ALMOST PERIODICITY AND EQUICONTINUITY OF THE TOPOLOGICAL TRANSFORMATION GROUP

In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.

A Topological Transformation of Quantum Dynamics

In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K?× S1?defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K?× S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.

Topological Transformation during Normal Grain Growth

This paper investigates topological transformation during normal grain growth by carrying out a computer vertex simulation. Results show that topological correlation agrees with the models proposed by Blanc et al. and Weaire. Topological transformation occurs more often on grains with some topological classes instead of equal probability on each boundary. This can be qualitatively explained by topological correlation.

Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering function

In this paper, a model of topology optimization with linear buckling constraints is established based on an independent and continuous mapping method to minimize the plate/shell structure weight. A composite exponential function（CEF） is selected as filtering functions for element weight, the element stiffness matrix and the element geometric stiffness matrix, which recognize the design variables, and to implement the changing process of design variables from“discrete” to “continuous” and back to “discrete”. The buckling constraints are approximated as explicit formulations based on the Taylor expansion and the filtering function. The optimization model is transformed to dual programming and solved by the dual sequence quadratic programming algorithm. Finally, three numerical examples with power function and CEF as filter function are analyzed and discussed to demonstrate the feasibility and efficiency of the proposed method.

One-pot topotactic synthesis of Ti^(3+) self-doped 3D TiO_2 hollow nanoboxes with enhanced visible light response

Ti^（3＋） self-doped anatase three-dimensional（3D） TiO_2 hollow nanoboxes were synthesized via a topological transformation process involving template participation by a facile one-pot hydrothermal treatment with an ethanol solution of zinc powder and TiOF_2. It is worth noting that the 3D TiO_2 hollow nanoboxes are assembled from six single-crystal nanosheets and have dominant exposure of the {001} facets. It is found from EPR spectra that adding zinc powder is an environment-friendly and effective strategy to introduce Ti^（3＋） and oxygen vacancy（Ov） into the bulk of 3D hollow nanoboxes rather than the surface, which is responsible for their enhanced visible photocatalytic properties.The photocatalytic activity was evaluated by measuring the formation rate of hydroxide free radicals using 7-hydroxycoumarin as a probe. The sample prepared with zinc/TiOF_2 mass ratio of0.25 exhibited the highest RhB photodegradation activity under visible-light irradiation with a degradation rate of 96%, which is 4.0-times higher than that of pure TiO_2. The results suggest a novel approach to construct in-situ 3D hierarchical TiO_2 hollow nanoboxes doped with Ti^（3＋） and Ov without introducing any impurity elements for superior visible-light photocatalytic activity.

A universal strategy for the synthesis of porous twodimensional transition metal oxide nanosheets based on chemical topology transformation

Two-dimensional(2D)transition metal oxides(TMOs)have attracted much attention for various applications,owing to the abundance of active sites,rapid ion transmission speed,and short carrier migration distance.However,the current preparation strategies are usually limited to producing intrinsically layered compounds or sacrificing template.Here,we report a universal strategy for preparing ultrathin porous 2D TMO nanosheets by chemical topological transformation of the corresponding transition metal selenides.We observed that the as-prepared 2D TMO nanosheets not only perfectly inherit the transition metal selenides’2D topological structure,but also possess numerous pore structures formed as a way to release the structural stress associated with lattice growth.As a proof of concept,ultrathin porous WO_(3),Mo O_(3),and Co_(3)O_(4) nanosheets were successfully prepared based on the in-situ oxidation of the corresponding ultrathin WSe_(2),MoSe_(2),and Co_(0.85)Se,respectively.The outstanding sensing properties and photodetector performance displayed by the as-prepared porous 2D WO_(3) nanosheets further indicate the promising prospects of topology transformation for the preparation of porous 2D TMO nanosheets.