Two-Generator Free Kleinian Groups and Hyperbolic Displacements

The $\log 3$ Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least $\log 3$ by one of the non-commuting isometries $\xi$ or $\eta$ provided that $\xi$ and $\eta$ generate a torsion-free, discrete group which is not co-compact and contains no parabolic. This theorem lies in the foundation of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds whose fundamental group has no 2-generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds. In this paper, we show that every point in the hyperbolic 3-space is moved a distance at least $(1/2)\log(5+3\sqrt{2})$ by one of the isometries in $\{\xi,\eta,\xi\eta\}$ when $\xi$ and $\eta$ satisfy the conditions given in the $\log 3$ Theorem.Comment: 43 Pages. 2 figures. Almost completely rewritten in line with the referee's recommendations. By Lemma 4.10, suggested by the referee, the proofs of many lemmas are substantially shortened. The main theorem, Theorem 5.1, is reproved with a more geometric approac

Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $\Gamma$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds for the maximum of hyperbolic displacements given by symmetric subsets of isometries in purely loxodromic finitely generated free Kleinian groups are conjectured.Comment: 62 pages, 3 Figures, Comments are welcom

Jorgensen's Inequality and Purely Loxodromic 2-Generator Free Kleinian Groups

Let $\xi$ and $\eta$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $\Gamma=\langle\xi,\eta\rangle$ is a purely loxodromic free Kleinian group. For $\gamma\in\Gamma$ and $z\in\mathbb{H}^3$, let $d_{\gamma}z$ denote the distance between $z$ and $\gamma\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geodesic segments connecting the axes of $\xi$, $\eta\xi\eta^{-1}$ and $\eta^{-1}\xi\eta$, respectively. In this manuscript it is proved that if $d_{\gamma}z_2<1.6068...$ for every $\gamma\in\{\eta, \xi^{-1}\eta\xi, \xi\eta\xi^{-1}\}$ and $d_{\eta\xi\eta^{-1}}z_2\leq d_{\eta\xi\eta^{-1}}z_1$, then $$|\text{trace}^2(\xi)-4|+|\text{trace}(\xi\eta\xi^{-1}\eta^{-1})-2|\geq 2\sinh^2\left(\tfrac{1}{4}\log\alpha\right) = 1.5937....$$ Above $\alpha=24.8692...$ is the unique real root of the polynomial $21 x^4 - 496 x^3 - 654 x^2 + 24 x + 81$ that is greater than $9$. Also generalisations of this inequality for finitely generated purely loxodromic free Kleinian groups are conjectured.Comment: A contradiction with Theorem 4.1 in v3, named as Theorem 4.2 in this version, arose while rephrasing of Theorem 4.2 in v3. This was fixed by restating Theorem 4.2, which was named as Theorem 4.3 in this version. Lemma 4.1 is due to the anonymous referee. Conjecture 4.2 was also restated accordingly. No changes occurred in the computations otherwise. 26 pages, 3 Figure

TRANSATLANTIC TRADE AND INVESTMENT PARTNERSHIP (TTIP): IMPACT AND CORRELATION ANALYSIS BASED ON TRADE AND INVESTMENT BETWEEN TURKEY AND EU, AND THE USA

The main objective of this study is to ascertain the importance and necessity of Turkey, based on trade and investment, in TTIP agreement as a partner country with both European Union (EU) and the USA. Here, it uses secondary data sources — Central Bank of the Republic of Turkey (CBRT) and Turkish Statistical Institute — from 2001 to 2016. To obtain vibrant findings, the study introduces two statistical analyses: correlation and regression. In addition, it elucidates descriptive analysis for comprehending the general trade and investment scenario. The results derived from correlation suggest that there is a strong positive correlation between Turkey’s total foreign trade and its export to and import from EU and the USA, Turkey’s total FDI and EU’s FDI in Turkey, as well as Turkey’s total FDI and the USA’s FDI in Turkey. In addition, a strong positive correlation has been found between Turkey’s total resident FDI in abroad and Turkey’s FDI in EU and the USA. On the other hand, the regression results indicate that Turkey’s trade and investment with EU and the USA have a significant impact on Turkey’s total foreign trade and investmen

Interplay between physical cleaning, membrane pore size and fluid rheology during the evolution of fouling in membrane bioreactors

