Solutions of the Einstein field equations

Solutions of the Einstein field equations are metrics of spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations gives a Lorentz manifold. Solutions are broadly classed as exact or non-exact. The Einstein field equations are :G_{\mu\nu} + \Lambda g_{\mu\nu} , = \kappa T_{\mu\nu} , where G_{\mu\nu} is the Einstein tensor, \Lambda is the cosmological constant (sometimes taken to be zero for simplicity), g_{\mu\nu} is the metric tensor, \kappa is a constant, and T_{\mu\nu} is the stress–energy tensor. The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of t

Complex networks

Cécile Caretta Cartozo

The last decade has witnessed a fundamental change in the role of network theory. Side by side with the increasing relevance of interdisciplinarity, this ensemble of mathematical tools and methods has known such a success and such a wide range of different applications to transform it in the proper scientific field of complex networks that groups researchers from different communities. Two different approaches can be distinguished. The first one is descriptive, tightly related to the experimental world. Reality is permeated with complex systems that lend themselves naturally to a network representation. The amount of data on natural, social and artificial systems is constantly increasing, and network theory provides a powerful framework for the analysis of such large datasets. The second is theoretical, aimed at producing models and analytical solutions for the reproduction and understanding of real networks properties. The results presented in this Thesis reflect the interdisciplinarity of the field of complex networks as well as its twofold nature. The first part of the manuscript describes new methods for the study of gene expression and protein interactions. Biological systems are highly complex and sensitive to the environment. Experimental data are affected by errors and poor reproducibility. We propose a method for the comparative analysis of similar datasets. We present an application to three independent studies on gene expression in the cell cycle of the fission yeast that show very poor agreement. In an attempt to reconcile them, we define a periodic genes network in which each node represents a gene identified as periodically regulated in the cell cycle and an edge between two nodes is drawn according to the phase difference between the expression peaks of the corresponding genes. The analysis of the topological structure of the network reveals a universal picture that overcomes the discrepancies and that is strongly related to the biological structure of the cell cycle and to the regulation of its progression. The method is able to group genes that are co-expressed during the cell cycle and to identify putative regulators of the transition from one phase to the next. Proteins are the fundamental product of gene expression. They are characterized by a modular structure comprising independent domains that act as mediators of protein interactions and are common to evolutionary related proteins. We propose the study of domain-centered interaction networks as a complementary tool for the understanding of protein interactions. We explore their topological structure looking for stable similarity groups and we investigate domain-peptide correlations that could be used as patterns for the prediction of the properties and functions to unknown proteins. The second part of this Thesis presents an exact theoretical solution for the navigability of small world networks with an inverse power-law distribution of the length of the connections. Optimal navigability is crucial for real networks involving transportation or communication processes (Internet, electric power grid, neural networks). Most of these systems are small worlds, characterized by a surprisingly small distance between any two nodes that grows only logarithmically with the size. How they are able to exploit this property for optimized performances without a complete knowledge of the structure is still an open question. It has been shown that a small world can arise from a proper compromise between local and long-range connections. In the specific case in which the length of the connections is power-law distributed and a greedy algorithm is used to forward a message from a source to a target, we obtained an original exact solution for the behavior of the expected delivery time in any dimension. We discussed the resulting scenario for the navigability and the implication of different structural parameters.

EPFL2009

On A Class Of Mean Field Solutions Of The Monge Problem For Perfect And Self-Interacting Systems

Philippe Choquard

The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum "cost" of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the "cost" of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.

2010

Complex networks

Cécile Caretta Cartozo

EPFL2009