The number of limit cycles for a family of polynomial systems

AbstractIn this paper, the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods. With the help of a computer algebra system (e.g., Maple 7.0), we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems (2.1) and (2.2) is 5n + 5 + (1 − (−1)n)/2 for c ≠ 0 and n for c ≡ 0

Numerical simulation on air distribution of a tennis hall in winter and evaluation on indoor thermal environment

Supplying air with ball spout air diffusers is a common air-conditioning system for air distribution in large space stadiums. When supplying hot air with ball spout diffusers in winter, the phenomenon of hot jet upturning may appear, so the design should consider adjusting the spout angle so as to control the rising airflow. The purpose of the paper is to predict and optimize the air distribution of a tennis hall in winter for the purpose of guiding the design and regulation of air-conditioning system. Based on the optimal scheme of summer conditions, using computational fluid dynamics (CFD) technique, the air distribution and indoor thermal environment of a tennis hall in winter were numerically simulated. Two conditions were considered discharging air with spouts downwards with a 30 degree slope and discharging air horizontally. Indoor thermal environment was evaluated from two case studies including the protection of the movement of the ball and thermal comfort of the human body, and consequently, the optimal design was then proposed. The results can provide some guidance for air distribution design and spout regulation in winter conditions of air-conditioning systems in similar tennis halls

Large Y3,2 Y_{3,2} -tilings in 3-uniform hypergraphs

Let $Y_{3,2}$ be the 3-graph with two edges intersecting in two vertices. We prove that every 3-graph $H$ on $n$ vertices with at least $\max \left \{ \binom{4\alpha n}{3}, \binom{n}{3}-\binom{n-\alpha n}{3} \right \}+o(n^3)$ edges contains a $Y_{3,2}$-tiling covering more than $4\alpha n$ vertices, for sufficiently large $n$ and $0<\alpha< 1/4$. The bound on the number of edges is asymptotically best possible and solves a conjecture of the authors for 3-graphs that generalizes the Matching Conjecture of Erd\H{o}s

Rainbow Hamilton cycle in hypergraph systems

R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di proved that every $n$-vertex $k$-graph $H$, $k\geq3, \gamma>0$ and $n$ is sufficiently large, with $\delta_{k-1}(H)\geq(1/2+\gamma)n$ contains a tight Hamilton cycle, which can be seen as a generalization of Dirac's theorem in hypergraphs. In this paper, we extend this result to the rainbow setting as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, a $k$-graph $G$ on $V$ is rainbow if $E(G)\subseteq\bigcup_{i\in[m]}E(H_i)$ and $|E(G)\cap E(H_i)|\leq 1$ for $i\in[m]$. Then we show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for $i\in[n]$, then there exists a rainbow tight Hamilton cycle.Comment: 20 pages,5 figure

Recirculating Light Phase Modulator

High efficiency and a compact footprint are desired properties for electro-optic modulators. In this paper, we propose, theoretically investigate and experimentally demonstrate a recirculating phase modulator, which increases the modulation efficiency by modulating the optical field several times in a non-resonant waveguide structure. The 'recycling' of light is achieved by looping the optical path that exits the phase modulator back and coupling it to a higher order waveguide mode, which then repeats its passage through the phase modulator. By looping the light back twice, we were able to demonstrate a recirculating phase modulator that requires nine times lower power to generate the same modulation index of a single pass phase modulator. This approach of modulation efficiency enhancement is promising for the design of advanced tunable electro optical frequency comb generators and other electro-optical devices with defined operational frequency bandwidths