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The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. [ABSTRACT FROM AUTHOR]

Massively multiplayer online games (MMOGs) played on the Web provide a new form of social, computer-mediated interactions that allow the connection of millions of players worldwide. The rules governing team-based MMOGs are typically complex and nondeterministic giving rise to an intricate dynamical behavior. However, due to the novelty and complexity of MMOGs, their behavior is understudied. In this article, we investigate the MMOG World of Tanks Blitz by using a combined approach based on data science and complex adaptive systems. We analyze data on the population level to get insights into organizational principles of the game and its game mechanics. For this reason, we study the scaling behavior and the predictability of system variables. As a result, we find a power-law behavior on the population level revealing long-range interactions between system variables. Furthermore, we identify and quantify the predictability of summary statistics of the game and its decomposition into explanatory variables. This reveals a heterogeneous progression through the tiers and identifies only a single system variable as key driver for the win rate. [ABSTRACT FROM AUTHOR]