The theory of relativity describes the laws of physics in a given space-time. However, a physical theory must provide observational predictions expressed in terms of measurements, which are the outcome of practical experiments and observations. Ideal for readers with a mathematical background and a basic knowledge of relativity, this book will help readers understand the physics behind the mathematical formalism of the theory of relativity. It explores the informative power of the theory of relativity, and highlights its uses in space physics, astrophysics and cosmology. Readers are given the tools to pick out from the mathematical formalism those quantities that have physical meaning and which can therefore be the result of a measurement. The book considers the complications that arise through the interpretation of a measurement, which is dependent on the observer who performs it. Specific examples of this are given to highlight the awkwardness of the problem.

Coalescence growth of droplets is a fundamental process for liquid cloud evolution. The initiation of collisions and coalescence occurs when a few droplets become large enough to fall. Gravitational collisions represent the most efficient mechanism for multi-disperse solutions, when droplets span a large variety of sizes. However, turbulence provides another mechanism for droplets coalescence, taking place also in the case of uniform condensational growth leading to narrow droplet-size spectra. We consider the problem of estimating the rate of collisions of small droplets dispersed in a highly turbulent medium. The problem is investigated by means of high-resolution direct numerical simulations of a three-dimensional turbulent flow, seeded with inertial particles, up to resolutions of 2048^3 grid points. Rate of collision is estimated in terms of the probability to find particles at close positions, and of the statistics of particles velocity. In particular, we show that the statistics of velocity differences between inertial particles suspended in an incompressible turbulent flow is extremely intermittent. When particles are separated by distances of the order of their diameter, the competition between quiet regular regions and multivalued caustics leads to a quasi bi-fractal behavior of the particle velocity statistics, with high-order moments bringing the signature of caustics. This results in large probabilities that close particles have important velocity differences. Together with preferential concentration of particles in low-vorticity regions, caustics contribute to speed-up collisions between inertial particles. Implications for the early stage of rain droplets formation are discussed.

We study the inverse problem of determining the relative orientations of the moving C- and N-terminal domains in a flexible protein from measurements of its mean magnetic susceptibility tensor ? ̄ . The latter is an integral average of rotations of the corresponding magnetic susceptibility tensor ?. The largest fraction of time that the two terminals can stay in a given orientation, still producing the ? ̄ measurements, is the maximal probability of that orientation. We extend this definition to any measurable subset of the rotation group. This extension permits a quantitative assessment of the results when the generating distribution is either continuous or discrete. We establish some properties of the maximal probability and present some numerical experiments.

We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in R3 acting on the space of pairs of anisotropic symmetric 3 × 3 tensors. This is motivated by the problem of determining the structure of some proteins in an aqueous solution.

Phase separation of a two-dimensional van der Waals fluid subject to a gravitational force is studied by numerical simulations based on lattice Boltzmann methods implemented with a finite difference scheme. A growth exponent alpha = 1 is measured in the direction of the external force.

Many important kernel methods in the machine learning area, such as kernel principal component analysis, feature approximation, denoising, compression and prediction require the computation of the dominant set of eigenvectors of the symmetric kernel Gram matrix. Recently, an efficient incremental approach was presented for the fast calculation of the dominant kernel eigenbasis. In this manuscript we propose faster algorithms for incrementally updating and downsizing the dominant kernel eigenbasis. These methods are well-suited for large scale problems since they are both efficient in terms of complexity and data management.

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.

Here we investigate the behavior of the analytical and numerical solution of a nonlinear second kind Volterra integral equation where the linear part of the kernel has a constant sign and we provide conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Runge-Kutta and Direct Quadrature methods.

In this paper, for the "critical case" with two delays, we establish two relations between any two solutions y(t) and y*(t) for the Volterra integral equation of non-convolution type y(t)=f(t)+\int_{t-\tau}^{t-\delta}k(t,s)g(y(s))ds and a solution z(t) of the first order differential equation \dot z(t)=\beta(t)[z(t-\delta)-z(t-\tau) , and offer a sufficient condition that limt->+?(y(t)-y*(t))=0.

A graphic processing unit (GPU) implementation of the multicomponent lattice Boltzmann equation with multirange interactions for soft-glassy materials ["glassy" lattice Boltzmann (LB)] is presented. Performance measurements for flows under shear indicate a GPU/CPU speed up in excess of 10 for 1024(2) grids. Such significant speed up permits to carry out multimillion time-steps simulations of 1024(2) grids within tens of hours of GPU time, thereby considerably expanding the scope of the glassy LB toward the investigation of long-time relaxation properties of soft-flowing glassy materials.

Phase separation of binary fluids quenched by contact with cold external walls is considered. Navier-Stokes, convection-diffusion, and energy equations are solved by lattice Boltzmann method coupled with finite-difference schemes. At high viscosity, different morphologies are observed by varying the thermal diffusivity. In the range of thermal diffusivities with domains growing parallel to the walls, temperature and phase separation fronts propagate toward the inner of the system with power-law behavior. At low viscosity hydrodynamics favors rounded shapes, and complex patterns with different length scales appear. Off-symmetrical systems behave similarly but with more ordered configurations.