Applications of incidence bounds in point covering problems

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed. First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.Comment: 39 pages, 8 figure

LHC constraints on M_1/2 and m_0 in the semi-constrained NMSSM

Constraints from searches for squarks and gluinos at the LHC at sqrt{s}=8 TeV are applied to the parameter space of the NMSSM with universal squark/slepton and gaugino masses at the GUT scale, but allowing for non-universal soft Higgs mass parameters (the sNMSSM). We confine ourselves to regions of the parameter space compatible with a 125 GeV Higgs boson with diphoton signal rates at least as large as the Standard Model ones, and a dark matter candidate compatible with WMAP and XENON100 constraints. Following the simulation of numerous points in the m_0-M_{1/2} plane, we compare the constraints on the sNMSSM from 3-5 jets + missing E_T channels as well as from multijet + missing E_T channels with the corresponding cMSSM constraints. Due to the longer squark decay cascades, lower bounds on M_{1/2} are alleviated by up to 50 GeV. For heavy squarks at large m_0, the dominant constraints originate from multijet + missing E_T channels due to gluino decays via stop pairs.Comment: 18 pages, 2 Tables, 3 Figure

The same-sign top signature of R-parity violation

Baryonic R-parity violation could explain why low-scale supersymmetry has not yet been discovered at colliders: sparticles would be hidden in the intense hadronic activity. However, if the known flavor structures are any guide, the largest baryon number violating couplings are those involving the top/stop, so a copious production of same-sign top-quark pairs is in principle possible. Such a signal, with its low irreducible background and efficient identification through same-sign dileptons, provides us with tell-tale signs of baryon number violating supersymmetry. Interestingly, this statement is mostly independent of the details of the supersymmetric mass spectrum. So, in this paper, after analyzing the sparticle decay chains and lifetimes, we formulate a simplified benchmark strategy that covers most supersymmetric scenarios. We then use this information to interpret the same-sign dilepton searches of CMS, draw approximate bounds on the gluino and squark masses, and extrapolate the reach of the future 14 TeV runs.Comment: 32 pages, 12 figures, 3 tables, 1 appendi

Positive Gorenstein Ideals

We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring \RR[x] with socle in degree 2d, which when viewed as a linear functional on \RR[x]_{2d} is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work, our main tools are Cayley-Bacharach duality and elementary convex geometry