Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the apriori estimates for the Weyl equations, associated to the "Bel-Robinson norms". In particular if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a "geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.Comment: 68 page

The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur

A neural network based technique for short-term forecasting of anomalous load periods

The paper illustrates a part of the research activity conducted by authors in the field of electric Short Term Load Forecasting (STLF) based on Artificial Neural Network (ANN) architectures. Previous experiences with basic ANN architectures have shown that, even though these architectures provide results comparable with those obtained by human operators for most normal days, they evidence some accuracy deficiencies when applied to ''anomalous'' load conditions occurring during holidays and long weekends. For these periods a specific procedure based upon a combined (unsupervised/supervised) approach has been proposed. The unsupervised stage provides a preventive classification of the historical load data by means of a Kohonen's Self Organizing Map (SOM) The supervised stage, performing the proper forecasting activity, is obtained by using a multi-layer perceptron with a backpropagation learning algorithm similar to the ones above mentioned. The unconventional use of information deriving from the classification stage permits the proposed procedure to obtain a relevant enhancement of the forecast accuracy for anomalous load situations

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

Environment and Electrical Energy Intermational Conference 2012

In the Conference the latest development in energy systems, are examinated, associated with equipment and environmentat aspect

Informative Calibration of the Instrumentation

The calibration cost of an instrument is so expensive that is necessary to reduce the number of calibrated points in the scale. There is the necessity to exploit as well as possible the metrological valence of a few calibrated points to exend all over the instrumental range. The choice of the number andthe position of such few points is the problem faced in the work. Utilizing the Shannon sampling procedure in the transposed domain of the input entities of the instrument, with respectto the time, the truncation error of the cardinal series has faced utilizing the Helms and Thomas upper bound evaluation. To enhance the efficiency of the procedure,has been utilized the minimax procedure searching the optimal indication on the better determination. In every case the efficiency of the procedure is lower on the edge of instrumental scale. To upgrade the edge calibration, making as even as possible the calibration, upon the scale,one has been utilazed the generalized sampling theorem by Papoulis that allows to chos the calibration points more frequent atthe edge, with respec of the central part of thescale. An algorithm to guide the calibration procedure is tempted