The contact polytope of the Leech lattice

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikiric)

The main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to different toroidal compactifications of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactification, but we also apply this to other toroidal compactifications, in particular the perfect cone compactification, for which we obtain a combinatorial characterization of the indeterminacy locus, as well as a geometric description up to codimension six, and an explicit toroidal resolution of the Prym map up to codimension four. © 2017 European Mathematical Society

Dicationic Alkylammonium Bromide Gemini Surfactants. Membrane Perturbation and Skin Irritation

Dicationic alkylammonium bromide gemini surfactants represent a class of amphiphiles potentially effective as skin permeation enhancers. However, only a limited number of studies has been dedicated to the evaluation of the respective cytotoxicity, and none directed to skin irritation endpoints. Supported on a cell viability study, the cytotoxicity of gemini surfactants of variable tail and spacer length was assessed. For this purpose, keratinocyte cells from human skin (NCTC 2544 cell line), frequently used as a model for skin irritation, were employed. The impact of the different gemini surfactants on the permeability and morphology of model vesicles was additionally investigated by measuring the leakage of calcein fluorescent dye and analyzing the NMR spectra of 31P, respectively. Detail on the interaction of gemini molecules with model membranes was also provided by a systematic differential scanning calorimetry (DSC) and molecular dynamics (MD) simulation. An irreversible impact on the viability of the NCTC 2544 cell line was observed for gemini concentrations higher than 25 mM, while no cytotoxicity was found for any of the surfactants in a concentration range up to 10 mM. A higher cytotoxicity was also found for gemini surfactants presenting longer spacer and shorter tails. The same trend was obtained in the calorimetric and permeability studies, with the gemini of longest spacer promoting the highest degree of membrane destabilization. Additional structural and dynamical characterization of the various systems, obtained by 31P NMR and MD, provide some insight on the relationship between the architecture of gemini surfactants and the respective perturbation mechanism

Optimización del contenido en macroelementos en hojas de frambueso mediante encalado en suelos muy ácidos

Raspberries thrive best in an acid soil. However, if the soil pH is lower than 5.6, lime application is necessary. In this study the effects of lime and dolomite application in combination with NPK 15:15:15 fertilizer (600 kg ha–1) on the macroelement contents (N, P, K, Ca, and Mg) in an extremely acid soil (pH 4.35) and raspberry leaves were evaluated during a three-year period. Optimal pH value for raspberry cultivation (5.84) was achieved with the application of 9 t ha–1 of lime. The mineral nitrogen (NH4+NO3)-N content and P content in the soil after liming increased significantly, but P concentration stayed below the optimal values. Additionally, no changes were noted in K concentration in the soil. The Ca concentration increased significantly in all treatments, while the Mg content increased significantly only in the treatment with dolomite. The N content in the raspberry leaves increased, but K content decreased after liming. There was no change in P content in the leaves affected by liming. Lime increased Ca content in the leaves above the optimal values, while it did not affect the initially optimal Mg content in the leaves. After liming and NPK fertilization, the concentrations of N, P and K in the leaves were still below the optimal values, indicating a need for the combination of higher rates of fertilizer with lime in raspberry cultivation in very acid soils.Los frambuesos crecen mejor en suelos ácidos. Sin embargo, si el pH del suelo es inferior a 5,6, es necesario aplicar cal. En este estudio se evaluaron durante un período de tres años los efectos de aplicar cal y dolomita, en combinación con fertilizantes NPK 15:15:15 (600 kg ha–1), en el contenido de macroelementos (N, P, K, Ca y Mg) en un suelo extremadamente ácido (pH 4,35) y en hojas de frambueso. El pH óptimo para el cultivo del frambueso (5,84) se logró con la aplicación de 9 t ha–1 de cal. En el caso del suelo, después del encalado el contenido en nitrógeno mineral (NH4+NO3)-N y de P aumentó significativamente, pero la concentración de P se mantuvo por debajo de los valores óptimos; además, no se observaron cambios en la concentración de K; la concentración de Ca aumentó significativamente en todos los tratamientos, pero el contenido de Mg aumentó significativamente sólo en el tratamiento con dolomita. En las hojas de frambueso, después del encalado el contenido de N aumentó, pero el contenido de K disminuyó; no hubo cambios en el contenido de P; el contenido de Ca aumentó por encima de los valores óptimos, mientras que no se afectó el contenido inicial óptimo de Mg. Después del encalado y la fertilización NPK, las concentraciones de N, P y K en las hojas estaban todavía por debajo de los valores óptimos, lo que indica la necesidad de combinar mayores tasas de abono con cal en el cultivo de frambueso en suelos muy ácidos

Inhomogeneous extreme forms

G.F. Voronoi wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify the covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem

A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15