Existence and multiple solutions for a nonlinear system with a parameter

In this paper, the critical point theory and the Morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u)$Au=\lambda f\left(u\right)$, where the matrix A$A$ is not necessarily positive definite. Some existence and multiple solutions are obtained.     

Existence of solutions for some discrete boundary value problems with a parameter

In this paper, the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem (BVP) ; u(0)=0=Δu(T), where T is a positive integer, Z(1,T)={1,2,…,T}, Δ is the forward difference operator defined by Δu(k)=u(k+1)-u(k) and f:Z(1,T)×R→R is continuous, λ∈R⁺ is a parameter. By using the critical point theory and Morse theory, we obtain that the above (BVP) has solutions for λ being in some different intervals.     

Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences

In this paper, we consider the following nonlinear q-fractional three-point boundary value problem $$\begin{array}{l}(D_{q}^{\alpha}u)(t) + f(t,u(t))=0, \quad 0 < t < 1, 2 < \alpha< 3,\\ [2pt]u(0) = (D_qu)(0) = 0, \quad(D_qu)(1) = \beta(D_qu)(\eta),\end{array}$$ where 0<?|? ^??-2<1. By using a fixed-point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.     

多普勒测风激光雷达风场探测结果分析

为了验证自行研制的瑞利散射测风激光雷达的性能及应用价值,将激光雷达测量的风速风向廓线与探空气球测量结果进行对比,结果吻合得较好,高度10 km以下风速最大差距3.0 m/s,20 km以下最大差距4.5 m/s;风向除拐点外标准差最大32。连续探测结果显示了强劲的西风急流和风向转换特征,急流中心的高度一般在10~12 km,中心最大风速接近70 m/s,最小风速不低于30 m/s,20 km左右拐点最小风速不足1 m/s,20 km以上风速逐渐增加 ;在20 km以下风向为西风,在270左右变化,20 km以上为东风,在90左右变化。探测结果表明瑞利散射测风激光雷达既能跟踪大尺度季节性气候特征的变化规律,又能突出小尺度瞬态气候特征的形成、演化过程。     

Chiral PSiSi-Ligand Enabled Iridium-Catalyzed Atroposelective Intermolecular C−H Silylation

Catalytic enantioselective intermolecular C−H silylation offers an efficient approach for the rapid construction of chiral organosilicon compounds, but remains a significant challenge. Herein, a new type of chiral silyl ligand is developed, which enables the first iridium-catalyzed atroposelective intermolecular C−H silylation reaction of 2-arylisoquinolines. This protocol features mild reaction conditions, high atom economy, and remarkable yield with excellent stereoselectivity (up to 99 % yield, 99 % ee), delivering enantioenriched axially chiral silane platform molecules with facile convertibility. Key to the success of this unprecedented transformation relies on a novel chiral PSiSi-ligand, which facilitates the intermolecular C−H silylation process with perfect chem-, regio- and stereo-control via a multi-coordinated silyl iridium complex.