Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scaleT=ℝorℤ, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales.
Hydrogen‐treated TiO2 nanotube (H–TNT) arrays serve as highly ordered nanostructured electrode supports, which are able to significantly improve the electrochemical performance and durability of fuel cells. The electrical conductivity of H–TNTs increases by approximately one order of magnitude in comparison to air‐treated TNTs. The increase in the number of oxygen vacancies and hydroxyl groups on the H–TNTs help to anchor a greater number of Pt atoms during Pt electrodeposition. The H–TNTs are pretreated by using a successive ion adsorption and reaction (SIAR) method that enhances the loading and dispersion of Pt catalysts when electrodeposited. In the SIAR method a Pd activator can be used to provide uniform nucleation sites for Pt and leads to increased Pt loading on the H‐TNTs. Furthermore, fabricated Pt nanoparticles with a diameter of 3.4 nm are located uniformly around the pretreated H–TNT support. The as‐prepared and highly ordered electrodes exhibit excellent stability during accelerated durability tests, particularly for the H–TNT‐loaded Pt catalysts that have been annealed in ultrahigh purity H2 for a second time. There is minimal decrease in the electrochemical surface area of the as‐prepared electrode after 1000 cycles compared to a 68 % decrease for the commercial JM 20 % Pt/C electrode after 800 cycles. X‐ray photoelectron spectroscopy shows that after the H–TNT‐loaded Pt catalysts are annealed in H2 for the second time, the strong metal–support interaction between the H–TNTs and the Pt catalysts enhances the electrochemical stability of the electrodes. Fuel‐cell testing shows that the power density reaches a maximum of 500 mW cm−2 when this highly ordered electrode is used as the anode. When used as the cathode in a fuel cell with extra‐low Pt loading, the new electrode generates a specific power density of 2.68 kW gPt−1. It is indicated that H–TNT arrays, which have highly ordered nanostructures, could be used as ordered electrode supports.
Graphlets are induced subgraph patterns and have been frequently applied to characterize the local topology structures of graphs across various domains, e.g., online social networks (OSNs) and biological networks. Discovering and computing graphlet statistics are highly challenging. First, the massive size of real-world graphs makes the exact computation of graphlets extremely expensive. Secondly, the graph topology may not be readily available so one has to resort to web crawling using the available application programming interfaces (APIs). In this work, we propose a general and novel framework to estimate graphlet statistics of "any size". Our framework is based on collecting samples through consecutive steps of random walks. We derive an analytical bound on the sample size (via the Chernoff-Hoeffding technique) to guarantee the convergence of our unbiased estimator. To further improve the accuracy, we introduce two novel optimization techniques to reduce the lower bound on the sample size. Experimental evaluations demonstrate that our methods outperform the state-of-the-art method up to an order of magnitude both in terms of accuracy and time cost.
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