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1. Juvenile rank acquisition is associated with fitness independent of adult rank

   https://doi.org/10.1098/rspb.2019.2969  

   Strauss, Eli D.; Shizuka, Daizaburo; Holekamp, Kay E. (March 2020, Proceedings of the Royal Society B: Biological Sciences)

   Social rank is a significant determinant of fitness in a variety of species. The importance of social rank suggests that the process by which juveniles come to establish their position in the social hierarchy is a critical component of development. Here, we use the highly predictable process of rank acquisition in spotted hyenas to study the consequences of variation in rank acquisition in early life. In spotted hyenas, rank is ‘inherited’ through a learning process called ‘maternal rank inheritance.’ This pattern is very consistent: approximately 80% of juveniles acquire the exact rank expected under the rules of maternal rank inheritance. The predictable nature of rank acquisition in these societies allows the process of rank acquisition to be studied independently from the ultimate rank that each juvenile attains. In this study, we use Elo-deviance scores, a novel application of the Elo-rating method, to calculate each juvenile's deviation from the expected pattern of maternal rank inheritance during development. Despite variability in rank acquisition among juveniles, most of these juveniles come to attain the exact rank expected of them according to the rules of maternal rank inheritance. Nevertheless, we find that transient variation in rank acquisition in early life is associated with long-term fitness consequences for these individuals: juveniles ‘underperforming’ their expected ranks show reduced survival and lower lifetime reproductive success than better-performing peers, and this relationship is independent of both maternal rank and rank achieved in adulthood. We also find that multiple sources of early life adversity have cumulative, but not compounding, effects on fitness. Future work is needed to determine if variation in rank acquisition directly affects fitness, or if some other variable, such as maternal investment or juvenile condition, causes variation in both of these outcomes. 

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2. Graphs with large rank numbers and rank numbers of subdivided stars

   Hayley Boynton, Ethan Burroughs (April 2019, AKCE International Journal of Graphs and Combinatorics)

   A k-ranking of a graph G is a labeling with k labels such that if f(u) = f(v) then every uv path contains a vertex w such that f(w) > f(u). The rank number of G, denoted Xr (G), is the minimum k such that a k-ranking exists for G. In this paper we characterize graphs with large rank numbers. In addition, we characterize subdivided stars based on their rank numbers. 

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3. RANK AND BORDER RANK OF KRONECKER POWERS OF TENSORS AND STRASSEN’S LASER METHOD

   https://doi.org/10.1007/s00037-021-00217-y  

   Conner, Austin; Gesmundo, Fulvio; Landsberg, Joseph M.; Ventura, Emanuele (April 2022, Computational complexity)

   We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for q > 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bl¨aser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen’s laser method, introducing a skew-symmetric version of the Coppersmith– Winograd tensor, Tskewcw,q. For q = 2, the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det3 ∈ C9 ⊗ C9 ⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 ⊗ C3 ⊗ C3. 

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4. Statistical Rank Selection for Incomplete Low-rank Matrices

   https://doi.org/10.1109/ICASSP.2019.8682540  

   Zhang, Rui; Shapiro, Alexander; Xie, Yao (May 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))
5. Extreme Learning to Rank via Low Rank Assumption

   Cheng, Minhao; Davidson, Ian; Hsieh, Cho-Jui (July 2018, International Conference on Machine Learning (ICML))