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The objective of this review is to provide some insight into trends in the development of one of the most important fields of theoretical astrophysics – the theory of radiative transfer. The foundation of the field goes back to the pioneering works of Schuster, Schwarzschild, Milne and Eddington appearing at the beginning of the last century in connection with modeling the stellar atmospheres. Since then the approaches proposed in these works have made rapid progress forming a theory usually referred to as ‘classical’. Parallel to this an alternative approach was offered by Ambartsumian with his methods based on the principle of invariance and the laws of addition of layers. With their deep physical content the methods have been extremely flexible in applications and efficient in numerical computations. Their major role in the theory is well-known. However, the modern fundamental results in this direction make expedient reviewing the most important of them to elucidate their place in existing theory and significance from the point of view of its further progress.

The main motivation of this work is the implementation of a general finite element solver for some of the improved Boussinesq models. Here, we use an extension of the model proposed by Zhao et al. [2004] to investigate the behavior of surface water waves. The equations in this model do not contain spatial derivatives with an order higher than two. Some effects like energy dissipation and wave generation by natural phenomena or external physical mechanisms are also included. As a consequence, some modified dispersion relations are derived for this extended model. A matrix-based linear stability analysis of the proposed model is presented. It is shown that this model is robust with respect to instabilities related to steep bottom gradients.

This chapter deals with the source function (SF) descriptor, originally put forth by Bader and Gatti back in 1998. After a brief review on how this descriptor is defined and what it physically represents, the various forms through which the SF may be analyzed are presented in some detail. The relationships between atomic SF contributions and chemical bond nature are analyzed in some prototypical cases, and the capability of the SF to neatly reveal π-electron conjugation directly from the electron distribution and independently from any MO scheme or decomposition is introduced. Applications of the SF to chemistry from the literature are reviewed and critically discussed, including the use of the SF to assess chemical transferability or to describe chemical bonding in challenging situations, like for instance the short-strong hydrogen bonds in π-conjugated frameworks or the metal–metal and metal–ligand interactions in the organometallic complexes. Comparison with the insight obtained from other bond topological descriptors is given, emphasizing the special role the SF has of being directly derivable from experimental electron density distributions and to so provide an ideal tool to compare experiment and theory. The robustness of the SF descriptor against changes in the models used to derive electron densities from theory of experiment is detailed. First results on using the SF to define an unambiguous full population analysis are outlined. The possible ways of further decomposing the atomic SF in chemically meaningful additive pieces, such as core and valence atomic contributions, are analyzed in view of their potential insight and degree of arbitrariness.

For the modelling of the radiation fields in stellar winds and the resulting forces new efficient algorithms are presented. In the first one, the radiative transfer equation for moving 3D media is solved analytically with the assumption that the source function is known, eg. from the solution of the NLTE rate equations. For a wind with inhomogeneities an a-posteriori error controlled finite element code is described that takes scattering explicitly into account. Finally, we present possibilities for the accurate inclusion of an arbitrary number of spectral lines in a deterministic and in a stochastic way.

Accelerated Lambda Iteration, or ALI, methods are reviewed. An emphasis is given to the critical evaluation of various methods, analysing their physical and mathematical meaning, and recommending the most advantageous methods to interested non-specialists who consider applying these methods to solving actual line formation and model stellar atmosphere problems.

Many stars continuously eject material and thus surround themselves with a stellar wind. We can observe this wind indirectly through its characteristic effects on the stellar spectrum. In particular, this mass flow influences the line profiles. These so called P-Cygni profiles can be observed in the visual part of the spectrum of early-type stars and Wolf-Rayet stars, as well as - and even more clearly - in the ultraviolet part. In order to study the stellar wind structure, a computer programme was written which calculates theoretical P-Cygni profiles using the Comoving Frame Method. As input, this code can use the results obtained from other programmes, such as a non-LTE code. Through parameter adjustments a fit to an observed profile can be made.

