Counting Labelled Trees with Given Indegree Sequence

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $\lambda$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page

Dynamical evolution of an effective two-level system with PT symmetry

We investigate the dynamics of parity- and time-reversal (PT ) symmetric two-energy-level atoms in the presence of two optical and a radio-frequency (rf) fields. The strength and relative phase of fields can drive the system from unbroken to broken PT symmetric regions. Compared with the Hermitian model, Rabi-type oscillation is still observed, and the oscillation characteristics are also adjusted by the strength and relative phase in the region of unbroken PT symmetry. At exception point (EP), the oscillation breaks down. To better understand the underlying properties we study the effective Bloch dynamics and find the emergence of the z components of the fixed points is the feature of the PT symmetry breaking and the projections in x-y plane can be controlled with high flexibility compared with the standard two-level system with PT symmetry. It helps to study the dynamic behavior of the complex PT symmetric model.Comment: 10 pages, 6 figures,to appear in CP

Numerical simulation of two-phase cross flow in the gas diffusion layer microstructure of proton exchange membrane fuel cells

The cross flow in the under-land gas diffusion layer (GDL) between 2 adjacent channels plays an important role on water transport in proton exchange membrane fuel cell. A 3-dimensional (3D) two-phase model that is based on volume of fluid is developed to study the liquid water-air cross flow within the GDL between 2 adjacent channels. By considering the detailed GDL microstructures, various types of air-water cross flows are investigated by 3D numerical simulation. Liquid water at 4 locations is studied, including droplets at the GDL surface and liquid at the GDL-catalyst layer interface. It is found that the water droplet at the higher-pressure channel corner is easier to be removed by cross flow compared with droplets at other locations. Large pressure difference Δp facilitates the faster water removal from the higher-pressure channel. The contact angle of the GDL fiber is the key parameter that determines the cross flow of the droplet in the higher-pressure channel. It is observed that the droplet in the higher-pressure channel is difficult to flow through the hydrophobic GDL. Numerical simulations are also performed to investigate the water emerging process from different pores of the GDL bottom. It is found that the amount of liquid water removed by cross flow mainly depends on the pore's location, and the water under the land is removed entirely into the lower-pressure channel by cross flow

Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast and the diffusion (white noise) term is small. The system is modeled by $$dX^{\epsilon,\delta}(t)=f(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dt+\sqrt{\delta}\sigma(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dW(t) , \ X^\epsilon(0)=x,$$ where $\alpha^\epsilon(t)$ is a finite state space Markov chain with irreducible generator $Q=(q_{ij})$. The relative changing rates of the switching and the diffusion are highlighted by the two small parameters $\epsilon$ and $\delta$. We associate to the system the averaged ODE $$d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x,$$ where $\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i$ and $(\nu_1,\dots,\nu_{m_0})$ is the unique invariant probability measure of the Markov chain with generator $Q$. Suppose that for each pair $(\epsilon,\delta)$ of parameters, the process has an invariant probability measure $\mu^{\epsilon,\delta}$, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. We are able to prove that if $\bar f$ has finitely many unstable or hyperbolic fixed points, then $\mu^{\epsilon,\delta}$ converges weakly to $\mu^0$ as $\epsilon\to 0$ and $\delta \to 0$. Our results generalize to the setting of state-dependent switching $$\mathbb{P}\{\alpha^\epsilon(t+\Delta)=j~|~\alpha^\epsilon=i, X^{\epsilon,\delta}(s),\alpha^\epsilon(s), s\leq t\}=q_{ij}(X^{\epsilon,\delta}(t))\Delta+o(\Delta),~~ i\neq j$$ as long as the generator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible for all $x\in\mathbb{R}^d$. We conclude our analysis by studying a predator-prey model.Comment: 40 page