On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space ${\mathbb{R}}_1^{n+2}$, classifying all rotation null hypersurfaces in ${\mathbb{R}}_1^{n+2}$. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the $(n+1)$-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the $3$-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of ${\mathbb{R}}_1^4$ and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.     

A new characterization of the Berger sphere in complex projective space

We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group ${\text{Nil}}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb{R})$ and the Lie group ${\text{Sol}}_3$) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space.     

Numerically anticanonical divisors on Kato surfaces

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb{Q})$. We present a way to compute the associated rational coefficients of the first Chern class $c_1\left(S\right)$. We get in particular a simple geometric obstruction for $c_1\left(S\right)$ to be an integral class, or equivalently index $\left(S\right)=1$. In the final part we discuss relations with some recent work of Dloussky (2011) and Oeljeklaus and Toma (2009).     

Bulk and shear mechanical loss of titania-doped tantala

We report on the mechanical loss from bulk and shear stresses in thin film, ion beam deposited, titania-doped tantala. The numerical values of these mechanical losses are necessary to fully calculate the Brownian thermal noise in precision optical cavities, including interferometric gravitational wave detectors like LIGO. We found the values from measuring the normal mode mechanical quality factors, Q's, in the frequency range of about 2000-10,000 Hz, of silica disks coated with titania-doped tantala coupled with calculating the elastic energy in shear and bulk stresses in the coating using a finite element model. We fit the results to both a frequency independent and frequency dependent model and find ${?}_{\mathrm{shear}}=(8.3\pm 1.1)\times {10}^{?4}$, ${?}_{\mathrm{bulk}}=(6.6\pm 3.8)\times {10}^{?4}$ with a frequency independent model and ${?}_{\mathrm{shear}}\left(f\right)=(5.0\pm 0.7)\times {10}^{?4}+(5.4\pm 1.1)\times {10}^{?8}f$, ${?}_{\mathrm{bulk}}\left(f\right)=(11\pm 2.8)\times {10}^{?4}?(8.7\pm 4.7)\times {10}^{?8}f$ with a frequency dependent (linear) model. The ratio of these values suggest that modest improvement in the coating thermal noise may be possible in future gravitational wave detector optics made with titania-doped tantala as the high index coating material by optimizing the coating design to take advantage of the two different mechanical loss angles.     

Luminescent, optical and color properties of natural rose quartz

Rose quartz is an interesting mineral with numerous impurities that have been studied by scanning electron microscopy (SEM), X-ray fluorescence (XRF), X-ray diffraction (XRD), cathodoluminescence (CL), ion beam luminescence (IBL), radioluminescence (RL), thermoluminescence (TL) and optical absorption (OA). After HF etching, rose quartz from Oliva de Plasencia (Caceres, Spain) shows under SEM the presence of other silicate phases such as dumortierite $[{\mathrm{Al}}_{6.5-7}({\mathrm{BO}}_3)({\mathrm{SiO}}_4{)}_3(\mathrm{O},\mathrm{OH}{)}_3]$. The OA spectrum of rose quartz suggests that these inclusions are the cause of coloration of rose quartz. The luminescence (CL, IBL, RL, TL) spectra behavior, at both room temperature and lower, confirms that the $\sim 340\mathrm{nm}$ emission could be associated with Si–O strain structures, including non-bridging oxygen or silicon vacancy–hole centers; the observed $\sim 400\mathrm{nm}$ emission could be associated with recombination of a hole trapped adjacent to a substitutional, charge-compensated aluminum alkali ion center; the $\sim 500\mathrm{nm}$ emission could be associated with substitutional ${\mathrm{Al}}^{3+}$ and the $\sim 700\mathrm{nm}$ peak could be associated with ${\mathrm{Fe}}^{3+}$ point defects in ${\mathrm{Si}}^{4+}$ sites. These results suggest that, while defect properties of rose quartz are not greatly dissimilar to those of purer forms of quartz and silica, further research seems necessary to determine criteria for the evolution of the newly-formed self-organized microstructures in the rose quartz lattice under irradiation.     

Pressure dependence of negative thermal expansion in Zr2(WO4)(PO4)2

Negative thermal expansion materials can experience significant stresses when they are used in composites. Under ambient conditions Zr₂(WO₄)(PO₄)₂ displays anisotropic negative thermal expansion (NTE) (${\alpha }_v=?14.0\left(10\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_a=?7.9\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_b=2.5\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_c=?8.7\left(2\right)\times 10^{?6}{K}^{?1}$ at 0 GPa). The effect of hydrostatic pressure on its thermal expansion characteristics was investigated by neutron diffraction between 300 and 60 K at pressures up to 0.3 GPa. No phase transitions were observed in the pressure and temperature range examined. The material was found to have a bulk modulus, B₀, of 61.3(8) GPa at ambient temperature, and unlike some other NTE materials, pressure had no detectable effect on thermal expansion (${\alpha }_v=?14.2\left(8\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_a=?7.9\left(3\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_b=2.9\left(5\right)\times 10^{?6}{K}^{?1}$, ${\alpha }_c=?9.2\left(2\right)\times 10^{?6}{K}^{?1}$ at 0.3 GPa).     

On the non-integrability of a generalized Darboux Halphen system

In this paper we study a generalized Darboux Halphen system given by ${\dot{x}}_1=x_2{x}_3-x_1(x_2+x_3)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$${\dot{x}}_2=x_3{x}_1-x_2(x_3+x_1)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$${\dot{x}}_3=x_1{x}_2-x_3(x_1+x_2)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$ where $x_1$, $x_2$, $x_3$ are real variables, ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are real constants and ${\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)={\alpha }_1^2(x_1-x_2)(x_3-x_1)+{\alpha }_2^2(x_2-x_3)(x_1-x_2)+{\alpha }_3^2(x_3-x_1)(x_2-x_3)\text{.}$ We prove that, for any $({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{R}}^3\setminus \left\{(0,0,0)\right\}$, this system does not admit any non-constant global first integral that can be described by a formal power series. Furthermore, restricting the values of $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ to a full Lebesgue measure set, we prove that this system does not admit any non-constant rational or Darbouxian global first integral. This is a first step toward proving that this system is chaotic.