Self-affine Manifolds

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors. A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody.Comment: 40 pages, 13 figures, 2 table

Soluble Adenylyl Cyclase Is Localized to Cilia and Contributes to Ciliary Beat Frequency Regulation via Production of cAMP

Ciliated airway epithelial cells are subject to sustained changes in intracellular CO2/HCO3− during exacerbations of airway diseases, but the role of CO2/HCO3−-sensitive soluble adenylyl cyclase (sAC) in ciliary beat regulation is unknown. We now show not only sAC expression in human airway epithelia (by RT-PCR, Western blotting, and immunofluorescence) but also its specific localization to the axoneme (Western blotting and immunofluorescence). Real time estimations of [cAMP] changes in ciliated cells, using FRET between fluorescently tagged PKA subunits (expressed under the foxj1 promoter solely in ciliated cells), revealed CO2/HCO3−-mediated cAMP production. This cAMP production was specifically blocked by sAC inhibitors but not by transmembrane adenylyl cyclase (tmAC) inhibitors. In addition, this cAMP production stimulated ciliary beat frequency (CBF) independently of intracellular pH because PKA and sAC inhibitors were uniquely able to block CO2/HCO3−-mediated changes in CBF (while tmAC inhibitors had no effect). Thus, sAC is localized to motile airway cilia and it contributes to the regulation of human airway CBF. In addition, CO2/HCO3− increases indeed reversibly stimulate intracellular cAMP production by sAC in intact cells

Quantum erasure using entangled surface acoustic phonons

Using the deterministic, on-demand generation of two entangled phonons, we demonstrate a quantum eraser protocol in a phononic interferometer where the which-path information can be heralded during the interference process. Omitting the heralding step yields a clear interference pattern in the interfering half-quanta pathways; including the heralding step suppresses this pattern. If we erase the heralded information after the interference has been measured, the interference pattern is recovered, thereby implementing a delayed-choice quantum erasure. The test is implemented using a closed surface-acoustic-wave communication channel into which one superconducting qubit can emit itinerant phonons that the same or a second qubit can later re-capture. If the first qubit releases only half of a phonon, the system follows a superposition of paths during the phonon propagation: either an itinerant phonon is in the channel, or the first qubit remains in its excited state. These two paths are made to constructively or destructively interfere by changing the relative phase of the two intermediate states, resulting in a phase-dependent modulation of the first qubit's final state, following interaction with the half-phonon. A heralding mechanism is added to this construct, entangling a heralding phonon with the signalling phonon. The first qubit emits a phonon herald conditioned on the qubit being in its excited state, with no signaling phonon, and the second qubit catches this heralding phonon, storing which-path information which can either be read out, destroying the signaling phonon's self-interference, or erased.Comment: 16 pages, 8 figure