Continued fractions in non-Euclidean imaginary quadratic fields

In the Euclidean imaginary quadratic fields, continued fractions have been used to give rational approximations to complex numbers since the late 19th century. A variety of algorithms have been proposed in the 130 years following their introduction, but none are applicable outside of the same five fields. Here we overcome the non-Euclidean obstacle. We show how continued fractions can be produced in any imaginary quadratic field, and we prove that they share many of the properties enjoyed by their classical forebear. The inspiration for the algorithm is a fractal arrangement of circles arising from subsets of $\text{GL}_2(\mathbb{C})$ acting on the Riemann sphere. The geometry of these arrangements reveals an analog of the Euclidean algorithm that points us toward a more general continued fraction.Comment: 47 pages, 20 figure

Fundamental polyhedra of projective elementary groups

For $O$ an imaginary quadratic ring, we compute a fundamental polyhedron of $\text{PE}_2(O)$, the projective elementary subgroup of $\text{PSL}_2(O)$. This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for $\text{PE}_2(O)$, show that it has infinite-index and is its own normalizer in $\text{PSL}_2(O)$, and split $\text{PSL}_2(O)$ into a free product with amalgamation that has $\text{PE}_2(O)$ as one of its factors.Comment: 6 pages, 1 figure, citations added in version

Probing the Protosolar Disk Using Dust Filtering at Gaps in the Early Solar System

Jupiter and Saturn formed early, before the gas disk dispersed. The presence of gap-opening planets affects the dynamics of the gas and embedded solids and halts the inward drift of grains above a certain size. A drift barrier can explain the absence of calcium aluminium rich inclusions (CAIs) in chondrites originating from parent bodies that accreted in the inner solar system. Employing an interdisciplinary approach, we use a $\mu$-X-Ray-fluorescence scanner to search for large CAIs and a scanning electron microscope to search for small CAIs in the ordinary chondrite NWA 5697. We carry out long-term, two-dimensional simulations including gas, dust, and planets to characterize the transport of grains within the viscous $\alpha$-disk framework exploring the scenarios of a stand-alone Jupiter, Jupiter and Saturn \textit{in situ}, or Jupiter and Saturn in a 3:2 resonance. In each case, we find a critical grain size above which drift is halted as a function of the physical conditions in the disk. From the laboratory search we find four CAIs with a largest size of $\approx$200$\,\mu$m. \Combining models and data, we provide an estimate for the upper limit of the $\alpha$-viscosity and the surface density at the location of Jupiter, using reasonable assumptions about the stellar accretion rate during inward transport of CAIs, and assuming angular momentum transport to happen exclusively through viscous effects. Moreover, we find that the compound gap structure in the presence of Saturn in a 3:2 resonance favors inward transport of grains larger than CAIs currently detected in ordinary chondrites.Comment: 16 pages, 10 figures, updated to match published version in Astrophysical Journa

Imaging through turbulence with a quadrature-phase optical interferometer

We present an improved technique for imaging through turbulence at visible wavelengths using a rotation shearing pupil-plane interferometer, intended for astronomical and terrestrial imaging applications. While previous astronomical rotation shearing interferometers have made only visibility modulus measurements, this interferometer makes four simultaneous measurements on each interferometric baseline, with phase differences of π/2 between each measurement, allowing complex visibility measurements (modulus and phase) across the entire input pupil in a single exposure. This technique offers excellent wavefront resolution, allowing operation at visible wavelengths on large apertures, is potentially immune to amplitude fluctuations (scintillation), and may offer superior calibration capabilities to other imaging techniques. The interferometer has been tested in the laboratory under weakly aberrating conditions and at Palomar Observatory under ordinary astronomical observing conditions. This research is based partly on observations obtained at the Hale Telescope

Diphyllobothrium sp. en Canis familiaris de la región del subtrópico argentino (Puerto Iguazú, Misiones)

This paper reports the first finding of Diphyllobothrium sp. eggs in Canis familiaris(domestic dog) from Puerto Iguazú, a subtropical city of Misiones province, Argentina. In 2013,two positive cases of Diphyllobothrium sp. eggs were detected during an annual parasitological survey of dogs. Dog feces were collected in vials containing 10 % formalin and processed using Telemann?s sedimentation and Sheather?s flotation techniques. The two cases were detected in rural areas of the municipality. Since Misiones is not a part of the endemic area of diphyllobothriasis and given the fact that it is located in the three-border area of Argentina, Brazil and Paraguay, we consider this finding of great importance to public health. We stress the need for updating the current knowledge about the life cycle of these parasites considering the range of intermediate and definitive hosts, their zoonotic potential, and the epidemiological situation in non-endemic areas.This paper reports the first finding of Diphyllobothrium sp. eggs in Canis familiaris (domestic dog) from Puerto Iguazú, a subtropical city of Misiones province, Argentina. In 2013, two positive cases of Diphyllobothrium sp. eggs were detected during an annual parasitological survey of dogs. Dog feces were collected in vials containing 10 % formalin and processed using Telemann's sedimentation and Sheather's flotation techniques. The two cases were detected in rural areas of the municipality. Since Misiones is not a part of the endemic area of diphyllobothriasis and given the fact that it is located in the three-border area of Argentina, Brazil and Paraguay, we consider this finding of great importance to public health. We stress the need for updating the current knowledge about the life cycle of these parasites considering the range of intermediate and definitive hosts, their zoonotic potential, and the epidemiological situation in non-endemic areas.Fil: Rivero, Maria Romina. Ministerio de Salud. Instituto Nacional de Medicina Tropical; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Motta, Carlos E.. Universidad Nacional de Rio Cuarto. Facultad de Agronomia y Veterinaria. Departamento de Patología Animal; ArgentinaFil: Salas, Martin Miguel. Ministerio de Salud. Instituto Nacional de Medicina Tropical; ArgentinaFil: Chiaretta, Alicia. Universidad Nacional de Rio Cuarto. Facultad de Agronomia y Veterinaria. Departamento de Patología Animal; ArgentinaFil: Salomón, Oscar Daniel. Ministerio de Salud. Instituto Nacional de Medicina Tropical; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

From stable to chaotic juggling: theory, simulation, and experiments

Recent results of dynamical systems theory are used to derive strong predictions concerning the global properties of a simplified model of a planar juggling robot. In particular, it is found that certain lower-order local (linearized) stability properties determine the essential global (nonlinear) stability properties, and that successive increments in the controller gain settings give rise to a cascade of stable period-doubling bifurcations that comprise a universal route to chaos. The theoretical predictions are verified by simulation and corroborated by experimental data from the juggling robot