Metric connections in projective differential geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.Comment: 10 page

Borehole climatology: a discussion based on contributions from climate modeling

Progress in understanding climate variability through the last millennium leans on simulation and reconstruction efforts. Exercises blending both approaches present a great potential for answering questions relevant both for the simulation and reconstruction of past climate, and depend on the specific peculiarities of proxies and methods involved in climate reconstructions, as well as on the realism and limitations of model simulations. This paper explores research specifically related to paleoclimate modeling and borehole climatology as a branch of climate reconstruction that has contributed significantly to our knowledge of the low frequency climate evolution during the last five centuries. The text flows around three main issues that group most of the interaction between model and geothermal efforts: the use of models as a validation tool for borehole climate reconstructions; comparison of geothermal information and model simulations as a means of either model validation or inference about past climate; and implications of the degree of realism on simulating subsurface climate on estimations of future climate change. The use of multi-centennial simulations as a surrogate reality for past climate suggests that within the simplified reality of climate models, methods and assumptions in borehole reconstructions deliver a consistent picture of past climate evolution at long time scales. Comparison of model simulations and borehole profiles indicate that borehole temperatures are responding to past external forcing and that more realism in the development of the soil model components in climate models is desirable. Such an improved degree of realism is important for the simulation of subsurface climate and air-ground interaction; results indicate it could also be crucial for simulating the adequate energy balance within climate change scenario experiments

Perturbation Theory for Metastable States of the Dirac Equation with Quadratic Vector Interaction

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Pad\'e approximants for calculating the distributional Borel transform. A complete agreement is found with previous non-perturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum.Comment: 10 pages, 1 figur

Active Mass Under Pressure

After a historical introduction to Poisson's equation for Newtonian gravity, its analog for static gravitational fields in Einstein's theory is reviewed. It appears that the pressure contribution to the active mass density in Einstein's theory might also be noticeable at the Newtonian level. A form of its surprising appearance, first noticed by Richard Chase Tolman, was discussed half a century ago in the Hamburg Relativity Seminar and is resolved here.Comment: 28 pages, 4 figure

Conduction‐dominated heat transport of the annual temperature signal in soil

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/95182/1/jgrb13727.pd

Harmonic fields on the extended projective disc and a problem in optics

The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1-fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the non-characteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptic-hyperbolic boundary-value problems is formulated for those equations as well. For both classes of boundary-value problems, an arbitrarily small lower-order perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.Comment: 30 pages; Section 3.3 has been revise

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

A quality assurance process of a surface wind database in Eastern Canada

Presentación realizada para las XXXII Jornadas Científicas de la Asociación Meteorológica Española y 13º Encuentro Hispano-Luso de Meteorología celebrados en Alcobendas (Madrid), del 28 al 30 de mayo de 2012

Cellular taxonomy and spatial organization of the murine ventral posterior hypothalamus.

The ventral posterior hypothalamus (VPH) is an anatomically complex brain region implicated in arousal, reproduction, energy balance, and memory processing. However, neuronal cell type diversity within the VPH is poorly understood, an impediment to deconstructing the roles of distinct VPH circuits in physiology and behavior. To address this question, we employed a droplet-based single-cell RNA sequencing (scRNA-seq) approach to systematically classify molecularly distinct cell populations in the mouse VPH. Analysis of \u3e16,000 single cells revealed 20 neuronal and 18 non-neuronal cell populations, defined by suites of discriminatory markers. We validated differentially expressed genes in selected neuronal populations through fluorescence in situ hybridization (FISH). Focusing on the mammillary bodies (MB), we discovered transcriptionally-distinct clusters that exhibit neuroanatomical parcellation within MB subdivisions and topographic projections to the thalamus. This single-cell transcriptomic atlas of VPH cell types provides a resource for interrogating the circuit-level mechanisms underlying the diverse functions of VPH circuits

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant sectional curvature Riemannian manifold. In three-dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.Comment: arXiv admin note: substantial text overlap with arXiv:1201.440