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1. Classifying primitive solvable permutation groups of rank 5 and 6

   https://doi.org/10.1515/jgth-2023-0205  

   Dey, Anakin; O’Neal, Kolton; Tran, Duc_Van Khanh; Upshur, Camron; Yang, Yong (April 2024, Journal of Group Theory)

   Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω.Let $G_0$G_{0}be the stabilizer of a point 𝛼 in Ω.The rank of 𝐺 is defined as the number of orbits of $G_0$G_{0}in Ω, including the trivial orbit $\left\{\alpha \right\}$\{\alpha\}.In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower. 

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2. GROUPS OF MORLEY RANK 4

   https://doi.org/10.1017/jsl.2014.81  

   WISCONS, JOSHUA (March 2016, The Journal of Symbolic Logic)

   Abstract We show that any simple group of Morley rank 4 must be a bad group with no proper definable subgroups of rank larger than 1. We also give an application to groups acting on sets of Morley rank 2. 

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3. SIMPLE GROUPS OF MORLEY RANK 5 ARE BAD

   https://doi.org/10.1017/jsl.2017.86  

   DELORO, ADRIEN; WISCONS, JOSHUA (September 2018, The Journal of Symbolic Logic)

   Abstract We show that any simple group of Morley rank 5 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 2. The main result is then used to catalog the nonsoluble connected groups of Morley rank 5. 

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4. Coloring graphs with no induced five‐vertex path or gem

   https://doi.org/10.1002/jgt.22572  

   Chudnovsky, Maria; Karthick, T.; Maceli, Peter; Maffray, Frédéric (April 2020, Journal of Graph Theory)

   For a graph 𝐺 , let 𝜒(𝐺) and 𝜔(𝐺) , respectively, denote the chromatic number and clique number of 𝐺 . We give an explicit structural description of ( 𝑃5 , gem)‐free graphs, and show that every such graph 𝐺 satisfies 𝜒(𝐺)≤⌈5𝜔(𝐺)4⌉ . Moreover, this bound is best possible. Here a gem is the graph that consists of an induced four‐vertex path plus a vertex which is adjacent to all the vertices of that path. 

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5. Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions

   Asi, Hilal; Liu, Daogao; Tian, Kevin (June 2024, https://doi.org/10.48550/arXiv.2406.02789)

   This paper studies differentially private stochastic convex optimization (DP-SCO) in the presence of heavy-tailed gradients, where only a 𝑘 kth-moment bound on sample Lipschitz constants is assumed, instead of a uniform bound. The authors propose a reduction-based approach that achieves the first near-optimal error rates (up to logarithmic factors) in this setting. Specifically, under ( 𝜖 , 𝛿 ) (ϵ,δ)-approximate differential privacy, they achieve an error bound of 𝐺 2 𝑛 + 𝐺 𝑘 ⋅ ( 𝑑 𝑛 𝜖 ) 1 − 1 𝑘 , n ​ G 2 ​ ​ +G k ​ ⋅( nϵ d ​ ​ ) 1− k 1 ​ , up to a mild polylogarithmic factor in 1 𝛿 δ 1 ​ , where 𝐺 2 G 2 ​ and 𝐺 𝑘 G k ​ are the 2nd and 𝑘 kth moment bounds on sample Lipschitz constants. This nearly matches the lower bound established by Lowy and Razaviyayn (2023). Beyond the basic result, the authors introduce a suite of private algorithms that further improve performance under additional assumptions: an optimal algorithm under a known-Lipschitz constant, a near-linear time algorithm for smooth functions, and an optimal linear-time algorithm for smooth generalized linear models. 

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