Tiling Spaces are Inverse Limits

Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology. Then Omega is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Gamma. This result extends previous results of Anderson and Putnam, of Ormes, Radin and Sadun, of Bellissard, Benedetti and Gambaudo, and of G\"ahler. In particular, the construction in this paper is a natural generalization of G\"ahler's.Comment: Latex, 6 pages, including one embedded figur

Tilings, tiling spaces and topology

To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical properties of the space (such as mixing, or the spectrum of the translation operator) are closely related to bulk properties of the individual tilings (such as the diffraction pattern). The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed. Tiling spaces can be constructed as inverse limits of branched manifolds.Comment: 8 pages, including 2 figures, talk given at ICQ

Optical Coherence Tomography Angiography of the Optic Disc; an Overview.

Different diseases of the optic disc may be caused by or lead to abnormal vasculature at the optic nerve head. Optical coherence tomography angiography (OCTA) is a novel technology that provides high resolution mapping of the retinal and optic disc vessels. Recent studies have shown the ability of OCTA to visualize vascular abnormalities in different optic neuropathies. In addition, quantified OCTA measurements were found promising for differentiating optic neuropathies from healthy eyes

Distinguishing wet from dry age-related macular degeneration using three-dimensional computer-automated threshold Amsler grid testing

Background/aims: With the increased efficacy of current therapy for wet age-related macular degeneration (AMD), better ways to detect wet AMD are needed. This study was designed to test the ability of three-dimensional contrast threshold Amsler grid (3D-CTAG) testing to distinguish wet AMD from dry AMD. Methods: Conventional paper Amsler grid and 3D-CTAG tests were performed in 90 eyes: 63 with AMD (34 dry, 29 wet) and 27 controls. Qualitative comparisons were based upon the three-dimensional shapes of central visual field (VF) defects. Quantitative analyses considered the number and volume of the three-dimensional defects. Results: 25/34 (74%) dry AMD and 6/29 (21%) wet AMD eyes had no distortions on paper Amsler grid. Of these, 5/25 (20%) dry and 6/6 (100%) wet (p=0.03) AMD eyes exhibited central VF defects with 3D-CTAG. Wet AMD displayed stepped defects in 16/28 (57%) eyes, compared with only 2/34 (6%) of dry AMD eyes (p=0.002). All three volumetric indices of VF defects were two- to four-fold greater in wet than dry AMD (p<0.006). 3D-CTAG had 83.9% positive and 90.6% negative predictive values for wet AMD. Conclusions: 3D-CTAG has a higher likelihood of detecting central VF defects than conventional Amsler grid, especially in wet AMD. Wet AMD can be distinguished from dry AMD by qualitative and quantitative 3D-CTAG criteria. Thus, 3D-CTAG may be useful in screening for wet AMD, quantitating disease severity, and providing a quantitative outcome measure of therapy

Angioarchitectural evolution of clival dural arteriovenous fistulas in two patients.

Dural arteriovenous fistulas (dAVFs) may present in a variety of ways, including as carotid-cavernous sinus fistulas. The ophthalmologic sequelae of carotid-cavernous sinus fistulas are known and recognizable, but less commonly seen is the rare clival fistula. Clival dAVFs may have a variety of potential anatomical configurations but are defined by the involvement of the venous plexus just overlying the bony clivus. Here we present two cases of clival dAVFs that most likely evolved from carotid-cavernous sinus fistulas

The geometry of entanglement: metrics, connections and the geometric phase

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.Comment: 42 page