Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces

Let $X=X_1 \times X_2$ be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the Laplacian of $X$ can be lifted to a holomorphic function on a Riemann surface which is a branched covering of $\mathbb C$. In all other cases, the resolvent of the Laplacian of $X$ admits a singular meromorphic lift. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are finite dimensional and explicitly realized as direct sums of finite-dimensional irreducible spherical representations of the group of the isometries of $X$

Estimation of phosphorus emissions in the upper Iguazu basin (Brazil) using GIS and the MoRE Model

Pollution emissions into the drainage basin have direct impact on surface water quality. These emissions result from human activities that turn into pollution loads when they reach the water bodies, as point or diffuse sources. Their pollution potential depends on the characteristics and quantity of the transported materials. The estimation of pollution loads can assist decision-making in basin management. Knowledge about the potential pollution sources allows for a prioritization of pollution control policies to achieve the desired water quality. Consequently, it helps avoiding problems such as eutrophication of water bodies. The focus of the research described in this study is related to phosphorus emissions into river basins. The study area is the upper Iguazu basin that lies in the northeast region of the State of Paraná, Brazil, covering about 2,965 km² and around 4 million inhabitants live concentrated on just 16% of its area. The MoRE (Modeling of Regionalized Emissions) model was used to estimate phosphorus emissions. MoRE is a model that uses empirical approaches to model processes in analytical units, capable of using spatially distributed parameters, covering both, emissions from point sources as well as non-point sources. In order to model the processes, the basin was divided into 152 analytical units with an average size of 20 km². Available data was organized in a GIS environment. Using e.g. layers of precipitation, the Digital Terrain Model from a 1:10000 scale map as well as soils and land cover, which were derived from remote sensing imagery. Further data is used, such as point pollution discharges and statistical socio-economic data. The model shows that one of the main pollution sources in the upper Iguazu basin is the domestic sewage that enters the river as point source (effluents of treatment stations) and/or as diffuse pollution, caused by failures of sanitary sewer systems or clandestine sewer discharges, accounting for about 56% of the emissions. Second significant shares of emissions come from direct runoff or groundwater, being responsible for 32% of the total emissions. Finally, agricultural erosion and industry pathways represent 12% of emissions. This study shows that MoRE is capable of producing valid emission calculation on a relatively reduced input data basis

Special functions associated to a certain fourth order differential equation

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}dx)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations. This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors

Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group

This work is devoted to the relativistic generalization of Chasles' theorem, namely to the proof that every proper orthochronous isometry of Minkowski spacetime, which sends some point to its chronological future, is generated through the frame displacement of an observer which moves with constant acceleration and constant angular velocity. The acceleration and angular velocity can be chosen either aligned or perpendicular, and in the latter case the angular velocity can be chosen equal or smaller than than the acceleration. We start reviewing the classical Euler's and Chasles' theorems both in the Lie algebra and group versions. We recall the relativistic generalization of Euler's theorem and observe that every (infinitesimal) transformation can be recovered from information of algebraic and geometric type, the former being identified with the conjugacy class and the latter with some additional geometric ingredients (the screw axis in the usual non-relativistic version). Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in detail. We prove its exponentiality and identify a causal semigroup and the corresponding Lie cone. Through the identification of new Ad-invariants we classify the conjugacy classes, and show that those which admit a causal representative have special physical significance. These results imply a classification of the inequivalent Killing vector fields of Minkowski spacetime which we express through simple representatives. Finally, we arrive at the mentioned generalization of Chasles' theorem.Comment: Latex2e, 49 pages. v2: few typos correcte

Future asymptotic expansions of Bianchi VIII vacuum metrics

Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and in a previous article we analyzed the asymptotic behaviour of solutions in these variables. One objective of this paper is to give an asymptotic expansion for the metric. Furthermore, we relate this expansion to the topology of the compactified spatial hypersurfaces of homogeneity. The compactified spatial hypersurfaces have the topology of Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII spacetimes, the length of a circle fibre converges to a positive constant but that in the case of general Bianchi VIII solutions, the length tends to infinity at a rate we determine.Comment: 50 pages, no figures. Erronous definition of Seifert fibred spaces correcte