This paper presents a 2×2 dynamical system to study the cyclic appearance and disappearance of the gas phase in a two-component (CO2, H2O), two phase (gas, liquid) flow in a single pore. Depending on injection rate, linearization of the dynamical system around equilibrium shows that the gas phase can exhibit two behaviors, either cyclically vanishing and appearing, or approaching a steady-state volume value. Numerical simulations were also run on the fully non-linear dynamical system to verify the results of the linearized model.

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

In this paper, we prove that there exist noncollision singularites in a planar four-body problem, thus settles the Painleve conjecture.

This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B) 6= ∅ and P(B′) = ∅ whenever B′ $ B, where P(B) is the set of all periodic patterns on Z2 generated by B. Given a set of Wang tiles B, write B = C1 ∪ C2 ∪ · · · ∪ Ck ∪ N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1 ∪ C2 ∪ · · · ∪ Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1 ∪ C2 ∪ · · · ∪ Ck. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles B, h(B) is positive if and only if B contains an MPE set, and h(B) is zero if and only if B is a subset of an SZE set.

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