Local Gromov-Witten Invariants are Log Invariants

We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.Comment: 15 pages, version accepted for publication in Advances in Mathematic

Sketch of the flow domain adopted in this study.

<p>Sketch of the flow domain adopted in this study.</p

Time evolution of the lamella until splashing emerges.

<p>A. Shape evolution for the impact at speed <i>U</i>_<i>0</i> = 8 m/s and the liquid dynamic viscosity <i>μ</i>_<i>L</i> = 40 cP. The other parameters are the same as that for <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0177390#pone.0177390.g003" target="_blank">Fig 3</a>. B. Onset time for the splash <i>t</i>_<i>onset</i> as a function of <i>Re</i> (solid symbols) and <i>We</i> (open symbols), as the drop radius is fixed at <i>a</i> = 1.6 mm. To make the two relationships plotted in one figure, the horizontal axis is 10 times reduced for <i>We</i>. The different colors correspond to different liquid dynamic viscosity <i>μ</i>_<i>L</i> = 9.4 to 40 cP and surface tension <i>σ</i> = 10 to 80 dynes/cm. The inset plots <i>t</i>_<i>onset</i> vs drop radius at the same impact condition as that for <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0177390#pone.0177390.g003" target="_blank">Fig 3</a> except that <i>U</i>_<i>0</i> = 10 m/s.</p

Droplet volume fraction during an impact.

<p>Volume fraction contours for a viscous silicone oil drop with diameter of 1.6 mm impacting at 4 m/s onto a smooth dry solid surface under the surrounding pressure of 34 kPa. The successive times are t = 0, 0.6<i>τ</i>, <i>τ</i>, 2<i>τ</i>, 3<i>τ</i> and 7.4<i>τ</i> where <i>τ</i> ≡ <i>a</i>/<i>U</i>_<i>0</i> is the characteristic time of impact, as listed in order from A to F.</p

Temporal and spatial evolution of drop splashing in the early stages of impact.

<p>A. Three regimes are found to be responsible for the different splashing features shown at the successive times t = <i>t</i>_<i>onset</i>, <i>τ</i>, 1.5<i>τ</i> and 3<i>τ</i>. B. Normalized radial distance of the outermost ejected droplet as a function of normalized time at different <i>Re</i> (solid symbols, <i>Re</i> = 602 to 3840) and <i>We</i> (open symbols, <i>We</i> = 1354 to 30082). Different symbol shapes denote different values for <i>Re</i> or <i>We</i>. The inset plots the dimensional radial speed of the outermost ejected droplets <i>U</i>_<i>e</i> vs the drop impact speed <i>U</i>_<i>0</i>. The unit for both the speeds is m/s.</p

Additional file 1 of Removal of leftover feed shapes environmental microbiota and limits houseflies-mediated dispersion of pathogenic bacteria in sow breeding farms

Supplementary Material 1: Supplementary Table 1. Alpha diversity of lactating sow leftover microbiota at different time spot