Some nonlinear problems in servomechanisms

Summary A mathematical model for a hydraulic servomechanism is constructed. It is shown that the model, in general, reduces to a nonlinear third-order equation of the formxx+(₁+xx+–₂ x=p(t). Under certain conditions imposed on the constants involved, it is proved that above equation possesses a periodic solution.     

三级相互作用对聚亚甲基链末端距四次矩的影响

本文用三级相互作用近似理论导出了高分子链末端距的四次矩的计算公式。计算聚亚甲基链的末端距四次矩,并同二级相互作用的结果进行了比较。对于聚亚甲基链,三级相互作用近似的四次矩小于二级相互作用近似的四次矩。     

Synthesis and property of soluble poly(1,1′-dialkyl-3,3′-ferrocenylenevinylene)s via titanium-induced dicarbonyl-coupling reaction of 1,1′-dialkylferrocene-3,3′-dicarbaldehydes

1,1′-Dialkylferrocene-3,3′-dicarbaldehydes ( 1a–c ) with long alkyl chains such as ethyl, hexyl, and dodecyl groups were prepared in 13–25% yield via three-step reactions. The titanium-induced dicarbonyl-coupling reaction of 1a–c gave poly(1,1′-dialkyl-3,3′-ferrocenylenevi-nylene)s ( 2a–c ) in quantitative yields, which were the molecular weights of 3000–10,000 and highly soluble in chloroform, benzene, and hexane. The electrical conductivity and the third-order nonlinear optical susceptibility for poly(1,1′-dihexyl-3,3′-ferrocenylenevinylene) ( 2b ) were estimated to be 1 × 10^?2 S/cm on doping with iodine and 1–4 × 10^?12 esu at a wavelength of 1–2.4 μm, respectively. © 1995 John Wiley & Sons, Inc.     

C60-吡咯烷衍生物的合成及非线性光学性质的研究

通过富勒烯C₆₀与肌氨酸和有机醛化合物的1,3-偶极环加成反应, 获得了九种含不同有机功能基团的C₆₀吡咯烷衍生物1～9, 用¹H NMR, ¹³C NMR, FTIR, UV-vis和FAB-MS进行了结构表征; 利用皮秒激光光源, 采用z扫描技术测定了分子的三阶非线性超极化率γ⁽³⁾, 结果显示: 化合物3 (γ⁽³⁾＝4.14×10^－33 esu)具有最大的三阶非线性光学系数, 说明增加噻吩共轭链的长度, 使三阶非线性活性增加; 对具有相同共轭链的C₆₀-噻吩吡咯烷衍生物(2, 5, 1和4), 吸电子取代基减小了三阶光学非线性活性, 给电子基增大了三阶光学非线性活性; 同时发现喹啉环2-位键联(7)比4-位(8)有更好的三阶光学非线性活性.     

The use of Sinc‐collocation method for solving Falkner–Skan boundary‐layer equation

The MHD Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity on the semi‐infinite domain. The proposed approach is equipped by the orthogonal Sinc functions that have perfect properties. This method solves the problem on the semi‐infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, the governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by the Sinc‐collocation method. It is shown that the Sinc‐collocation method converges to the solution at an exponential rate. Copyright © 2010 John Wiley & Sons, Ltd.     

一类变号(k,n-k)共轭边值问题正解的存在性

通过构造一个特殊的锥,利用范数形式的锥拉伸锥压缩不动点定理,在允许非线性项变号无下界且没有任何单调性假设的条件下,得出了一类高阶(k,n-k)共轭两点边值问题方程组正解的存在性结论.     

一个奇摄动微分方程非线性混合边值问题

研究了一个三阶半线性微分方程的奇摄动非线性混合边值问题.利用边界层函数法构造了该问题的形式渐近解,并采用微分不等式理论证明了解的存在性,给出了渐近解的误差估计,最后得出了边界层函数指数型衰减的结论.     

金属有机骨架薄膜材料的三阶非线性光学性能研究进展

金属有机骨架(MOFs)材料因其可设计的结构以及灵活可控的配位模式,在三阶非线性光学(NLO)领域引起了广泛的关注。与液体分散状态相比,MOFs在固体状态下的三阶NLO性能更为重要,这不仅可以深入了解MOFs本身所固有的光学性能,还有助于实现MOFs在光学器件中的实际应用。然而,由于散射的存在和透光率的限制,单独的MOFs材料难以直接实现固体状态下的三阶NLO性能研究,将MOFs制备成具有较好光学透过性的薄膜是研究其NLO性能最为可行的一种策略。MOFs薄膜不仅很好地继承了MOFs所固有的三阶NLO性能,而且还结合了薄膜的高透光率以及灵活的机械性能。基于此,本文分析总结了MOFs薄膜的制备方法及其NLO性能研究方面的相关工作,并根据目前MOFs薄膜在三阶NLO性能方面的研究现状对其未来发展予以了展望。     

Novel mesophases in fluorine substituted banana-shaped mesogens

A new homologous series of achiral banana-shaped mesogens ('Dn') has been synthesized and studied by the classical techniques (optical microscopy, differential scanning calorimetry, X-ray diffraction, miscibility studies and electro-optic investigations). The short homologues (D6-D8) exhibit a two-dimensional phase 'B1_x' different from a B1 phase with a rectangular lattice. The longer homologues (D9-D14) present a mesophase which displays the defects of the B7 phase of the PIMB-NO₂ compounds. Nevertheless the D9-D14 mesophase is not miscible with the B7 phase, and contrary to B7, exhibits a bistable behaviour ('ferroelectric' type) suggesting at least a B7 variant.