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

Research of thermal deformation of a kinematic wave reducerwith a modified tooth profile during the work in low temperature conditions

In the conditions of the Extreme North working resource of mechanicaltools and machineelements is reduced because of bad weather conditions in this region. At a low temperature materials are exposed to deformation which is capable to break operability of the mechanism. In connection with the high requirements to the accuracy of a kinematic wave reducer, it is necessary to conduct a research for the purpose of comparison of value of thermal deformation and the appointed admission on a reducer detail. If value of thermal deformation is more admission, then it can lead to jamming of the mechanism. The research was conducted for a collected reducer and separately for not loaded driver gear

Research of the load distribution in the wave kinematic reducer with a modified tooth profile and dependence of the load abilities in proportion to its gear ratio and overall dimensions

Nowadays, there are many types of reducers based on work of gear trains, which transfer torque. The most popular reducers are with such type of gearing as an involute gear, a worm drive and an eccentrically cycloid gear. A new type of the reducer will be represented in this work. It is a wave reducer with the modified profile of the tooth close to the profile of the tooth of Novikov gearing. So such reducers can be widely used in drives of difficult technical mechanisms, for example, in mechatronics, robotics and in drives of exact positioning. In addition, the distribution of loading in gearing of teeth of a reducer was analyzed in this paper. It proves that the modified profile of the tooth allows distributing loading to several teeth in gearing. As a result, an admissible loading ability of a reducer becomes higher. The aim of the research is to define a possibility to reduce overall dimensions of a reducer without changing the gear ratio or to increase the gear ratio without changing overall dimensions. So, the result of this work will be used in further research

Formation of conserved charge at the de Sitter space

The article considers a new mechanism of charge accumulation in the early Universe in theories with compact extra dimensions. The relaxation processes in the extra space metric that take place during its formation lead to the establishment of symmetrical extra space configuration. As a result, the initial accumulation of the number associated with the symmetry occurs. We demonstrate this mechanism using a simple example of a two-dimensional apple-like extra space metric with $U(1)$-symmetry. The conceptual idea of the mechanism can be used to develop a model for the production of the baryon or lepton number in the early Universe.Comment: 9 pages, 3 figure

On Type IIB Vacua With Varying Coupling Constant

We describe type IIB compactifications with varying coupling constant in d=6,7,8,9 dimensions, where part of the ten-dimensional SL(2,Z) symmetry is broken by a background with Gamma_0(n) or Gamma(n) monodromy for n=2,3,4. This extends the known class of F-theory vacua to theories which are dual to heterotic compactifications with reduced rank. On compactifying on a further torus, we obtain a description of the heterotic moduli space of G bundles over elliptically fibered manifolds without vector structure in terms of complex geometries.Comment: 32 pages, 5 eps figure

Generation of Plasma in Low-Pressure Discharge

A hydrodynamic model of plasma has been developed, which takes into account both secondary and primary electrons. It has been shown that a solution with a plasma potential higher than the anode potential is possible if the ionization frequency is higher than some critical value. At lower ionization frequencies, it is possible to obtain a solution with a plasma potential below the anode potential. © 2021 Institute of Physics Publishing. All rights reserved

Properties and Structural Transformations of Mgb2-tapes under the Action of Plasma Shock Waves

The current-carrying properties (

K3-fibered Calabi-Yau threefolds I, the twist map

A construction of Calabi-Yaus as quotients of products of lower-dimensional spaces in the context of weighted hypersurfaces is discussed, including desingularisation. The construction leads to Calabi-Yaus which have a fiber structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus are of some interest in connection with Type II -heterotic string dualities in dimension 4. A section at the end of the paper summarises this for the non-expert mathematician.Comment: 31 pages LaTeX, 11pt, 2 figures. To appear in International Journal of Mathematics. On the web at http://personal-homepages.mis.mpg.de/bhunt/preprints.html , #

Mapping of multiple muscles with transcranial magnetic stimulation: Absolute and relative test-retest reliability

The spatial accuracy of transcranial magnetic stimulation (TMS) may be as small as a few millimeters. Despite such great potential, navigated TMS (nTMS) mapping is still underused for the assessment of motor plasticity, particularly in clinical settings. Here, we investigate the within‐limb somatotopy gradient as well as absolute and relative reliability of three hand muscle cortical representations (MCRs) using a comprehensive grid‐based sulcus‐informed nTMS motor mapping. We enrolled 22 young healthy male volunteers. Two nTMS mapping sessions were separated by 5–10 days. Motor evoked potentials were obtained from abductor pollicis brevis (APB), abductor digiti minimi, and extensor digitorum communis. In addition to individual MRI‐based analysis, we studied normalized MNI MCRs. For the reliability assessment, we calculated intraclass correlation and the smallest detectable change. Our results revealed a somatotopy gradient reflected by APB MCR having the most lateral location. Reliability analysis showed that the commonly used metrics of MCRs, such as areas, volumes, centers of gravity (COGs), and hotspots had a high relative and low absolute reliability for all three muscles. For within‐limb TMS somatotopy, the most common metrics such as the shifts between MCR COGs and hotspots had poor relative reliability. However, overlaps between different muscle MCRs were highly reliable. We, thus, provide novel evidence that inter‐muscle MCR interaction can be reliably traced using MCR overlaps while shifts between the COGs and hotspots of different MCRs are not suitable for this purpose. Our results have implications for the interpretation of nTMS motor mapping results in healthy subjects and patients with neurological conditions

Differential effects of anodal and dual tDCS on sensorimotor functions in chronic hemiparetic stroke patients

Background and purpose Previous tDCS studies in chronic stroke patients reported highly inconsistent effects on sensorimotor functions. Underlying reasons could be the selection of different kinematic parameters across studies and for different tDCS setups. We reasoned that tDCS may not simply induce global changes in a beneficial-adverse dichotomy, but rather that different sensorimotor kinematics are differentially affected. Furthermore, the often-postulated higher efficacy of bilateral-dual (bi-tDCS) over unilateral-anodal (ua-tDCS) could not yet be demonstrated consistently either. We investigated the effects of both setups on a wider range of kinematic parameters from standardized robotic tasks in patients with chronic stroke. Methods Twenty-four patients with arm hemiparesis received tDCS (20min, 1 mA) concurrent to kinematic assessments in a sham-controlled, cross-over and double-blind clinical trial. Performance was measured on four sensorimotor tasks (reaching, proprioception, cooperative and independent bimanual coordination) from which 30 parameters were extracted. On the group-level, the patterns of changes relative to sham were assessed using paired-samples t-tests and classified as (1) performance increases, (2) decreases and (3) non-significant differences. Correlations between parametric change scores were calculated for each task to assess effects on the individual-level. Results Both setups induced complex effect patterns with varying proportions of performance increases and decreases. On the group-level, more increases were induced in the reaching and coordination tasks while proprioception and bimanual cooperation were overall negatively affected. Bi-tDCS induced more performance increases and less decreases compared to ua-tDCS. Changes across parameters occurred more homogeneously under bi-tDCS than ua-tDCS, which induced a larger proportion of performance trade-offs. Conclusions Our data demonstrate profound tDCS effects on sensorimotor functions post-stroke, lending support for more pronounced and favorable effects of bi-tDCS compared to ua-tDCS. However, no uniformly beneficial pattern was identified. Instead, the modulations varied depending on the task and electrode setup, with increases in certain parameters occurring at the expense of decreases in others