Codes Cross-Correlation Impact on S-curve Bias and Data-Pilot Code Pairs Optimization for CBOC Signals

The aim of this paper is to analyze the impact of spreading codes cross-correlation on code tracking performance, and to optimize the data-pilot code pairs of Galileo E1 Open Service (OS) Composite Binary Offset Carrier (CBOC) signals. The distortion of the discriminator function (i.e., S-curve), due to data and pilot spreading codes cross-correlation properties, is evaluated when only the data or pilot components of CBOC signals are tracked, considering the features of the modulation schemes. Analyses show that the S-curve bias also depends on the receiver configuration (e.g., the tracking algorithm and correlator spacing). In this paper, two methods are proposed to optimize the data-pilot code pairs of Galileo E1 OS. The optimization goal is to obtain minimum average S-curve biases when tracking only the pilot components of CBOC signals for the specific correlator spacing. The S-curve biases after optimization processes are analyzed and compared with the un-optimized results. It is shown that the optimized data-pilot code pairs could significantly mitigate the intra-channel (i.e., data and pilot) codes cross-correlation，and then improve the code tracking performance of CBOC signals

KNbO3 single crystal growth by the radio frequency heating Czochralski method

A radio frequency heating Czochralski technique to obtain single crystal KNbO3 is first presented. The technological parameters of KNbO3 crystal growth by the Czochralski technique and its pulling conditions were studied in detail. The experiments on second harmonic generation using 1.06 micrometer Nd:YAG laser in KNbO3 have been conducted. The second harmonic efficiency for upconversion of KNbO3 is found to be as high as that of NaBa2Nb5O15. An automatic scanning measurement for the optical homogeneity of KNbO crystal is also described. KNbO3 is revealed to be a potentially useful nonlinear material for optical device applications

The Two Dimensional Liquid Crystal Droplet Problem with Tangential Boundary Condition

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove the boundary of the droplet is a chord-arc curve with its normal vector field in the VMO space, and its arc-length parametrization belongs to the Sobolev space $H^{3/2}$. In fact, the boundary curves of such droplets closely resemble the so-called Weil-Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is also studied

Uniform profile near the point defect of Landau-de Gennes model

For the Landau-de Gennes functional on 3D domains, $$I_\varepsilon(Q,\Omega):=\int_{\Omega}\left\{\frac12|\nabla Q|^2+\frac{1}{\varepsilon^2}\left( -\frac{a^2}{2}\mathrm{tr}(Q^2)-\frac{b^2}{3}\mathrm{tr}(Q^3)+\frac{c^2}{4}[\mathrm{tr}(Q^2)]^2 \right) \right\}\,dx,$$ it is well-known that under suitable boundary conditions, the global minimizer $Q_\varepsilon$ converges strongly in $H^1(\Omega)$ to a uniaxial minimizer $Q_*=s_+(n_*\otimes n_*-\frac13\mathrm{Id})$ up to some subsequence \e_n\rightarrow\infty , where $n_*\in H^1(\Omega,\mathbb{S}^2)$ is a minimizing harmonic map. In this paper we further investigate the structure of $Q_\varepsilon$ near the core of a point defect $x_0$ which is a singular point of the map $n_*$. The main strategy is to study the blow-up profile of $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ where $\{x_n\}$ are carefully chosen and converge to $x_0$. We prove that $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ converges in $C^2_{loc}(\mathbb{R}^n)$ to a tangent map $Q(x)$ which at infinity behaves like a ``hedgehog" solution that coincides with the asymptotic profile of $n_*$ near $x_0$. Moreover, such convergence result implies that the minimizer $Q_{\varepsilon_n}$ can be well approximated by the Oseen-Frank minimizer $n_*$ outside the $O(\varepsilon_n)$ neighborhood of the point defect