Conformal structures of static vacuum data

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$. We provide two criteria: If $h$ is real analytic, $\kappa$ is closed, and one of it integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.Comment: 68 pages, typos corrected, references and details adde

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise

Vegetation monitoring through retrieval of NDVI and LST time series from historical databases.

The PhD dissertation presented here falls into the Earth Observation field, specifically vegetation monitoring. This work consists in the extensive exploitation of historical databases of satellite images for vegetation monitoring through two parameters, which are the land surface temperature (LST) and a vegetation index (NDVI). Up to now, vegetation monitoring has been limited to the use of vegetation indices, so the addition of the land surface temperature parameter represents the main innovative character of this PhD study. This dissertation is divided into 5 chapters. The first chapter begins by introducing the theoretical aspects of NDVI and LST parameters, addressing the means for retrieving them from remotely sensed observations, as well as their main limitations. Then, an introduction to vegetal physiology is developed, which allows for understanding how NDVI and LST parameters are linked to plants. A bibliographical study is then presented, which stresses out the gaps in the exploitation of historical databases. The second describes the data used in this PhD. The instrument providing most of these data is embarked on the NOAA (National Oceanic and Atmospheric Administration) satellite series. This instrument is the AVHRR (Advanced Very High Resolution Radiometer). The AVHRR databases used in this work are the PAL (Pathfinder AVHRR Land) and GIMMS (Global Inventory Modeling and Mapping Studies) databases. Additional data used punctually are also described briefly. The third chapter describes the operations applied to the data to prepare their temporal analysis. These operations start with the calculations of vegetation index and land surface temperature parameters. The AVHRR data used in this work are contaminated by the orbital drift of NOAA satellites, so an important part of this doctorate consisted in developing a technique for correcting this effect. We chose to develop our own technique, which we validated by direct comparison with data retrieved by geostationary satellites. In the fourth chapter, the different methods used for data temporal analysis are presented. Those methods consist of trend detection, harmonic analysis, and fitting the temporal series to annual NDVI evolution curves. Then, a phenological analysis is presented, which allows for retrieval of trends in spring and autumn dates for most of the globe. These trends are validated by comparison with previous studies. The trend analysis for spring dates is then extended to the 1948-2006 period using air temperature data. The long-term observation of different NDVI indicators also allows for the detection of land vegetation changes, even in our case of coarse spatial resolution. Finally, two methods for NDVI temporal analysis are compared. In the fifth chapter, a quick presentation of simultaneous study of NDVI and LST is developed through a revision of previous results, followed by the observations carried out from the orbital drift corrected data. These observations allowed for the determination of indicators of NDVI and LST, thus enabling for the characterization of the vegetation at global scale. A harmonic analysis of NDVI and LST at European scale is also presented. The application of the developed indicators for simultaneous monitoring of NDVI and LST shows promising results. As a conclusion, the main results described above are summarized, and plans for a close future are presented. This PhD has also demonstrated that such work could be carried out in a small structure with limited resources. __________________________________________________________________________________________________ RESUMEN El trabajo de tesis doctoral aquí presentado consiste en el uso extensivo de bases de datos históricas de imágenes de satélite para el seguimiento de la vegetación terrestre, a través de dos parámetros; la temperatura de la superficie terrestre (LST por sus siglas en inglés) y el índice de vegetación NDVI. El primer capítulo de la memoria introduce las nociones de NDVI y LST desde una perspectiva teórica, así como sus principales limitaciones y sus vínculos con la fisiología vegetal. Un estudio bibliográfico permite poner el acento sobre las lagunas en el uso de las bases de datos históricas. El segundo capítulo describe los datos utilizados en este trabajo, proporcionados en su mayoría por el instrumento AVHRR (Advanced Very High Resolution Radiometer) a bordo de la serie de satélites de la NOAA (National Oceanic and Atmospheric Administration) a través de las bases de datos PAL (Pathfinder AVHRR Land) y GIMMS (Global Inventory Modeling and Mapping Studies). También se presentan datos adicionales que se usaron puntualmente. El tercer capítulo describe el proceso para obtener las series temporales de NDVI y LST, las cuales están contaminadas por la deriva orbital de los satélites NOAA. Hemos propuesto una técnica propia para su corrección, validada por comparación directa con datos obtenidos por satélites geoestacionarios. En el cuarto capítulo se introducen diferentes métodos utilizados para el análisis temporal de los datos. Se obtuvieron tendencias acerca de parámetros vinculados a la evolución anual de NDVI para la mayor parte del globo, validadas por comparación con estudios previos. En el quinto capítulo se presenta un análisis conjunto del NDVI y de la LST, seguido por la elaboración de indicadores de la evolución anual de estos dos parámetros. A continuación se presenta un análisis armónico del NDVI y de la LST para Europa. El uso de los indicadores desarrollados para el seguimiento simultáneo del NDVI y de la LST revela resultados prometedores. Por último se presentan las conclusiones más relevantes del trabajo realizado, así como planes de trabajo para un futuro próximo