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Graphene ripples possess peculiar essential properties owing to the strong chemical bonds, as an investigation using first principle calculations clearly revealed. Various charge distributions, bond lengths, energy bands, and densities of states strongly depend on the corrugation structures, ripple curvatures and periods. Armchair ripples belonging to a zero-gap semiconductor display split middle-energy states, while the zigzag ripples exhibit highly anisotropic energy bands, semi-metallic behavior implicated by the destruction of the Dirac cone, and the newly created critical points.Their density of states exhibit many low-lying prominent peaks and can explain the experimental measurements. There exist certain important similarities and differences between graphene ripples and carbon nanotubes.

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In this article, we propose an end-to-end adaptive framework for optimal trade execution based on Proximal Policy Optimization (PPO). We use two methods to account for the time dependencies in the market data based on two different neural network architecture: 1) Long short-term memory (LSTM) networks, 2) Fully-connected networks (FCN) by stacking the most recent limit orderbook (LOB) information as model inputs. The proposed framework can make trade execution decisions based on level-2 limit order book (LOB) information such as bid/ask prices and volumes directly without manually designed attributes as in previous research. Furthermore, we use a sparse reward function, which gives the agent reward signals at the end of each episode as an indicator of its relative performances against the baseline model, rather than implementation shortfall (IS) or a shaped reward function. The experimental results have demonstrated advantages over IS and the shaped reward function in terms of performance and simplicity. The proposed framework has outperformed the industry commonly used baseline models such as TWAP, VWAP, and AC as well as several Deep Reinforcement Learning (DRL) models on most of the 14 US equities in our experiments.