On the X-rays of permutations

The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete Tomography since many types of integral matrices can be written as linear combinations of permutation matrices. This paper is an invitation to the study of X-rays of permutations from a combinatorial point of view. We present connections between these objects and nondecreasing differences of permutations, zero-sum arrays, decomposable permutations, score sequences of tournaments, queens' problems and rooks' problems.Comment: 7 page

Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D>=2, whose points may be occupied by $c$ distinct types of atoms. To ``analyze'' T, we conduct ell measurements that we call_discrete X-rays_. A discrete X-ray in direction xi determines the number of atoms of each type on each line parallel to xi. Given ell such non-parallel X-rays, we wish to reconstruct T. The complexity of the problem for c=1 (one atom type) has been completely determined by Gardner, Gritzmann and Prangenberg, who proved that the problem is NP-complete for any dimension D>=2 and ell>=3 non-parallel X-rays, and that it can be solved in polynomial time otherwise. The NP-completeness result above clearly extends to any c>=2, and therefore when studying the polyatomic case we can assume that ell=2. As shown in another article by the same authors, this problem is also NP-complete for c>=6 atoms, even for dimension D=2 and axis-parallel X-rays. They conjecture that the problem remains NP-complete for c=3,4,5, although, as they point out, the proof idea does not seem to extend to c<=5. We resolve the conjecture by proving that the problem is indeed NP-complete for c>=3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0-1 matrices with given row and column sums

AGN behind the SMC selected from radio and X-ray surveys

The XMM-Newton survey of the Small Magellanic Cloud (SMC) revealed 3053 X-ray sources with the majority expected to be active galactic nuclei (AGN) behind the SMC. However, the high stellar density in this field often does not allow assigning unique optical counterparts and hinders source classification. On the other hand, the association of X-ray point sources with radio emission can be used to select background AGN with high confidence, and to constrain other object classes like pulsar wind nebula. To classify X-ray and radio sources, we use clear correlations of X-ray sources found in the XMM-Newton survey with radio-continuum sources detected with ATCA and MOST. Deep radio-continuum images were searched for correlations with X-ray sources of the XMM-Newton SMC-survey point-source catalogue as well as galaxy clusters seen with extended X-ray emission. Eighty eight discrete radio sources were found in common with the X-ray point-source catalogue in addition to six correlations with extended X-ray sources. One source is identified as a Galactic star and eight as galaxies. Eight radio sources likely originate in AGN that are associated with clusters of galaxies seen in X-rays. One source is a PWN candidate. We obtain 43 new candidates for background sources located behind the SMC. A total of 24 X-ray sources show jet-like radio structures.Comment: 9 pages, 6 figures, accepted for publication in A&

The smallest sets of points not determined by their X-rays

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\psi_{\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\mathbb{R}^d$) such that two $n$-point sets in $\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\psi_{\mathbb{Z}^d}(m)\leq 2^{m-1}$ has been observed many times in the literature. In this note we show $\psi_{\mathbb{K}^d}(m)=O(m^{d+1+\varepsilon})$ for $\varepsilon>0$. For the cases $\mathbb{K}^d=\mathbb{Z}^d$ and $\mathbb{K}^d=\mathbb{R}^d$, $d>2$, this represents the first upper bound on $\psi_{\mathbb{K}^d}(m)$ that is polynomial in $m$. As a corollary we derive bounds on the sizes of solutions to both the classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we establish lower bounds on $\psi_{\mathbb{K}^d}$ that enable us to prove a strengthened version of R\'enyi's theorem for points in $\mathbb{Z}^2$

Cold X-ray Effects on Satellite Solar Panels in Orbit

An exo-atmospheric nuclear detonation releases up to 80 percent of its’ energy as X-rays. Satellite’s solar cells and their protective coatings are vulnerable to low energy X-ray radiation. Cold X-rays (~1-1.5 keV) are absorbed close to the surface of materials causing the blow-off and rapid formation of Warm Dense Plasmas (WDPs), particularly in a gap between the unshielded active elements of solar cells. To understand how WDPs are created, it is necessary to investigate the power density distribution produced by cold X-rays for typical solar panel surface materials. The Monte Carlo stepping model implemented in the GEANT4 software toolkit is utilized to determine the power density created by cold X-rays in a multi-layered target composed of a layer of an active cell shielded by layers of cover glass and anti-reflective coating. The power density generated by cold X-rays in the unshielded semiconductor layer at different incidence angles is also investigated in order to account for different orientations of the satellite’s solar panels with respect to the point of nuclear detonation. The flux spectrum of X-rays originating from a nuclear blast is described by the Planck\u27s blackbody function with the temperature from 0.1 keV to 10 keV. The secondary radiation (photo-electrons, fluorescence photons, Auger- and Compton-electrons) resulting from absorption and scattering of primary X-rays is taken into account in the redistribution of energy deposition within slabs. The profiles of power density within the slab system produced by primary cold X-rays, secondary photons and electrons are calculated as a function of depth. The discontinuity in power density profiles is observed at the interfaces of slabs due to discrete changes in stopping power between slab materials. The power density is found to be higher in slab materials with higher mass density. The power density profiles are then used in the atomistic Momentum Scaling Model (MSM) coupled with the Molecular Dynamics (MD) method (MSM-MD) to predict the spatiotemporal evolution of WDP in vacuum. The spatial and temporal distribution of density and temperature fields of expanding WDP is evaluated from the MSM-MD simulations. These modeling results provide insights into the underlining physics of the formation and spatiotemporal evolution of WDPs induced by cold X-rays

From Polygons to Ultradiscrete Painlev\'e Equations

The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the B\"acklund transformations and the evolution equations for the ultradiscrete Painlev\'e equations

Band-specific phase engineering for curving and focusing light in waveguide arrays

Band specific design of curved light caustics and focusing in optical waveguide arrays is introduced. Going beyond the discrete, tight-binding model, which we examined recently, we show how the exact band structure and the associated diffraction relations of a periodic waveguide lattice can be exploited to phase-engineer caustics with predetermined convex trajectories or to achieve optimum aberration-free focal spots. We numerically demonstrate the formation of convex caustics involving the excitation of Floquet-Bloch modes within the first or the second band and even multi-band caustics created by the simultaneous excitation of more than one bands. Interference of caustics in abruptly autofocusing or collision scenarios are also examined. The experimental implementation of these ideas should be straightforward since the required input conditions involve phase-only modulation of otherwise simple optical wavefronts. By direct extension to more complex periodic lattices, possibilities open up for band specific curving and focusing of light inside 2D or even 3D photonic crystals

Anosov subgroups: Dynamical and geometric characterizations

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.Comment: 88 page