Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential

In this article, we prove the existence and multiplicity of non-trivial solutions for an indefinite fractional elliptic equation with magnetic field and concave–convex nonlinearities. Our multiplicity results are based on studying the decomposition of the Nehari manifold.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.     

从机械能守恒的角度推导单摆简谐振动的周期

本文给出了利用基本的物理原理结合高等数学公式的方法推导单摆做简谐运动的周期公式,可以简明扼要地理解单摆简谐运动的过程.     

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.     

The contact Yamabe flow on K-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.     

Bi-f-harmonic map equations on singly warped product manifolds

Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps.In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensional manifolds and some special maps from or into a warped product manifold.     

pH值对Bi2MoO6晶体形貌和可见光催化性能的影响

以(NH₄)₆Mo₇O₂₄·4H₂O和Bi(NO₃)₃·5H₂O为原料,采用普通水热法制备Bi₂MoO₆光催化剂,研究pH值对制备该光催化剂的影响.对所制备的系列样品,采用X-射线衍射仪(XRD)、扫描电子显微镜(SEM)、比表面积分析仪、X射线光电子能谱仪(XPS)和紫外-可见漫反射(UV-Vis DRS)进行表征.结果表明:pH值对Bi₂MoO₆晶体的物相组成、形貌和光催化性能均有显著影响.pH值为1~7时,所制备的样品为纯相Bi₂MoO₆,pH值为9或11时,出现第二相Bi_3.64Mo_0.36O_6.55;随着pH值的升高,形貌依次为纳米棒、纳米片和无规则纳米颗粒.在可见光(λ≥420 nm)照射下,通过光催化降解罗丹明B(Rhodamine B,RhB),探讨了制备Bi₂MoO₆的pH值对其可见光催化活性的影响.当pH=7时,制备的样品光催化效果最好,光照50 min后对初始浓度为5 mg·L^-1的罗丹明B溶液的降解率为85%.     

A Novel Chemo‐selective Epoxidation across Acrylate Ester: Synthesis of New Water‐Soluble Forskolin Analogues

While removing the TBDMS group from OH protection, a novel epoxidation reaction occurred across acrylate ester attached to a forskolin fragment. Besides spectroscopic data, the epoxide formation was confirmed by ring opening with a secondary amine. This unique epoxidation reaction, to our knowledge, is not known in the literature. This reaction led us to discover a simple deblocking protocol. The epoxide and the desired Michael substrates were used to introduce imidazole into forskolin.