While prominent occipital alpha rhythms result in less heterogeneous spatial mixing on posterior electrodes, central electrodes show a diversity of rhythms present. This makes the individual contributions, such as the sensorimotor mu-rhythm and temporal alpha rhythms, hard to disentangle from the dominant occipital alpha. Additionally, we show how strong occipital rhythms can contribute the majority of activity to frontal channels, potentially compromising analyses that are solely conducted in sensor space. We also outline specific consequences of signal mixing for frequently used assessment of power, power ratios and connectivity profiles in basic research and for neurofeedback application. With this work, we hope to illustrate the effects of volume conduction in a concrete way, such that the provided practical illustrations may be of use to EEG researchers to in order to evaluate whether sensor space is an appropriate choice for their topic of investigation

Research of interconnected wind turbines with intelligent control

Development of renewable energy sources in Kazakhstan, including wind energy, brings the needs to design and develop control and integration to a power system. The aim of the Project is to design a power converter supplied by intelligent controllers that provide a stable 3-phase output voltage in the presence of uncertain fluctuations of a wind speed

Kahlerian K3 surfaces and Niemeier lattices

Using results (especially see Remark 1.14.7) of our paper "Integral symmetric bilinear forms and some of their applications", 1979, we clarify relation between Kahlerian K3 surfaces and Niemeier lattices. We want to emphasise that all twenty four Niemeier lattices are important for K3 surfaces, not only the one which is related to the Mathieu group.Comment: Var7: 88 pages. We added last case

Predicting the effects of climate change on Schistosoma mansoni transmission in eastern Africa

Background: Survival and fitness attributes of free-living and sporocyst schistosome life-stages and their intermediate host snails are sensitive to water temperature. Climate change may alter the geographical distribution of schistosomiasis by affecting the suitability of freshwater bodies for hosting parasite and snail populations. Methods: We have developed an agent-based model of the temperature-sensitive stages of the Schistosoma mansoni and intermediate host snail lifecycles. The model was run using low, moderate and high warming climate projections over eastern Africa. For each climate projection, eight model scenarios were used to determine the sensitivity of predictions to different relationships between air and water temperature, and different snail mortality rates. Maps were produced showing predicted changes in risk as a result of increasing temperatures over the next 20 and 50 years. Results Baseline model output compared to prevalence data indicates suitable temperatures are necessary but not sufficient for both S. mansoni transmission and high infection prevalences. All else being equal, infection risk may increase by up to 20% over most of eastern Africa over the next 20 and 50 years. Increases may be higher in Rwanda, Burundi, south-west Kenya and eastern Zambia, and S. mansoni may become newly endemic in some areas. Results for 20-year projections are robust to changes in simulated intermediate host snail habitat conditions. There is greater uncertainty about the effects of different habitats on changes in risk in 50 years’ time. Conclusions: Temperatures are likely to become suitable for increased S. mansoni transmission over much of eastern Africa. This may reduce the impact of control and elimination programmes. S. mansoni may also spread to new areas outside existing control programmes. We call for increased surveillance in areas defined as potentially suitable for emergent transmission

Research of interconnected wind turbines with intelligent control

Development of renewable energy sources in Kazakhstan, including wind energy, brings the needs to design and develop control and integration to a power system. The aim of the Project is to design a power converter supplied by intelligent controllers that provide a stable 3-phase output voltage in the presence of uncertain fluctuations of a wind speed

The Geometry and Moduli of K3 Surfaces

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.Comment: 34 pages, 2 figures. (R. Laza, M. Schutt and N. Yui, eds.

Interferometry of hyper-Rayleigh scattering by inhomogeneous thin films

The use of specific symmetry properties of the optical second-harmonic generation (the s,s-exclusion rule) has allowed us to observe high-contrast hyper-Rayleigh interference patterns in a completely diffuse light - an effect having no analog in case of linear (Rayleigh) scattering.Comment: 8 pages, 3 figure

Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with respect to +2 and -2 classes. As a consequence, we get counter examples to a version of the weight 2 Torelli question, when n-1 is not a prime power.Comment: Version 3: Latex, 54 pages. Expository change