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  1. Free, publicly-accessible full text available April 17, 2024
  2. The ladle furnace plays a critical role in the secondary steelmaking stage, where many processes take place in the ladle such as steel property and temperature homogenization, inclusion removal, degassing, and desulfurization. Although many research has been conducted to study these aspects, due to the complicated heat and mass transfer process inside the ladle, many details about the physical process are still not quite clear. For example, the efficacy of plug/injector designs in turbulent mixing of molten steel were not fully understood. Due to its complex three dimensional flow phenomena inside the ladle, previous two dimensional flow measurement of water ladle models provided little insight into understanding the three dimensional flow phenomenon of turbulent mixing. Therefore, to achieve a better understanding on the efficacy of plug/injector designs in turbulent mixing, we implemented an advanced volumetric flow measurement instrument of Shake-the-Box system to measure the three-dimensional flow field inside a water ladle model. Totally, three different plug/injector designs were tested under two different flow rates (8 LPM and 11.5 LPM) of gas injection within a volumetric flow measurement region of 4.8 cm × 4.8 cm × 2.4 cm. The flow measurement results suggest the double slits injector produces the highest turbulence kinetic energy comparing the three injectors. 
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    Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks. 
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  5. Since its publication, the authors of Wang et al. (2021) have brought to our attention an error in their article. A grant awarded by the National Science Foundation (grant no. MCB 1817985) to author Elizabeth Vierling was omitted from the Acknowledgements section. The correct Acknowledgements section is shown below. Acknowledgements We thank Suiwen Hou (Lanzhou University) and Zhaojun Ding (Shandong University) for providing the seeds used in this study. We thank Xiaoping Gou (Lanzhou University) and Ravishankar Palanivelu (University of Arizona) for critically reading the manuscript and for suggestions regarding the article. This work was supported by grants from National Natural Science Foundation of China (31870298) to SX, the US Department of Agriculture (USDA-CSREES-NRI-001030) and the National Science Foundation (MCB 1817985) to EV, and the Youth 1000-Talent Program of China (A279021801) to LY. 
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