The line-driven instability may cause pronounced structure in winds of hot, luminous stars, e.g., fragments of dense shells, strong reverse shocks, and fast cloudlets. We discuss the linear stability theory, including the line-drag effect, phase reversal due to the diffuse radiation field, and the relevance of so-called Abbott waves. Recent hydrodynamic simulations focuss on the influence of a time-dependent source function on the flow structure, and on the X-ray emission from wind shocks and cloud collisions.

Observations of the solar chromosphere are often interpreted using methods derived from static modeling (e.g., the Vernazza et al. 1981 model atmospheres and work based on such models) or linear theory (e.g., phase relations). Recent numerical simulations have shown that such an analysis can be very misleading. It is found that enhanced chromospheric emission, which corresponds to an outwardly increasing semi-empirical temperature structure, can be produced by wave motions without any increase in the mean gas temperature. Thus, despite long held beliefs, the Sun may not have a classical chromosphere in magnetic field free internetwork regions. This dynamic picture is consistent with observations in CO lines and the calcium H and K bright grains. More opaque lines, on the other hand, seem to show emission all of the time. This indicates the existence of a hotter, magnetic, component that increases in importance with height.

Radiation plays an important role, firstly in determining the structure of stars through the dominant role radiation plays in the energy balance (in some objects also in the momentum balance), secondly because we diagnose astrophysical plasma through the emitted electromagnetic radiation.

Pulse mode sonar operation is analyzed under the physical hypotheses that the medium is a stationary homogenous fluid; both the sonar system and the scattering objects are stationary; the scattering objects are rigid bodies; and the scattering objects lie in the far field of the transmitter and receiver.

The image principle in electromagnetics replaces a physical structure by an equivalent source, the image of the primary source. The number of structures with analytic image expressions is limited but increasing. In finding new image expressions, the Heaviside operational calculus can be applied by first transforming the three-dimensional problem to one concerning transmission lines. The image problem can be reduced to the compact form in terms of a pseudo-differential operator applied to the original source function. To interprete the result in terms of computable functions, approximations can be done at this stage. As a simple example of the theory the interface of two isotropic media with smooth or slightly rough interface is considered. A table of operational formulas is given as an appendix.

The starting point of this work is the inversion of vertical seismic profiling (VSP) data. The usual processing of VSP data by inverse techniques is restricted to 1D propagation model. In this case, the parameters to identify are the acoustic impedance as function of travel time and the seismic source so that we have as unknowns two functions of one variable and as data a function of two variables, the time and the depth positions of geophones. The problem is thus largely overdetermined and an elementary mathematical analysis can be made. The source is modelled as a boundary condition at the top of the geophones zone. So this boundary condition replaces the true source function and the medium parameters above the geophones zone. The question asked by V. Richard from IFP was the “management” of this unknown source when 3D propagation effects are taken into account in horizontally layered medium where the propagation equations are parametrized by the k parameter of the Hankel transform. Now we think that the answer is that it is impossible to work round the fact that there are at least two unknown functions, the source and the medium parameters above the geophones zone. During this study, we have searched for some non local boundary conditions and this was the opportunity to obtain some results on exact transparent conditions for 3D propagation in 1D media (preliminary communication was made by Petit and Cuer (1994)) and on the discretization of such conditions in the acoustic case (preliminary communication was made by Cuer and Petit (1995)). This is the mathematical substance of this work in which the Poisson summation formula is used to prove the stability of a discrete non local boundary condition.

Blind source separation has found applications in various areas including biomedical signal processing and genomic signal processing. Often, blind source separation is performed via independent component analysis (ICA) under the assumption of mutual independence among source signals. However, in bio-signal and genomic signal processing, the assumption of independence is often untrue, and the performance of the ICA approach is not so good. Much effort has been devoted to searching alternative approaches to blind source separation without the independence assumption. In this paper we present a sparse component analysis method, which exploits the sparseness of the source signals and makes the separated signals as sparse as possible according to a properly defined sparsity function, to reliably extract source signals from their mixtures. Some related theoretical and practical issues are investigated, with support and validation by simulation results.