Point interactions for 3D sub-Laplacians

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point $q_0\in M$ exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on $C_0^{\infty }(M?\{q_0\})$ is essentially self-adjoint in $L^2(M)$. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on $C_0^{\infty }(N?\{q_0\})$ is never essentially self-adjoint in $L^2(N)$, if $\dim ?N\le 3$. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.     

直接碳燃料电池燃料的研究进展

直接碳燃料电池(DCFC)是一种清洁高效利用碳资源发电的装置。其因能量转换率高,对环境污染小,燃料选择范围广等优点获得了越来越多的关注。DCFC的性能与使用的燃料密切相关,为了探究燃料对DCFC的影响,本文分别阐述了石墨、炭黑、中密度纤维板、生物质、煤、活性炭的特性及改性方法,分析讨论了燃料表面含氧官能团以及燃料中的金属催化剂对阳极电化学反应的促进作用,发现燃料表面化学性质要比比表面积更加重要。同时,本文也提出了对生物质这一优良的可再生资源的期待,为未来DCFC燃料的发展提供参考。     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

Extremely Photostable Electron-Deficient Phthalocyanines that Generate High Levels of Singlet Oxygen

A robust lead-mediated synthetic procedure for the generation of phthalocyanines substituted with electron-withdrawing groups has been developed. The free-base phthalocyanine and various metal complexes were prepared without discernible degradation of the peripheral electron-withdrawing substituents. Upon irradiation with red light, some of the thus-obtained metal complexes generated high levels of singlet oxygen. In particular, a palladium complex exhibited attractive photostability upon exposure to singlet oxygen as a bleaching agent. The photostability of such complexes that may manifest concomitantly to the generation of high levels of singlet oxygen was attributed to the presence of the electron-withdrawing groups, which results in energetically low-lying highest occupied molecular orbitals.     

Interaction between the end groups and the main chain of conjugated polymers by time-resolved EPR and fluorescence spectroscopy

ABSTRACT

Interaction between a zinc porphyrin (ZnPor) as the end-group and poly(9,9-di-n-octylfluorene-2,7-vinylene) (PFV) as the main chain in a porphyrin end-modified fluorescent conjugated polymer, ZnPFV, was studied by time-resolved electron paramagnetic resonance (EPR) and fluorescence spectroscopy. While fluorescence from the PFV part of ZnPFV showed a spectral profile almost identical to that of a PFV oligomer without end-modification, the emission spectrum of the ZnPor part exhibited a much broader profile compared to that of the reference zinc porphyrin monomer. Based on the analysis of lifetimes and quantum yields, it was found that radiative rate constant of the ZnPor part was enhanced by nearly three times. The observed unusual enhancement in the radiative rate constant was rationalised in terms of a partial π-conjugation between the end group and the main chain, as a result of co-planarisation in fluid solution. On the other hand, the time-resolved EPR spectrum of ZnPFV at 100?K basically showed a similar spectral pattern to that of the reference zinc porphyrin, but with significant differences in zero-field spitting parameters and initial population ratios. The π-system of the excited triplet state is deduced to deviate from D_4h symmetry in the end zinc porphyrin groups. The obtained results show that interaction of the porphyrin end group with the main chain of the polymer significantly influences the excited singlet state properties of the porphyrin, while its triplet state properties were affected to a lesser extent.