Multi-scale theoretical approach to X-ray absorption spectra in disordered systems: an application to the study of Zn(II) in water

We develop a multi-scale theoretical approach aimed at calculating from first principles X-ray absorption spectra of liquid solutions and disordered systems. We test the method by considering the paradigmatic case of Zn(II) in water which, besides being relevant in itself, is also of interest for biology. With the help of classical molecular dynamics simulations we start by producing bunches of configurations differing for the Zn(II)-water coordination mode. Different coordination modes are obtained by making use of the so-called dummy atoms method. From the collected molecular dynamics trajectories, snapshots of a more manageable subsystem encompassing the metal site and two solvation layers are cut out. Density functional theory is used to optimize and relax these reduced system configurations employing a uniform dielectric to mimic the surrounding bulk liquid water. On the resulting structures, fully quantum mechanical X-ray absorption spectra calculations are performed by including core-hole effects and core-level shifts. The proposed approach does not rely on any guessing or fitting of the force field or of the atomic positions of the system. The comparison of the theoretically computed spectrum with the experimental Zn K-edge XANES data unambiguously demonstrates that among the different a priori possible geometries, Zn(II) in water lives in an octahedral coordination mode.Comment: 8 pages, 3 figure

Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

Circular groups, planar groups, and the Euler class

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientation-preserving C^1 action of Z oplus Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R^2 in every degree of smoothness.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper15.abs.htm

Optimized Planar Penning Traps for Quantum Information Studies

A one-electron qubit would offer a new option for quantum information science, including the possibility of extremely long coherence times. One-quantum cyclotron transitions and spin flips have been observed for a single electron in a cylindrical Penning trap. However, an electron suspended in a planar Penning trap is a more promising building block for the array of coupled qubits needed for quantum information studies. The optimized design configurations identified here promise to make it possible to realize the elusive goal of one trapped electron in a planar Penning trap for the first time - a substantial step toward a one-electron qubit

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017