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

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

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

A Flexible Parametric Modelling Framework for Survival Analysis

We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (cure models). This generality is achieved using four distributional parameters: two scale-type parameters – which, respectively, relate to accelerated failure time (AFT) and proportional hazards (PH) modelling – and two shape parameters. Furthermore, we advocate “multi-parameter regression” whereby more than one distributional parameter depends on covariates – rather than the usual convention of having a single covariate-dependent (scale) parameter. This general formulation unifies the most popular survival models, allowing us to consider the practical value of possible modelling choices. In particular, we suggest introducing covariates through just one or other of the two scale parameters (covering AFT and PH models), and through a “power” shape parameter (covering more complex non-AFT/non-PH effects); the other shape parameter remains covariate-independent, and handles automatic selection of the baseline distribution. We explore inferential issues and compare with alternative models through various simulation studies, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework using data from lung cancer, melanoma, and kidney function studies. Censoring is accommodated throughout

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 , #

M-Theory on (K3 X S^1)/Z_2

We analyze $M$-theory compactified on $(K3\times S^1)/Z_2$ where the $Z_2$ changes the sign of the three form gauge field, acts on $S^1$ as a parity transformation and on K3 as an involution with eight fixed points preserving SU(2) holonomy. At a generic point in the moduli space the resulting theory has as its low energy limit N=1 supergravity theory in six dimensions with eight vector, nine tensor and twenty hypermultiplets. The gauge symmetry can be enhanced (e.g. to $E_8$) at special points in the moduli space. At other special points in the moduli space tensionless strings appear in the theory.Comment: LaTeX file, 11 page

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah-Shiga, Dolgachev and Nagura-Sugiyama.Comment: 29 pages, 3 figure

Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries

At special loci in their moduli spaces, Calabi-Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi-Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez-Villegas, we find a calculationally tractable means of finding the Picard-Fuchs equations satisfied by the periods of all 3-forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self-contained.Comment: 54 pages, 3 figure