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The Frobenius-Perron theory of an endofunctor of a k \Bbbk -linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras. 

To study the sensing mechanism of bat's biosonar system, we propose a fast simulation algorithm to generate natural-looking trees and forest---the primary living habitat of bats. We adopt 3D Lindenmayer system to create the fractal geometry of the trees, and add additional parameters, both globally and locally, to enable random variations of the tree structures. Random forest is then formed by placing simulated trees at random locations of a field according to a spatial point process. By employing a single algorithmic model with different numeric parameters, we can rapidly simulate 3D virtual environments with a wide variety of trees, producing detailed geometry of the foliage such as the leaf locations, sizes, and orientations. Written in C++ and visualized with openGL, our algorithm is fast to implement, easily parallable, and more adaptive to real-time visualization compared with existing alternative approaches. Our simulated environment can be used for general purposes such as studying new sensors or training remote sensing algorithms. 

Summary This paper develops a functional hybrid factor regression modelling framework to handle the heterogeneity of many large-scale imaging studies, such as the Alzheimer’s disease neuroimaging initiative study. Despite the numerous successes of those imaging studies, such heterogeneity may be caused by the differences in study environment, population, design, protocols or other hidden factors, and it has posed major challenges in integrative analysis of imaging data collected from multicentres or multistudies. We propose both estimation and inference procedures for estimating unknown parameters and detecting unknown factors under our new model. The asymptotic properties of both estimation and inference procedures are systematically investigated. The finite-sample performance of our proposed procedures is assessed by using Monte Carlo simulations and a real data example on hippocampal surface data from the Alzheimer’s disease study. 

Cybercrime was estimated to cost the global economy $945 billion in 2020. Increasingly, law enforcement agencies are using social network analysis (SNA) to identify key hackers from Dark Web hacker forums for targeted investigations. However, past approaches have primarily focused on analyzing key hackers at a single point in time and use a hacker’s structural features only. In this study, we propose a novel Hacker Evolution Identification Framework to identify how hackers evolve within hacker forums. The proposed framework has two novelties in its design. First, the framework captures features such as user statistics, node-level metrics, lexical measures, and post style, when representing each hacker with unsupervised graph embedding methods. Second, the framework incorporates mechanisms to align embedding spaces across multiple time-spells of data to facilitate analysis of how hackers evolve over time. Two experiments were conducted to assess the performance of prevailing graph embedding algorithms and nodal feature variations in the task of graph reconstruction in five timespells. Results of our experiments indicate that Text- Associated Deep-Walk (TADW) with all of the proposed nodal features outperforms methods without nodal features in terms of Mean Average Precision in each time-spell. We illustrate the potential practical utility of the proposed framework with a case study on an English forum with 51,612 posts. The results produced by the framework in this case study identified key hackers posting piracy assets.