A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

We study novel variations of Voronoi games (and associated random processes) that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios in a geometric setting. Our work can be seen as a natural variation of the classical framework of Hotelling. While games based on Voronoi diagrams have appeared recently in the literature, our variations appear to be the first that take into account a notion of limited choices and the corresponding asymmetries in player strategies. As a problem example, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points in the unit square (or, for symmetry, the unit torus), so the kth miner will have associated points pk1, pk2, . . . , pkm. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Generally the points will be distinct for each miner. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player’s goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games? We provide several results on games in this setting. For example, we show that a correlated equilibrium can be found in polynomial time for the natural 1, 2, and 3-dimensional variations of the problem. For the one-dimensional version of the problem on the unit circle, we show that a pure Nash equilibrium exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists when each player owns the arc starting from their point clockwise to the next point. We also consider a random version of the game, where players have m points chosen uniformly at random. We derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game in this setting using interesting properties of random arc lengths on circles.

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation symmetries. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a Z2 index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via vortex-line condensation from a 3d superfluid to achieve all three root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are rigorously derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to ZN symmetries and discuss potential SPT phases beyond the group cohomology classification.

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the spacetime includes horizons, where the Hamiltonian fails to be elliptic.

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.