Magic numbers in the discrete tomography of cyclotomic model sets

We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. It turns out that for any of these model sets $\varLambda$ there exists a `magic number' $m_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-directions. In particular, for pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible numbers are in that very order 11, 9, 11 and 13.Comment: 6 pages, 1 figure; based on the results of arXiv:1101.4149 [math.MG]; presented at Aperiodic 2012 (Cairns, Australia

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.

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

N=1 Supersymmetric Gauge Theories and Supersymmetric 3-cycles

In this paper we discuss the strong coupling limit of chiral N=1 supersymmetric gauge theory via their embedding into M-theory. In particular we focus on the brane box models of Hanany and Zaffaroni and show that after a T-duality transformation their M-theory embedding is described by supersymmetric 3-cycles; its geometry will encode the holomorphic non-perturbative information about the gauge theory.Comment: 36 pages, LaTeX2e, 10 figures, additional references added, minor correction

Discrete tomography: Magic numbers for NN-fold symmetry

We consider the problem of distinguishing convex subsets of $n$-cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. In this context, a `magic number' $m_{\varLambda}$ has the property that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-directions. Recent calculations suggest that (with one exception in the case $n=4$) the least possible magic number for $n$-cyclotomic model sets might just be $N+1$, where $N=\operatorname{lcm}(n,2)$.Comment: 5 pages, 2 figures; new computer calculations based on the results of arXiv:1101.4149 and arXiv:1211.6318; presented at ICQ 12 (Cracow, Poland

Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets

We consider algebraic Delone sets $\varLambda$ in the Euclidean plane and address the problem of distinguishing convex subsets of $\varLambda$ by X-rays in prescribed $\varLambda$-directions, i.e., directions parallel to nonzero interpoint vectors of $\varLambda$. Here, an X-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. It is shown that for any algebraic Delone set $\varLambda$ there are four prescribed $\varLambda$-directions such that any two convex subsets of $\varLambda$ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number $c_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $c_{\varLambda}$ prescribed $\varLambda$-directions. In particular, this extends a well-known result of Gardner and Gritzmann on the corresponding problem for planar lattices to nonperiodic cases that are relevant in quasicrystallography.Comment: 21 pages, 1 figur

Critical behaviors near the (tri-)critical end point of QCD within the NJL model

We investigate the dynamical chiral symmetry breaking and its restoration at finite density and temperature within the two-flavor Nambu-Jona-Lasinio model, and mainly focus on the critical behaviors near the critical end point (CEP) and tricritical point (TCP) of quantum chromodynamics. The multi-solution region of the Nambu and Wigner ones is determined in the phase diagram for the massive and massless current quark, respectively. We use the various susceptibilities to locate the CEP/TCP and then extract the critical exponents near them. Our calculations reveal that the various susceptibilities share the same critical behaviors for the physical current quark mass, while they show different features in the chiral limit