Solutions for conformally invariant fractional Laplacian equations with multi-bumps centered in lattices

In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:

${(?\mathrm{\Delta })}^{s}u=K(x)u^{\frac{N+2s}{N?2s}},u>0\mathrm{in}{\mathbb{R}}^N,$

where $s\in (0,1)$ and $N>2+2s$, $K>0$ is periodic in $(x_1,\dots ,x_k)$ with $1\le k<\frac{N?2s}{2}$. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in ${\mathbb{R}}^k$, including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in ${\mathbb{R}}^k$, the restriction on $1\le k<\frac{N?2s}{2}$ is in some sense optimal, since we can show that for $k\ge \frac{N?2s}{2}$, no such solutions exist.     

Magnetically Recoverable Core–Shell Nanocomposites with Enhanced Photocatalytic Activity

Core–shell structured Fe₃O₄/SiO₂/TiO₂ nanocomposites with enhanced photocatalytic activity that are capable of fast magnetic separation have been successfully synthesized by combining two steps of a sol–gel process with calcination. The as‐obtained core–shell structure is composed of a central magnetite core with a strong response to external fields, an interlayer of SiO₂, and an outer layer of TiO₂ nanocrystals with a tunable average size. The convenient control over the size and crystallinity of the TiO₂ nanocatalysts makes it possible to achieve higher photocatalytic efficiency than that of commercial photocatalyst Degussa P25. The photocatalytic activity increases as the thickness of the TiO₂ nanocrystal shell decreases. The presence of SiO₂ interlayer helps to enhance the photocatalytic efficiency of the TiO₂ nanocrystal shell as well as the chemical and thermal stability of Fe₃O₄ core. In addition, the TiO₂ nanocrystals strongly adhere to the magnetic supports through covalent bonds. We demonstrate that this photocatalyst can be easily recycled by applying an external magnetic field while maintaining their photocatalytic activity during at least eighteen cycles of use.     

Asymptotics for the finite-time ruin probability in a discrete-time risk model with dependent insurance and financial risks<Superscript>*</Superscript>

We consider a discrete-time risk model with insurance and financial risks. Within period i ≥ 1, the real-valued net insurance loss caused by claims is the insurance risk, denoted by X _i , and the positive stochastic discount factor over the same time period is the financial risk, denoted by Y _i . Assume that {(X, Y), (X _i , Y _i ), i ≥ 1} form a sequence of independent identically distributed random vectors. In this paper, we investigate a discrete-time risk model allowing a dependence structure between the two risks. When (X, Y ) follows a bivariate Sarmanov distribution and the distribution of the insurance risk belongs to the class ?(γ) for some γ > 0, we derive the asymptotics for the finite-time ruin probability of this discrete-time risk model.     

Novel binding and efficient cellular uptake of guanidine-based peptide nucleic acids (GPNA)

Incorporation of a guanidine functional group into the PNA backbone facilitates cellular uptake of PNA into mammalian cells with efficiency comparable to that of the TAT transduction domain. The modified PNA recognizes and binds to the complementary DNA strand in accordance with Watson-Crick recognition rules. However, unlike polypyrimidine PNA which binds to DNA in 2:1 stoichiometry, the modified PNA binds to complementary DNA in a 1:1 ratio to form a highly stable duplex.     

化学探究性实验活动表现评价的设计探讨

活动表现评价是新课程改革倡导的一种新型的评价方式。本文对化学新课程中探究性实验的一般程序、内容选择、活动表现评价的方法、工具设计和成绩评定等方面进行了探讨。     

Bromination of Isothianaphthene Derivatives towards the Application in Organic Electronics

Different approaches to brominate dialkyl isothianaphthene-5,6-dicarboxylate were tried and single crystals of target molecule and side product were obtained. A planar copolymer with the trans-ethene has been prepared, which shows good solubility in common organic solvents and broad absorption with edge to 967 nm in film.     

The qualitative boundary behavior of blow-up solutions of the super-Liouville equations

Continuing our work on the boundary value problem for super-Liouville equation, we study the qualitative behavior of boundary blow-ups. The boundary condition is derived from the chirality conditions in the physics literature, and is geometrically natural. In technical terms, we derive a new Pohozaev type identity and provide a new alternative, which also works at the boundary, to the classical method of Brézis–Merle.     

Coalescence of 3-phenyl-propynenitrile on Cu(111) into interlocking pinwheel chains

3-phenyl-propynenitrile (PPN) adsorbs on Cu(111) in a hexagonal network of molecular trimers formed through intermolecular interaction of the cyano group of one molecule with the aromatic ring of its neighbor. Heptamers of trimers coalesce into interlocking pinwheel-shaped structures that, by percolating across islands of the original trimer coverage, create the appearance of gear chains. Density functional theory aids in identifying substrate stress associated with the chemisorption of PPN's acetylene group as the cause of this transition.