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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.

Mobile augmented reality (AR) has the potential to enable immersive, natural interactions between humans and cyber-physical systems. In particular markerless AR, by not relying on fiducial markers or predefined images, provides great convenience and flexibility for users. However, unwanted virtual object movement frequently occurs in markerless smartphone AR due to inaccurate scene understanding, and resulting errors in device pose tracking. We examine the factors which may affect virtual object stability, design experiments to measure it, and conduct systematic quantitative characterizations across six different user actions and five different smartphone configurations. Our study demonstrates noticeable instances of spatial instability in virtual objects in all but the simplest settings (with position errors of greater than 10cm even on the best-performing smartphones), and underscores the need for further enhancements to pose tracking algorithms for smartphone-based markerless AR.