Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$ points in \RR^3 is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k,d\in \NN, $1\leq k \leq d$, the maximum number of $k$-dimensional simplices of minimum (nonzero) volume spanned by $n$ points in \RR^d is $\Theta(n^k)$. (ii) The number of unit-volume tetrahedra determined by $n$ points in \RR^3 is $O(n^{7/2})$, and there are point sets for which this number is $\Omega(n^3 \log \log{n})$. (iii) For every d\in \NN, the minimum number of distinct volumes of all full-dimensional simplices determined by $n$ points in \RR^d, not all on a hyperplane, is $\Theta(n)$.Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 200

Magnetic Field Response and Chiral Symmetry of Time Reversal Invariant Topological Superconductors

We study the magnetic field response of the Majorana Kramers pairs of a one-dimensional time-reversal invariant (TRI) superconductors (class DIII) with or without a coexisting chirality symmetry. For unbroken TR and chirality invariance the parameter regimes for nontrivial values of the (Z_2) DIII-invariant and the (Z) chiral invariant coincide. However, broken TR may or may not be accompanied by broken chirality, and if chiral symmetry is unbroken, the pair of Majorana fermions (MFs) at a given end survives the loss of TR symmetry in an entire plane perpendicular to the spin-orbit coupling field. Conversely, we show that broken chirality may or may not be accompanied by broken TR, and if TR is unbroken, the pair of MFs survives the loss of broken chirality. In addition to explaining the anomalous magnetic field response of all the DIII class TS systems proposed in the literature, we provide a realistic route to engineer a "true" TR-invariant TS, whose pair of MFs at each end is split by an applied Zeeman field in arbitrary direction. We also prove that, quite generally, the splitting of the MFs by TR-breaking fields in TRI superconductors is highly anisotropic in spin space, even in the absence of the topological chiral symmetry.Comment: 4+ pages, 3 figures, slightly re-written, citations adde

Cognitive Radio Simultaneous Spectrum Access/ One-shot Game Modelling

The aim of this work is to asses simultaneous spectrum access situations that may occur in Cognitive Radio (CR) environments. The approach is that of one shot, noncooperative games describing CR interactions. Open spectrum access scenarios are modelled based on continuous and discrete reformulations of the Cournot game theoretical model. CR interaction situations are described by Nash and Pareto equilibria. Also, the heterogeneity of players is captured by the new concept of joint Nash-Pareto equilibrium, allowing CRs to be biased toward different types of equilibrium. Numerical simulations reveal equilibrium situations that may be reached in simultaneous access scenarios of two and three users.Comment: 6 double-column pages, 8 figures, CSNDSP 2012. arXiv admin note: substantial text overlap with arXiv:1207.3365, arXiv:1209.5387, arXiv:1209.501

Topological thermoelectric effects in spin-orbit coupled electron and hole doped semiconductors

We compute the intrinsic contributions to the Berry-phase mediated anomalous Hall and Nernst effects in electron- and hole-doped semiconductors in the presence of an in-plane magnetic field as well as Rashba and Dresselhaus spin orbit couplings. For both systems we find that the regime of chemical potential which supports the topological superconducting state in the presence of superconducting proximity effect can be characterized by plateaus in the topological Hall and Nernst coefficients flanked by well-defined peaks marking the emergence of the topological regime. The plateaus arise from a clear momentum space separation between the region where the Berry curvature is peaked (at the `near-band-degeneracy' points) and the region where the single (or odd number of) Fermi surface lies in the Brillouin zone. The plateau for the Nernst coefficient is at vanishing magnitudes surrounded by two peaks of opposite signs as a function of the chemical potential. These results could be useful for experimentally deducing the chemical potential regime suitable for realizing topological states in the presence of proximity effect.Comment: 8 pages, 8 figure

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and Combinatorics; 18 pages, 13 figure

Equivalence of topological mirror and chiral superconductivity in one dimension

Recently it has been proposed that a unitary topological mirror symmetry can stabilize multiple zero energy Majorana fermion modes in one dimensional (1D) time reversal (TR) invariant topological superconductors. Here we establish an exact equivalence between 1D "topological mirror superconductivity" and chiral topological superconductivity in BDI class which can also stabilize multiple Majorana-Kramers pairs in 1D TR-invariant topological superconductors. The equivalence proves that topological mirror superconductivity can be understood as chiral superconductivity in the BDI symmetry class co-existing with time-reversal symmetry. Furthermore, we show that the mirror Berry phase coincides with the chiral winding invariant of the BDI symmetry class, which is independent of the presence of the time-reversal symmetry. Thus, the time-reversal invariant topological mirror superconducting state may be viewed as a special case of the BDI symmetry class in the well-known Altland-Zirnbauer periodic table of free fermionic phases. We illustrate the results with the examples of 1D spin-orbit coupled quantum wires in the presence of nodeless s_{\pm} superconductivity and the recently discussed experimental system of ferromagnetic atom (Fe) chains embedded on a lead (Pb) superconductor.Comment: 5+ pages, 1 figur