We introduce Codex, a GPT language model finetuned on publicly available code from GitHub, and study its Python code-writing capabilities. A distinct production version of Codex powers GitHub Copilot. On HumanEval, a new evaluation set we release to measure functional correctness for synthesizing programs from docstrings, our model solves 28.8% of the problems, while GPT-3 solves 0% and GPT-J solves 11.4%. Furthermore, we find that repeated sampling from the model is a surprisingly effective strategy for producing working solutions to difficult prompts. Using this method, we solve 70.2% of our problems with 100 samples per problem. Careful investigation of our model reveals its limitations, including difficulty with docstrings describing long chains of operations and with binding operations to variables. Finally, we discuss the potential broader impacts of deploying powerful code generation technologies, covering safety, security, and economics.

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Small TiO2 crystals in the anatase phase are in high demand as photocatalysts. Stable TiO2 crystals in the anatase phase were obtained using a silica nanoparticle as a support. The focus of this study was to investigate the nanoscale effect of the silica support on the formation and properties of small anatase crystals. The experiments were carried out using powder X-ray diffraction, differential thermal analysis, transmission electron microscopy, and energy dispersion spectroscopy. The results showed that the size of the silica support played a crucial role in crystallization of TiO2 and regulation of TiO2 properties, including phase transition, crystal size, thermodynamic property and catalytic activity. A nanoscale curvature model of the spherical silica support was proposed to explain these size effects. Finally, the developed TiO2 catalysts were applied to the oxidation of methanol using a high-throughput photochemical reactor. The size effect of the silica supports on the TiO2 catalytic efficiency was demonstrated using this system.

A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs. arXiv:0808.3610v2 [math.CO]Abstract A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs.

An {\it Omnibus Sequence} of length $n$ is one that has each possible "message" of length $k$ embedded in it as a subsequence. We study various properties of Omnibus Sequences in this paper, making connections, whenever possible, to the classical coupon collector problem.Comment: 26 page