From 2-Dimensional Surfaces to Cosmological Solutions

We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.Comment: 12 pages, no figures, LaTeX2e, to be published in Class. Quant. Gra

Webs, Nijenhuis operators, and heavenly PDEs

In 1989 Mason and Newman proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields $X_1(\lambda),X_2(\lambda)$ which are divergence free with respect to some volume form. Earlier (in 1975) Pleba\'nski showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Pleba\'nski heavenly PDEs. Other PDEs leading to Mason--Newman vector fields are also known in the literature: Husain--Park (1992--94), Schief (1996). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new ``heavenly'' PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko--Schief--Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the ``heavelny'' PDEs and the existence of related vacuum Einstein metrics in 4D-case.Comment: 32

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Backlund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalisation of the integrable discrete Tzitzeica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretisation of affine spheres in affine differential geometry

Where have all the Linkages Gone? Staples in the Ghanaian Economy

SUMMARY There has been renewed interest in the relevance of the growth of a staple export trade to development. The staple is supposed to stimulate growth through technical, market and fiscal linkages. Some of this theoretical literature is reviewed in the light of a recent article by A. O. Hirschman. Ghana contains both ‘peasant’ and ‘enclave’ type export industries and their linkage effects are compared and contrasted. RESUME Où sont passèes toutes les liaisons? Matières premières dans I'economie ghaneenne II y a un relent d'intérêt pour le développement des exportations de matières premières en tant que facteur de développement. Les matières premières sont supposées stimuler le développement en créant des liaisons techniques, commerciales et fiscales. Certains de ces concepts théoriques sont revus à la lumière d'un récent article par A. O. Hirschman. Le Ghana a des industries d'exportation de type “paysan” et de type “enclave” et l'auteur compare leurs effets de liaison. RESUMEN Donde han ido a parar todos los enlaces? Productos basicos en la economí de Ghana Ha habido un interés renovado en la importancia del crecimiento de un comercio básico de exportación en función al desarrollo. Se supone que esta base estimule el crecimiento a través de enlaces técnicos, comerciales y fiscales. Se vuelve a examinar parte de esta literatura teórica en vista de un artículo publicado recientemente por A. O. Hirschman. Ghana tiene industrias de exportación tanto tradicionales como modernas, y se compara y contrasta sus efectos de enlace

GPR54 Regulates ERK1/2 Activity and Hypothalamic Gene Expression in a Gαq/11 and β-Arrestin-Dependent Manner

G protein-coupled receptor 54 (GPR54) is a Gq/11-coupled 7 transmembrane-spanning receptor (7TMR). Activation of GPR54 by kisspeptin (Kp) stimulates PIP2 hydrolysis, Ca2+ mobilization and ERK1/2 MAPK phosphorylation. Kp and GPR54 are established regulators of the hypothalamic-pituitary-gonadal (HPG) axis and loss-of-function mutations in GPR54 are associated with an absence of puberty and hypogonadotropic hypogonadism, thus defining an important role of the Kp/GPR54 signaling system in reproductive function. Given the tremendous physiological and clinical importance of the Kp/GPR54 signaling system, we explored the contributions of the GPR54-coupled Gq/11 and β-arrestin pathways on the activation of a major downstream signaling molecule, ERK, using Gq/11 and β-arrestin knockout mouse embryonic fibroblasts. Our study revealed that GPR54 employs the Gq/11 and β-arrestin-2 pathways in a co-dependent and temporally overlapping manner to positively regulate ERK activity and pERK nuclear localization. We also show that while β-arrestin-2 potentiates GPR54 signaling to ERK, β-arrestin-1 inhibits it. Our data also revealed that diminished β-arrestin-1 and -2 expression in the GT1-7 GnRH hypothalamic neuronal cell line triggered distinct patterns of gene expression following Kp-10 treatment. Thus, β-arrestin-1 and -2 also regulate distinct downstream responses in gene expression. Finally, we showed that GPR54, when uncoupled from the Gq/11 pathway, as is the case for several naturally occurring GPR54 mutants associated with hypogonadotropic hypogonadism, continues to regulate gene expression in a G protein-independent manner. These new and exciting findings add significantly to our mechanistic understanding of how this important receptor signals intracellularly in response to kisspeptin stimulation

D-dimensional metrics with D-3 symmetries

Hidden symmetry transformations of D-dimensional vacuum metrics with D-3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We relate the 4-dimensional Reissner-Nordstr\"om solution to a particular case of the 5-dimensional Gross-Perry metric.Comment: 8 page