A [001]-Oriented Hittorf's Phosphorus Nanorods/Polymeric Carbon Nitride Heterostructure for Boosting Wide-Spectrum-Responsive Photocatalytic Hydrogen Evolution from Pure Water

Red phosphorus is a promising photocatalyst with wide visible-light absorption up to 700 nm, but the fast charge recombination limits its photocatalytic hydrogen evolution reaction (HER) activity. Now, [001]-oriented Hittorf's phosphorus (HP) nanorods were successfully grown on polymeric carbon nitride (PCN) by a chemical vapor deposition strategy. Compared with the bare PCN and HP, the optimized PCN@HP hybrid exhibited a significantly enhanced photocatalytic activity, with HER rates reaching 33.2 and 17.5 μmol h⁻¹ from pure water under simulated solar light and visible light irradiation, respectively. It was theoretically and experimentally indicated that the strong electronic coupling between PCN and [001]-oriented HP nanorods gave rise to the enhanced visible light absorption and the greatly accelerated photoinduced electron–hole separation and transfer, which benefited the photocatalytic HER performance.     

Fabrication of Carbon Nanotube Thin Films for Flexible Transistors by Using a Cross-Linked Amine Polymer

Owing to their remarkable properties, single-walled carbon nanotube thin-film transistors (SWCNT-TFTs) are expected to be used in various flexible electronics applications. To fabricate SWCNT channel layers for TFTs, solution-based film formation on a self-assembled monolayer (SAM) covered with amino groups is commonly used. However, this method uses highly oxidized surfaces, which is not suitable for flexible polymeric substrates. In this work, a solution-based SWCNT film fabrication using methoxycarbonyl polyallylamine (Moc-PAA) is reported. The NH₂-terminated surface of the cross-linked Moc-PAA layer enables the formation of highly dense and uniform SWCNT networks on both rigid and flexible substrates. TFTs that use the fabricated SWCNT thin film exhibited excellent performance with small variations. The presented simple method to access SWCNT thin film accelerates the realization of flexible nanoelectronics.     

Inverse spectral problems for non-self-adjoint Sturm–Liouville operators with discontinuous boundary conditions

This paper deals with the inverse spectral problem for a non-self-adjoint Sturm–Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential q is known a priori on a subinterval $$ \left[ b,\pi \right] $$ with $$b\in \left( d,\pi \right] $$ or $$b=d$$, then $$h,\,\beta ,\,\gamma $$ and q on $$\left[ 0,\pi \right] $$ can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $$b\in \left( 0,d\right) $$, a similar statement holds if $$\beta ,\,\gamma $$ are also known a priori. Moreover, if q satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl m-function to solve the problem of missing eigenvalues and norming constants.     

Recyclable MoO3 nanobelts for photocatalytic degradation of Rhodamine B by near infrared irradiation

Molybdenum trioxide (MoO₃) represented an excellent photocatalytic performance with many applications, including degradation of organic contaminants and splitting of water. This paper presented a new route to synthesize MoO₃ nanobelts with high aspect ratios and crystallinity by a hydrothermal technique. This work showed that the as-synthesized nanobelts exhibited strong photocatalytic activity to degrade an organic dye of Rhodamine B (RhB) in aqueous solution under the exposure of the light source in the near infrared wavelength range, significantly improving the photocatalytic activity of the nanobelts. The results also showed that for a small concentration of RhB at 7.5 mg/L a complete photodegradation (for a given MoO₃ nanobelts quantity of 0.1 g) can be reached after exposing for 60 min. For all concentrations of the RhB solution, the photodegradation exhibited an exponential dependence on the exposure time followed by a sudden shutdown, but no complete photodegradation can be reached. Also, the residual quantity of RhB in solution after the photocatalytic reaction was determined by the initial RhB concentration. The photocatalytic degradation can be interpreted by the pseudo–first-order equation for the absorption of liquid/solid based on solid capacity; thus, photocatalytic degradation can be attributed to the interaction between the photoexcited electrons in the substrate and the antibonding orbital of the RhB in solution. The sudden shutdown was due to the inability of the photoexcited electrons in the substrate hopping to the antibonding orbital of RhB in the presence of the RhB intermediate products from the degraded RhB. In addition, this work showed that the photocatalytic reaction can be recovered after a thermal treatment of postreacted MoO₃ nanobelts, enhancing the utilization efficiency of the catalysis.     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

In this paper, the superconvergence analysis of a two‐grid method (TGM) with low‐order finite elements is presented for the fourth‐order dispersive‐dissipative wave equations for a second order fully discrete scheme. The superclose estimates in the H¹‐norm on the two grids are obtained by the combination technique of the interpolation and Ritz projection. Then, with the help of the interpolated postprocessing technique, the global superconvergence properties are deduced. Finally, numerical results are provided to show the performance of the proposed TGM for conforming bilinear element and nonconforming element, respectively. It shows that the TGM is an effective method to the problem considered of our paper compared with the traditional Galerkin finite element method (FEM).     

Characterizations of ( m,n )-Jordan Derivations and ( m,n )-Jordan Derivable Mappings on Some Algebras

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A~2) = 2 mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every(m, n)-Jordan derivation with m = n from a C*-algebra into its Banach bimodule is zero. An additive mappingδ from R into M is called a(m, n)-Jordan derivable mapping at W in R if(m + n)δ(AB + BA) =2mδ(A)B + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left(right) separating set generated algebraically by all idempotents in A, then every(m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital(A, B)-bimodule and U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.     

Molecular Acceptors Based on a Triarylborane Core Unit for Organic Solar Cells

Triarylboranes that exhibit p–π* conjugation serve as versatile building blocks to design n-type organic/polymer semiconductors. A series of new molecular acceptors based on triarylborane is reported here. These molecules are designed with a boron atom that bears a bulky 2,4,6-tri-tert-butylphenyl (Mes*) substituent at the core and strong electron-withdrawing 2-(3-oxo-2,3-dihydroinden-1-ylidene)malononitrile (IC) units as the end-capping groups that are linked to the core by bithiophene bridges. Butyl or butoxy groups are introduced to the bithiophene units to tune the optoelectronic properties. These molecules show nearly planar backbones with highly localized steric hindrance at the core, low LUMO/HOMO energy levels, and broad absorption bands spanning the visible region, which are all very desirable characteristics for use as electron acceptors in organic solar cell (OSC) applications. The attachment of butyl groups to the bithiophene bridges brings about a slightly twisted backbone, which in turn promotes good solubility and homogeneous donor/acceptor blend morphology, whereas the introduction of butoxy groups leads to improved planarity, favorable stacking in the film state, and a greatly reduced band gap. OSC devices based on these molecules exhibit encouraging photovoltaic performances with power conversion efficiencies reaching up to 4.07 %. These results further substantiate the strong potential of triarylboranes as the core unit of small molecule acceptors for OSC applications.