Extension of a theorem of Duffin and Schaeffer

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$ $r_s(n)\}$ for each $n\geqslant 0$. We prove that if $F(z)$ is bounded in a sector of its disk of convergence, it is a rational function. This extends a very recent result of Tang and Wang, who gave the analogous result when the sequence $f(n)$ takes on values of finitely many polynomials.Comment: 2 page

Multiplicity of Closed Reeb Orbits on Prequantization Bundles

We establish multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. The results hold under certain index requirements on the contact form and are sharp for unit cotangent bundles of CROSS's. In particular, we generalize and put in the symplectic-topological context a theorem of Duan, Liu, Long, and Wang for the standard contact sphere. We also prove similar results for non-hyperbolic contractible closed orbits and briefly touch upon the multiplicity problem for degenerate forms. On the combinatorial side of the question, we revisit and reprove the enhanced common jump theorem of Duan, Long and Wang, and interpret it as an index recurrence result.Comment: 31 page

Removal of nitrogen pollutant from domestic wastewater

Water as a medium for waste transport would be easily contaminated by human activities. Many methods have been proposed to treat contaminated water to protect human health and biodiversity (Z. Daud et al., 2017). Due to upgrade the existing wastewater treatment plant facilities, the typically advanced technologies have been proposed to remove many types of pollutant, effectively (Tchobanoglous, Burton, & Stensel, 2004). The development of wastewater treatment plant needs to be considered leading economic indicators to have low operational and maintenance costs (Lewandowski, 2015; Shammas, Wang, & Wu, 2009). Aerobic digestion (AD) has been known since 1950 as biological wastewater treatment process to treat wastewater by removing the pollutants for instance colloids, organic compounds and suspended solids to avoid the excessive pollutants released into the receiving water (Shammas and Wang, 2007)

Comments on Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation prepresentation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim (cond-mat/0201285), the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49(1); the three-dimensional Ising model has z = 0.37(2).Comment: RevTeX, 4 pages, 5 figure