Perturbation methods in semilinear elliptic problems involving critical hardy-sobolev exponent

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.     

Ramanujan-style proof of $$p_{-3}(11n+7) \equiv 0\ (\mathrm{mod\ }11)$$

In this note, we establish two identities of \((q;\,q)_\infty ^8\) by using Jacobi’s four-square theorem and two of Ramanujan’s identities. As an important consequence, we present one Ramanujan-style proof of the congruence \(p_{-3}(11n+7)\equiv 0\ (\mathrm{mod\ }11)\), where \(p_{-3}(n)\) denotes the number of 3-color partitions of n.     

Energy band alignment of 2D/3D MoS2/4H-SiC heterostructure modulated by multiple interfacial interactions

The interfacial properties of MoS₂/4H-SiC heterostructures were studied by combining first-principles calculations and X-ray photoelectron spectroscopy. Experimental (theoretical) valence band offsets (VBOs) increase from 1.49 (1.46) to 2.19 (2.36) eV with increasing MoS₂ monolayer (1L) up to 4 layers (4L). A strong interlayer interaction was revealed at 1L MoS₂/SiC interface. Fermi level pinning and totally surface passivation were realized for 4H-SiC (0001) surface. About 0.96e per unit cell transferring forms an electric field from SiC to MoS₂. Then, 1L MoS₂/SiC interface exhibits type I band alignment with the asymmetric conduction band offset (CBO) and VBO. For 2L and 4L MoS₂/SiC, Fermi level was just pinning at the lower MoS₂ 1L. The interaction keeps weak vdW interaction between upper and lower MoS₂ layers. They exhibit the type II band alignments and the enlarged CBOs and VBOs, which is attributed to weak vdW interaction and strong interlayer orbital coupling in the multilayer MoS₂. High efficiency of charge separation will emerge due to the asymmetric band alignment and built-in electric field for all the MoS₂/SiC interfaces. The multiple interfacial interactions provide a new modulated perspective for the next-generation electronics and optoelectronics based on the 2D/3D semiconductors heterojunctions.     

H6[P2Mo18O62]的钒取代多金属氧酸盐对小鼠黑色素瘤细胞B16黑色素生成的调控及其机制研究

合成了五种多金属氧酸盐. 以B16小鼠黑色素瘤细胞为模型, 用噻唑蓝(MTT)法检测细胞存活率, 酶标法测定抗氧化活性和内黑色素含量及酪氨酸酶活性, 最后用分子对接模拟多金属氧酸盐和酪氨酸酶结合的机制. 研究结果表明, 两种磷钼酸(H₇[P₂Mo₁₇VO₆₂]和H₈[P₂Mo₁₆V₂O₆₂])是高效的黑色素生成抑制剂, 在200 μmol/L的浓度下对黑色素合成的抑制率为74.40%和75.14%, 对细胞内酪氨酸酶活性的抑制率为35.71%和40.00%, 随着钒原子取代个数的增加, 两种抑制活性均逐渐降低. H₇[P₂Mo₁₇VO₆₂]和H₈[P₂Mo₁₆V₂O₆₂]对细胞没有毒性. 两种多酸都有较好的1,1-二苯基-2-三硝基苯肼(DPPH)清除能力, IC₅₀分别为1.683和2.800 mg/mL. 分子对接分数低于–146 kJ/mol. 综上所述, H₇[P₂Mo₁₇VO₆₂]和H₈[P₂Mo₁₆V₂O₆₂]能抑制B16细胞黑色素的生成, 其机制与抑制酪氨酸酶的活性有关.     

一类退化抛物方程解的几何性质

本文讨论非线性退化抛物方程u_t=△φ(u)的Cauchy问题弱解u(x,t)的正则性与几何性质.本文证明:若正数β足够大,则曲面ψ=ψ(x,t)=[φ(u)]~β是随时间t的连续变化而漂浮于空间R~(n+1)中的n维完备黎曼流形,它与实欧氏空R~n相切于低维流形(?)H_n(t),而H_u(t)={x∈R~n:u(x,t)0);函数ψ(x,t)在经典的意义下满足另一退化抛物方程.     

On uniform convexity of Banach spaces

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.     

格值上下文无关文法的范式

给出了格值上下文无关文法（LCFG）,Chomsky范式文法,Greibach范式文法的定义.证明了对任意的LCFG存在与之等价的Chomsky范式文法;给出了对任意的LCFG,存在与之等价的Greibach范式文法的条件.文中结论表明了LCFG的特性与其取值格的代数性质密切相关.     

Removable edges in a spanning tree of a k-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G O e denotes the graph obtained from G by the following way： deleting e to get G - e, and for any end vertex of e with degree k - 1 in G - e, say x, deleting x, and then adding edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of k-connected graphs have been investigated. In the present paper, we investigate the distribution of removable edges on a spanning tree of a k-connected graph （k ≥ 4）.