Berge Sorting

In 1966, Claude Berge proposed the following sorting problem. Given a string of $n$ alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, rearrange the pegs into a string consisting of $\lceil\frac{n}{2}\rceil$ white pegs followed immediately by $\lfloor\frac{n}{2}\rfloor$ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ such {\em Berge 2-moves} for $n\geq 5$. Extending Berge's original problem, we consider the same sorting problem using {\em Berge $k$-moves}, i.e., moves which take $k$ adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge 3-moves for $n\not\equiv 0\pmod{4}$ and in $\lceil\frac{n}{2}\rceil+1$ Berge 3-moves for $n\equiv 0\pmod{4}$, for $n\geq 5$. In general, we conjecture that, for any $k$ and large enough $n$, the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge $k$-moves. This estimate is tight as $\lceil\frac{n}{2}\rceil$ is a lower bound for the minimum number of required Berge $k$-moves for $k\geq 2$ and $n\geq 5$.Comment: 10 pages, 2 figure

QM(DFT) and MD studies on formation mechanisms of C_(60) fullerenes

One of the most puzzling aspects of fullerenes is how such complicated symmetric molecules are formed from a gas of atomic carbons, namely, the atomistic or chemical mechanisms. Are the atoms added one by one or as molecules (C2, C3)? Is there a critical nucleus beyond which formation proceeds at gas kinetic rates? What determines the balance between forming buckyballs, buckytubes, graphite and soot? The answer to these questions is extremely important in manipulating the systems to achieve particular products. A difficulty in current experiments is that the products can only be detected on time scales of microseconds long after many of the important formation steps have been completed. Consequently, it is necessary to use simulations, quantum mechanics and molecular dynamics, to determine these initial states. Experiments serve to provide the boundary conditions that severely limit the possibilities. Using quantum mechanical methods (density functional theory (DFT)) we derived a force field (MSXX FF) to describe one-dimensional (rings) and two-dimensional (fullerene) carbon molecules. Combining DFT with the MSXX FF, we calculated the energetics for the ring fusion spiral zipper (RFSZ) mechanism for formation of C60 fullerenes. Our results shows that the RFSZ mechanism is consistent with the quantum mechanics (with a slight modification for some of the intermediates)

Fitting 3D Morphable Models using Local Features

In this paper, we propose a novel fitting method that uses local image features to fit a 3D Morphable Model to 2D images. To overcome the obstacle of optimising a cost function that contains a non-differentiable feature extraction operator, we use a learning-based cascaded regression method that learns the gradient direction from data. The method allows to simultaneously solve for shape and pose parameters. Our method is thoroughly evaluated on Morphable Model generated data and first results on real data are presented. Compared to traditional fitting methods, which use simple raw features like pixel colour or edge maps, local features have been shown to be much more robust against variations in imaging conditions. Our approach is unique in that we are the first to use local features to fit a Morphable Model. Because of the speed of our method, it is applicable for realtime applications. Our cascaded regression framework is available as an open source library (https://github.com/patrikhuber).Comment: Submitted to ICIP 2015; 4 pages, 4 figure

Band structures of II-VI semiconductors using Gaussian basis functions with separable ab initio pseudopotentials: Application to prediction of band offsets

We describe the implementation of a separable pseudopotential into the dual space approach for ab initio density-functional calculations using Gaussian basis functions. We apply this Gaussian dual space method (GDS/DFT) to the study of II-VI semiconductors (II=Zn, Cd, Hg; VI=S, Se, Te, Po). The results compare well with experimental data and demonstrate the general transferability of the separable pseudopotential. We also introduce a band-consistent tight-binding (BC-TB) model for calculating the bulk contributions to the valence-band offsets (VBO’s). This BC-TB approach yields good agreement with all-electron ab initio GDS/DFT results. Comparisons between BC-TB results of VBO obtained with and without p-d coupling demonstrate quantitatively the importance of d electrons and cation-d–anion-p coupling in II-VI systems. Agreement between ab initio results and experimental results is excellent