A tight lower bound instance for k-means++ in constant dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the closest previously $(i-1)$ chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii. There are datasets on which this seeding algorithm gives an approximation factor of $\Omega(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say $1/poly(k)$). Brunsch and R\"{o}glin gave a dataset where the k-means++ seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability that is exponentially small in $k$. However, this and all other known lower-bound examples are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability exponentially small in $k$. This solves open problems posed by Mahajan et al. and by Brunsch and R\"{o}glin.Comment: To appear in TAMC 2014. arXiv admin note: text overlap with arXiv:1306.420

Impact of surface-polish on the angular and wavelength dependence of fiber focal ratio degradation

We present measurements of how multimode fiber focal-ratio degradation (FRD) and throughput vary with levels of fiber surface polish from 60 to 0.5 micron grit. Measurements used full-beam and laser injection methods at wavelengths between 0.4 and 0.8 microns on 17 meter lengths of Polymicro FBP 300 and 400 micron core fiber. Full-beam injection probed input focal-ratios between f/3 and f/13.5, while laser injection allowed us to isolate FRD at discrete injection angles up to 17 degrees (f/1.6 marginal ray). We find (1) FRD effects decrease as grit size decreases, with the largest gains in beam quality occurring at grit sizes above 5 microns; (2) total throughput increases as grit size decreases, reaching 90% at 790 nm with the finest polishing levels; (3) total throughput is higher at redder wavelengths for coarser polishing grit, indicating surface-scattering as the primary source of loss. We also quantify the angular dependence of FRD as a function of polishing level. Our results indicate that a commonly adopted micro-bending model for FRD is a poor descriptor of the observed phenomenon.Comment: 10 pages, 7 figures, presented at SPIE Astronomical Telescopes and Instrumentation, July 201

Towards a Coulomb gas of instantons in the SO(4)xU(1) Higgs model on R_4

The $SO(4)\times U(1)$ Higgs model on $\R_4$ is extended by a $F^3$ term so that the action receives a nonvanishing contribution from the interactions of 2-instantons and 3-instantons, and can be expressed as the inverse of the Laplacian on $\R_4$ in terms of the mutual distances of the instantons. The one-instanton solutions of both the basic and the extended models have been studied in detail numerically.Comment: 29 pages LaTeX, 4 Figures available from authors on reques

Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Plasmonic nanoparticle enhanced photocurrent in GaN/InGaN/GaN quantum well solar cells

We demonstrate enhanced external quantum efficiency and current-voltage characteristics due to scattering by 100 nm silver nanoparticles in a single 2.5 nm thick InGaN quantum well photovoltaic device. Nanoparticle arrays were fabricated on the surface of the device using an anodic alumina template masking process. The Ag nanoparticles increase light scattering, light trapping, and carrier collection in the III-N semiconductor layers leading to enhancement of the external quantum efficiency by up to 54%. Additionally, the short-circuit current in cells with 200 nm p-GaN emitter regions is increased by 6% under AM 1.5 illumination. AFORS-Het simulation software results were used to predict cell performance and optimize emitter layer thickness

Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2_2

The efficiency of radial transport of icy solid material from outer disk to the inner disk is currently unconstrained. Efficient radial transport of icy dust grains could significantly alter the composition of the gas in the inner disk. Our aim is to model the gaseous CO$_2$ abundance in the inner disk and use this to probe the efficiency of icy dust transport in a viscous disk. Features in the simulated CO$_2$ spectra are investigated for their dust flux tracing potential. We have developed a 1D viscous disk model that includes gas and grain motions as well as dust growth, sublimation and freeze-out and a parametrisation of the CO$_2$ chemistry. The thermo-chemical code DALI was used to model the mid-infrared spectrum of CO$_2$, as can be observed with JWST-MIRI. CO$_2$ ice sublimating at the iceline increases the gaseous CO$_2$ abundance to levels equal to the CO$_2$ ice abundance of $\sim 10^{-5}$, which is three orders of magnitude more than the gaseous CO$_2$ abundances of $\sim 10^{-8}$ observed by Spitzer. Grain growth and radial drift further increase the gaseous CO$_2$ abundance. A CO$_2$ destruction rate of at least $10^{-11}$ s$^{-1}$ is needed to reconcile model prediction with observations. This rate is at least two orders of magnitude higher than the fastest known chemical destruction rate. A range of potential physical mechanisms to explain the low observed CO$_2$ abundances are discussed. Transport processes in disks can have profound effects on the abundances of species in the inner disk. The discrepancy between our model and observations either suggests frequent shocks in the inner 10 AU that destroy CO$_2$, or that the abundant midplane CO$_2$ is hidden from our view by an optically thick column of low abundance CO$_2$ in to the disk surface XDR/PDR. Other molecules, such as CH$_4$ or NH$_3$, can give further handles on the rate of mass transport.Comment: Accepted for publication in A&A, 18 pages, 13 figures, abstract abridge

Storage stability of whole and nibbed, conventional and high oleic peanuts (<i>Arachis hypogeae </i>L.)

Peanuts are increasingly being used as nibbed ingredients in cereal bars, confectionery and breakfast cereals. However, studies on their oxidative stability in this format are limited. Storage trials to determine the stability to oxidation were carried out on whole and nibbed kernels of conventional (CP) and high oleic (HOP) peanuts, with respect to temperature and modified atmosphere packaging. HOP exhibited the highest oxidative stability, with a lag phase in whole kernels of 12–15 weeks before significant oxidation occurred. HOP also showed higher levels of intrinsic antioxidants, a trolox equivalent antioxidant capacity (TEAC) of 70 mMol equivalence and radical scavenging percentage (RSP) of 99.8 % at the beginning of storage trials, whereas CP showed values of 40 mMol and 81.2 %, respectively. The intrinsic antioxidants at the beginning of these storage trials were shown to affect the peroxide value (PV), where RSP and TEAC decreased, and PV increased. Therefore, in peanuts the processing format (nibbed or whole) had the highest influence on susceptibility of lipid oxidation, highest to lowest importance: processing format &gt; temperature &gt; atmospheric conditions

A Nonlinear Super-Exponential Rational Model of Speculative Financial Bubbles

Keeping a basic tenet of economic theory, rational expectations, we model the nonlinear positive feedback between agents in the stock market as an interplay between nonlinearity and multiplicative noise. The derived hyperbolic stochastic finite-time singularity formula transforms a Gaussian white noise into a rich time series possessing all the stylized facts of empirical prices, as well as accelerated speculative bubbles preceding crashes. We use the formula to invert the two years of price history prior to the recent crash on the Nasdaq (april 2000) and prior to the crash in the Hong Kong market associated with the Asian crisis in early 1994. These complex price dynamics are captured using only one exponent controlling the explosion, the variance and mean of the underlying random walk. This offers a new and powerful detection tool of speculative bubbles and herding behavior.Comment: Latex document of 24 pages including 5 eps figure