Third-order design of aspheric spectacle lenses

The theory of third-order aberrations for a system of rotationally symmetric aspheric thin lenses and relations for calculation of the shape of rotationally symmetric aspheric spectacle lenses with zero astigmatism are described. A comparison of aspheric and spherical spectacle lens design is presented. Possible applications of aspheric lens surfaces are proposed including the design of a spectacle lens with two types of primary aberrations corrected, an anastigmatic spectacle lens with minimum weight and a spectacle lens with the desired aesthetic shape. Our work performs a complex analysis of the third-order design of spectacle lenses considering not only astigmatism but all types of primary aberrations and their combinations.     

Self-adjoint realization of a class of third-order differential operators with an eigenparameter contained in the boundary conditions

The present paper deals with a class of third-order differential operators with eigenparameter dependent boundary conditions. Using operator theoretic formulation, the self-adjointness of this operator is proved, the properties of spectrum are investigated, its Green function and the resolvent operator are also obtained.     

Optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities

In this paper, optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities are investigated via Darboux transformation. We derive family of structures of rogue wave, including rogue waves with one peak and two valleys, bright rogue waves without valleys while with one peak or two peaks.     

Infinitely many solutions for a class of sublinear Schrodinger equation

In this paper, we investigate the Schr\"{o}dinger equation, which satisfies that the potential is asymptotical 0 at infinity in some measure-theoretic and the nonlinearity is sublinear growth. By using variant symmetric mountain lemma, we obtain infinitely many solutions for the problem. Moreover, if the nonlinearity is locally sublinear defined for $|u|$ small, we can also get the same result. In which, we show that these solutions tend to zero in $L^{\infty}(\mathbb{R}^{N})$ by the Br\"{e}zis-Kato estimate.     

Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties

This paper presents the concept of the Stochastic Multi-dimensional Harmonic Balance Method (Stochastic-MHBM) in order to solve dynamical problems with non-regular non linearities in presence of uncertainties. To treat the nonlinearity in the stochastic and frequency domains, the Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) is proposed. The approach is demonstrated using nonlinear two-degree-of-freedom model with different types of nonlinearities (cubic nonlinearity, contact/no contact, friction). The quasi-periodic stochastic dynamic response is evaluated considering uncertainties in linear and nonlinear parts of the mechanical system. The results are compared with those obtained from the classical Monte Carlo Simulation (MCS). For various numerical tests, it is found that the results agreed very well whilst requiring significantly less computation.     

Au:TiO2和Au:Al2O3纳米颗粒复合膜的线性和非线性光学特性

主要从实验和理论两个方面,探讨了不同Au颗粒尺寸和不同基质对Au：TiO₂和Au：Al₂O₃复合膜线性和非线性光学性质的影响.用吸收光谱研究了Au颗粒尺寸和基质与Au复合膜表面等离子体共振带之间的关系;用皮秒Z扫描技术研究了共振和非共振情况下(激发光波长分别为532nm和1064nm),Au颗粒尺寸和基质与复合膜三阶非线性极化率的关系.基于表面等离子体共振理论和局域场增强理论对复合膜进行了分析,得到了不同Au颗粒大小和不同基质时Au复合膜的 关键词： 金属纳米颗粒 复合膜 三阶非线性 表面等离子体共振