[EN] Fouling is one of the most pressing limitations during operation of membrane bioreactors, as it increases operating costs and is the cause of short membrane lifespans. Conducting effective physical cleanings is thus essential for keeping membrane operation above viable performance limits. The nature of organic foulants present in the sludge and the membrane properties are among the most influential factors determining fouling development and thus, efficiency of fouling mitigation approaches. The role of other factors like sludge viscosity on fouling is still unclear, given that contradictory effects have been reported in the literature. In the present study we use a new research approach by which the complex interplay between fouling type, levels of permeate flux, membrane material and feed properties is analyzed, and the influence of these factors on critical flux and membrane permeability is evaluated. A variety of systems including activated sludge and model solutions with distinct rheological behavior has been investigated for two membranes differing in pore size distribution. We present a novel method for assessing the efficiency of fouling removal by backwash and compare it with the efficiency achieved by means of relaxation. Results obtained have proven that backwash delays development of critical fouling as compared with relaxation and reduces fouling irreversibility regardless of fluid rheology. It was shown that backwash is especially effective for membranes for which internal fouling is the main cause of loss in permeability. Nonetheless, we found out that for membranes with tight pores, both relaxation and backwash are equally effective. The critical flux decreases significantly for high-viscosity fluids, such as activated sludge. This effect is mainly caused by an intensified concentration polarization at the feed side rather than by internal fouling events. However, membrane permeability has been proven to rely more on the permeate viscosity than on the feed viscosity: poor rejection of organic fractions showcasing high viscosity causes an acute decline in membrane permeability as a consequence of increased shear stress inside the membrane pores. (C) 2018 Elsevier Ltd. All rights reserved.M.W. acknowledges the support through an Alexander-von-Humboldt Professorship. M.C. Marti-Calatayud acknowledges the support to Generalitat Valenciana through the funding APOSTD2017. M.C. Marti-Calatayud thanks the contributions of Sybille Hanisch, Sanchita Khandelwal and Sara Vivanco. This work was supported by the German Federal Ministry of Education and Research (BMBF) through the project BRAMAR (02WCL1334A). We thank Synder Filtration for the supplied membranesMartí Calatayud, MC.; Schneider, S.; Yüce, S.; Wessling, M. (2018). Interplay between physical cleaning, membrane pore size and fluid rheology during the evolution of fouling in membrane bioreactors. Water Research. 147:393-402. https://doi.org/10.1016/j.watres.2018.10.017S39340214

Measurement of the profiles of disorder-induced localized resonances in photonic crystal waveguides by local tuning

Near the band edge of photonic crystal waveguides, localized modes appear due to disorder. We demonstrate a new method to elucidate spatial profile of the localized modes in such systems using precise local tuning. Using deconvolution with the known thermal profile, the spatial profile of a localized mode with quality factor ($Q$) $>10^5$ is successfully reconstructed with a resolution of $2.5 \ \mu$m

Lasing from single, stationary, dye-doped glycerol/water microdroplets located on a superhydrophobic surface

We report laser emission from single, stationary, Rhodamine B-doped glycerol/water microdroplets located on a superhydrophobic surface. In the experiments, a pulsed, frequency-doubled Nd:YAG laser operating at 532 nm was used as the excitation source. The microdroplets ranged in diameter from a few to 20 um. Lasing was achieved in the red-shifted portion of the dye emission spectrum with threshold fluences as low as 750 J/cm2. Photobleaching was observed when the microdroplets were pumped above threshold. In certain cases, multimode lasing was also observed and attributed to the simultaneous lasing of two modes belonging to different sets of whispering gallery modes.Comment: to appear in Optics Communication

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite{Mul}. One-parameter planar homothetic motion was defined in the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion is defined in the Hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale $h$ identically equal to 1, the results given in \cite{Yuc} are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system

Common coeliacomesenteric trunk: a computed tomography radiological study

Background: There is an increasing trend for administration of invasive radiological interventions, laparoscopic surgery, and transplantation procedures in recent years, and determining the vascular variations prior to these procedures is crucially important. Coeliacomesenteric trunk (CMT) is among these variations. This study aimed to retrospectively evaluate this rare anomaly by computed tomography (CT). Materials and methods: A total of 1000 CT angiography images were analysed retrospectively, and the patients with mesenteric and coeliac arteries arising from the abdominal aorta with a single root were identified. The level that CMT arose, and its branching patterns were determined individually for all patients. Results: Ten patients (6 males and 4 females) with a mean age of 50.2 years (17–87 years) had CMT in CT images. Conclusions: The knowledge of variations in the CMT prior to vascular or laparoscopic interventions will contribute to early intervention in case of a complication, or to avoid from a potential damage