Surface‐Enhanced Raman Scattering Based on Controllable‐Layer Graphene Shells Directly Synthesized on Cu Nanoparticles for Molecular Detection

Graphene shells with a controllable number of layers were directly synthesized on Cu nanoparticles (CuNPs) by chemical vapor deposition (CVD) to fabricate a graphene‐encapsulated CuNPs (G/CuNPs) hybrid system for surface‐enhanced Raman scattering (SERS). The enhanced Raman spectra of adenosine and rhodamine 6G (R6G) showed that the G/CuNPs hybrid system can strongly suppress background fluorescence and increase signal‐to‐noise ratio. In four different types of SERS systems, the G/CuNPs hybrid system exhibits more efficient SERS than a transferred graphene/CuNPs hybrid system and pure CuNPs and graphene substrates. The minimum detectable concentrations of adenosine and R6G by the G/CuNPs hybrid system can be as low as 10^?8 and 10^?10 M , respectively. The excellent linear relationship between Raman intensity and analyte concentration can be used for molecular detection. The graphene shell can also effectively prevent surface oxidation of Cu nanoparticles after exposure to ambient air and thus endow the hybrid system with a long lifetime. This work provides a basis for the fabrication of novel SERS substrates.     

One‐dimensional CuII coordination polymers containing C2h‐symmetric 1,1′:4′,1′′‐terphenyl‐3,3′‐dicarboxylate linkers

Two new one‐dimensional Cu^II coordination polymers (CPs) containing the C_2h‐symmetric terphenyl‐based dicarboxylate linker 1,1′:4′,1′′‐terphenyl‐3,3′‐dicarboxylate (3,3′‐TPDC), namely catena‐poly[[bis(dimethylamine‐κN)copper(II)]‐μ‐1,1′:4′,1′′‐terphenyl‐3,3′‐dicarboxylato‐κ⁴O,O′:O′′:O′′′] monohydrate], {[Cu(C₂₀H₁₂O₄)(C₂H₇N)₂]·H₂O}_n, (I), and catena‐poly[[aquabis(dimethylamine‐κN)copper(II)]‐μ‐1,1′:4′,1′′‐terphenyl‐3,3′‐dicarboxylato‐κ²O³:O^3′] monohydrate], {[Cu(C₂₀H₁₂O₄)(C₂H₇N)₂(H₂O)]·H₂O}_n, (II), were both obtained from two different methods of preparation: one reaction was performed in the presence of 1,4‐diazabicyclo[2.2.2]octane (DABCO) as a potential pillar ligand and the other was carried out in the absence of the DABCO pillar. Both reactions afforded crystals of different colours, i.e. violet plates for (I) and blue needles for (II), both of which were analysed by X‐ray crystallography. The 3,3′‐TPDC bridging ligands coordinate the Cu^II ions in asymmetric chelating modes in (I) and in monodenate binding modes in (II), forming one‐dimensional chains in each case. Both coordination polymers contain two coordinated dimethylamine ligands in mutually trans positions, and there is an additional aqua ligand in (II). The solvent water molecules are involved in hydrogen bonds between the one‐dimensional coordination polymer chains, forming a two‐dimensional network in (I) and a three‐dimensional network in (II).     

Estimates of densities for Lévy processes with lower intensity of large jumps

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.     

How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Let K and S be locally compact Hausdorff spaces and X be an abstract space. Suppose that T is a positive Banach lattice isomorphism from into . Then for each ordinal α the cardinalities of the αth derivatives and satisfy the following inequality Moreover, if then is a continuous image of a subset of which can be taken closed when K is compact. The first statement of this result for is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number is sharp in these vector‐valued theorems.     

On null sequences for Banach operator ideals,trace duality and approximation properties

Let be a Banach operator ideal and X be a Banach space. We undertake the study of the vector space of ‐null sequences of Carl and Stephani on X , , from a unified point of view after we introduce a norm which makes it a Banach space. To give accurate results we consider local versions of the different types of accessibility of Banach operator ideals. We show that in the most common situations, when is right‐accessible for , behaves much alike . When this is the case we give a geometric tensor product representation of . On the other hand, we show an example where the representation fails. Also, via a trace duality formula, we characterize the dual space of . We apply our results to study some problems related with the ‐approximation property giving a trace condition which is used to solve the remaining case () of a problem posed by Kim (2015). Namely, we prove that if a dual space has the ‐approximation property then the space has the ‐approximation property.     

Subspace arrangements as generalized star configurations

In this note, we show that any projective subspace arrangement can be realized as a generalized star configuration variety. This type of interpolation result may be useful in designing linear codes with prescribed codewords of minimum weight, as well as in answering a couple of questions about the number of equations needed to define a generalized star configuration variety.     

Square function estimates on layer potentials for higher‐order elliptic equations

In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m , with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.     

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

On the dynamics of a third order Newton's approximation method

We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function , by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at , has periodic points of any prime period and that the set of points a at which the approximation sequence does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.