The paper presents the results of numerical solution of the Allen-Cahn equation with a non-local term. This equation originally mentioned by Rubinstein and Sternberg in 1992 is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve. We study this motion approximately by the mentioned PDE, generalize the problem by including anisotropy and discuss the computational results obtained.
A study on photosynthetic and yield effects of waterlogging of winter wheat at four stages of growth was conducted in specially designed experimental tanks during the 2007-2008 and 2008-2009 seasons. Compared with the control, waterlogging treatments at tillering and jointing-booting stages reduced photosynthetic rate (PN) and transpiration (E) significantly, it also decreased average leaf water-use efficiency (WUE, defined as the ratio of PN to E) by 3.3% and 3.4% in both years. All parameters returned quickly to the control level after soil was drained. Damage to the photosynthetic apparatus during waterlogging resulted in a lower Fv/Fm ratio, especially at the first two stages. A strong reduction in root length, root mass, root/shoot ratio, total dry mass, and leaf area index were observed. The responses from vegetative plants at tillering and jointing-booting stages were greater than in generative plants at onset of flowering and at milky stages. The number of panicles per hectare at tillering stage and the spikelet per panicle at the stages of jointing-booting and at onset of flowering were also significantly reduced by waterlogging, giving 8.2-11.3% decrease of the grain yield relative to the control in both years. No significant difference in yield components and a grain yield was observed between the control and treatments applied at milky stages. These responses, modulated by the environmental conditions prevailing during and after waterlogging, included negative effects on the growth, photosynthetic apparatus, and the grain yield in winter wheat, but the effect was strongly stage-dependent. and G. C. Shao ... [et al.].
We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, $\int _R{\mathrm d}\alpha (t) f(t)$, where $R$ is a compact interval of $\mathbb{R}^n$, $\alpha $ and $f$ are functions with values on $L(Z,W)$ and $Z$ respectively, and $Z$ and $W$ are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, $\int _R\alpha (t)\mathrm{d}f(t)$, as well as to unbounded intervals $R$.