Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Development of an improved divergence‐free‐condition compensated coupled framework to solve flow problems with time‐varying geometries

In this article a coupled version of the improved divergence‐free‐condition compensated method will be proposed to simulate time‐varying geometries by direct forcing immersed boundary method. The proposed method can be seen as a quasi‐multi‐moment framework due to the fact that the momentum equations are discretized by both cell‐centered and cell‐face velocity. For simulating time‐varying geometries, a semi‐implicit iterative method is proposed for calculating the direct forcing terms. Treatments for suppressing spurious force oscillations, calculating drag/lift forces, and evaluating velocity and pressure for freshly cells will also be addressed. In order to show the applicability and accuracy, analytical as well as benchmark problems will be investigated by the present framework and compared with other numerical and experimental results.     

Gold-Catalyzed Divergent Ring-Opening Rearrangement of Cyclopropenes Enabled by Dichotomous Gold−Carbenes

The gold-catalyzed ring-opening rearrangement of cyclopropenes affords an efficient route to either polysubstituted naphthols or aryl-substituted furans. Owing to the unique dichotomy of gold−carbenes, this protocol provides a switchable reaction selectivity between naphthols and furans enabled by the use of TFP−Au(MeCN)SbF₆ (tri(2-furyl) phosphine) or PNP(AuNTf₂)₂ (bis(diphenylphosphino)(isopropyl) amine) as catalysts respectively. It is proposed that the gold−carbene intermediate might be involved in the cyclopropene→naphthol rearrangement while the gold-carbocation is more likely to be involved in the cyclopropene→furan rearrangement.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.     

A note on transverse divergence and taut Riemannian foliations

In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward consequence. In the final part of the paper we obtain a tautness characterization for transversally oriented foliations with dense leaves.     

On modeling of MPS‐based multiphase fluid flow with low density ratios

A divergence‐free moving particle semi‐implicit framework has been formulated for modeling of multiple miscible fluids having small density ratios (≤ 1.10) in the presence of free surface. A density‐weighted pressure gradient along with a hybrid free‐surface treatment is specifically adopted to incorporate the effect of marginal density difference through a higher‐order kernel. The hybrid free‐surface treatment reduces error in velocity divergence and resulting spurious velocity fluctuation in the vicinity of free surface for low‐velocity system. Scalar transport equation is resolved to update density associated with the particles at every time step. A total of 9 cases of lock‐exchange flow with different lock positions and density ratios have been utilized to validate the proposed framework. The model performs satisfactorily irrespective of lock positions and low density variations (3%‐9%).     

On Default Priors for Robust Bayesian Estimation with Divergences

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum γ-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the γ-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.     

Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.     

固体介质间隔层激光标准具性能的影响因素研究

针对固体介质间隔层的镀膜标准具, 以1 064 nm激光镀膜标准具为例, 首先研究了反射膜堆数对反射带宽、基底厚度以及对自由光谱区的影响:标准具的带宽随着膜层堆数增加而减小, 自由光谱区随着基板厚度的增加而减小;其次研究了基底误差对标准具中心波长定位和透过率的影响, 通过定量数值计算证明了基底误差可通过标准具的使用角度补偿;针对典型的H(LH)^m/Substrate/(HL)^mH和L(LH)^m/Substrate/(HL)^mL两个膜系结构, 研究了入射激光发散角对标准具中心波长偏移、通带半宽度、中心波长透过率和最大透过率的影响。随着激光发散角的增加, 中心波长向短波方向移动, 通带半宽度、中心波长透过率和最大透过率呈现下降的趋势, 并且第二个膜系结构的标准具性能优于第一个膜系结构的标准具。     

915 nm宽条形半导体激光器输出特性

为了研究温升对915 nm宽条形应变量子阱半导体激光器输出特性的影响,搭建了基于半导体制冷片(TEC)的双向温控平台对其进行了测试。首先,改变激光器的外表面温度,测量其在不同注入电流时的光功率和波长,并利用CCD相机测量其慢轴发散角。然后,利用计算机仿真软件对激光器的工作状态进行稳态模拟,从而获得了其对应的热分布情况,通过将模拟得到的数据与实验测量的结果进行比较,获得了两者趋于一致的结论:当热功率从2.1 W升高至20.0 W时,慢轴发散角从2.6°增大至5.0°,同时波长发生红移,热透镜焦距减小;激光器波长随温度变化关系的系数约为0.4 nm/℃,器件热阻为1.5 K/W。因此,为了同时获得高的输出功率和稳定的输出波长,有必要将激光器外表面温度精确控制在某一数值,否则波长将会发生漂移;此外,在设计制作高功率半导体激光器时,通过适当增加条宽并采用散热良好的封装结构,可以减小对慢轴发散角的影响。