Direction problems in affine spaces

This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October 5, 2012, Brussels, Belgiu

Jets in strongly-coupled N = 4 super Yang-Mills theory

We study jets of massless particles in N=4 super Yang-Mills using the AdS/CFT correspondence both at zero and finite temperature. We set up an initial state corresponding to a highly energetic quark/anti-quark pair and follow its time evolution into two jets. At finite temperature the jets stop after traveling a finite distance, whereas at zero temperature they travel and spread forever. We map out the corresponding baryon number charge density and identify the generic late time behavior of the jets as well as features that depend crucially on the initial conditions.Comment: 21 pages, 12 figures. Added discussion regarding string profiles in more than one spatial dimension. Refs adde

A Crash Course on Aging

In these lecture notes I describe some of the main theoretical ideas emerged to explain the aging dynamics. This is meant to be a very short introduction to aging dynamics and no previous knowledge is assumed. I will go through simple examples that allow one to grasp the main results and predictions.Comment: Lecture Notes (22 pages) given at "Unifying Concepts in Glass Physics III", Bangalore (2004); to be published in JSTA

The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces

The Swampland Distance Conjecture claims that effective theories derived from a consistent theory of quantum gravity only have a finite range of validity. This will imply drastic consequences for string theory model building. The refined version of this conjecture says that this range is of the order of the naturally built in scale, namely the Planck scale. It is investigated whether the Refined Swampland Distance Conjecture is consistent with proper field distances arising in the well understood moduli spaces of Calabi-Yau compactification. Investigating in particular the non-geometric phases of Kahler moduli spaces of dimension $h^{11}\in\{1,2,101\}$, we always found proper field distances that are smaller than the Planck-length.Comment: 71 pages, 11 figures, v2: refs added, typos correcte

Spontaneous-emission rates in finite photonic crystals of plane scatterers

The concept of a plane scatterer that was developed earlier for scalar waves is generalized so that polarization of light is included. Starting from a Lippmann-Schwinger formalism for vector waves, we show that the Green function has to be regularized before T-matrices can be defined in a consistent way. After the regularization, optical modes and Green functions are determined exactly for finite structures built up of an arbitrary number of parallel planes, at arbitrary positions, and where each plane can have different optical properties. The model is applied to the special case of finite crystals consisting of regularly spaced identical planes, where analytical methods can be taken further and only light numerical tasks remain. The formalism is used to calculate position- and orientation-dependent spontaneous-emission rates inside and near the finite photonic crystals. The results show that emission rates and reflection properties can differ strongly for scalar and for vector waves. The finite size of the crystal influences the emission rates. For parallel dipoles close to a plane, emission into guided modes gives rise to a peak in the frequency-dependent emission rate.Comment: 18 pages, 6 figures, to be published in Phys. Rev.

On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance

A new numerical method is introduced to study the problem of time evolution of generic non-linear dynamical systems in four-dimensional spacetimes. It is assumed that the time level surfaces are foliated by a one-parameter family of codimension two compact surfaces with no boundary and which are conformal to a Riemannian manifold C. The method is based on the use of a multipole expansion determined uniquely by the induced metric structure on C. The approach is fully spectral in the angular directions. The dynamics in the complementary 1+1 Lorentzian spacetime is followed by making use of a fourth order finite differencing scheme with adaptive mesh refinement. In checking the reliability of the introduced new method the evolution of a massless scalar field on a fixed Kerr spacetime is investigated. In particular, the angular distribution of the evolving field in to be superradiant scattering is studied. The primary aim was to check the validity of some of the recent arguments claiming that the Penrose process, or its field theoretical correspondence---superradiance---does play crucial role in jet formation in black hole spacetimes while matter accretes onto the central object. Our findings appear to be on contrary to these claims as the angular dependence of a to be superradiant scattering of a massless scalar field does not show any preference of the axis of rotation. In addition, the process of superradiance, in case of a massless scalar field, was also investigated. On contrary to the general expectations no energy extraction from black hole was found even though the incident wave packets was fine tuned to be maximally superradiant. Instead of energy extraction the to be superradiant part of the incident wave packet fails to reach the ergoregion rather it suffers a total reflection which appears to be a new phenomenon.Comment: 49 pages, 11 figure