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1. L-spaces, taut foliations, and graph manifolds

   https://doi.org/10.1112/S0010437X19007814  

   Hanselman, Jonathan; Rasmussen, Jacob; Rasmussen, Sarah Dean; Watson, Liam (March 2020, Compositio Mathematica)

   null (Ed.)

   If $$Y$$ is a closed orientable graph manifold, we show that $$Y$$ admits a coorientable taut foliation if and only if $$Y$$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $$Y$$ is an L-space if and only if $$\unicode[STIX]{x1D70B}_{1}(Y)$$ is not left-orderable. 

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2. Constructing nonproxy small test modules for the complete intersection property

   https://doi.org/10.1017/nmj.2021.7  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (June 2021, Nagoya Mathematical Journal)

   Abstract A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $$\mathsf {D}^{\mathsf f}(R)$$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $$\mathsf {D}^{\mathsf f}(R)$$ is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings. 

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3. Uniformly Bounded Arrays and Mutually Algebraic Structures

   https://doi.org/doi:10.1215/00294527-2020-0004  

   Laskowski, Michael C.; Terry, Caroline A. (January 2020, Notre Dame journal of formal logic)

   We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T. 

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4. Subject clustering by IF-PCA and several recent methods

   https://doi.org/10.3389/fgene.2023.1166404  

   Chen, Dieyi; Jin, Jiashun; Ke, Zheng Tracy (May 2023, Frontiers in Genetics)

   Subject clustering (i.e., the use of measured features to cluster subjects, such as patients or cells, into multiple groups) is a problem of significant interest. In recent years, many approaches have been proposed, among which unsupervised deep learning (UDL) has received much attention. Two interesting questions are 1) how to combine the strengths of UDL and other approaches and 2) how these approaches compare to each other. We combine the variational auto-encoder (VAE), a popular UDL approach, with the recent idea of influential feature-principal component analysis (IF-PCA) and propose IF-VAE as a new method for subject clustering. We study IF-VAE and compare it with several other methods (including IF-PCA, VAE, Seurat, and SC3) on 10 gene microarray data sets and eight single-cell RNA-seq data sets. We find that IF-VAE shows significant improvement over VAE, but still underperforms compared to IF-PCA. We also find that IF-PCA is quite competitive, slightly outperforming Seurat and SC3 over the eight single-cell data sets. IF-PCA is conceptually simple and permits delicate analysis. We demonstrate that IF-PCA is capable of achieving phase transition in a rare/weak model. Comparatively, Seurat and SC3 are more complex and theoretically difficult to analyze (for these reasons, their optimality remains unclear). 

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5. EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS

   https://doi.org/10.1017/S0004972721000332  

   BEIKE, NICOLAS F.; CARLETON, RACHEL; COSTANZO, DAVID G.; HEATH, COLIN; LEWIS, MARK L.; LU, KAIWEN; PEARCE, JAMIE D. (January 2021, Bulletin of the Australian Mathematical Society)

   null (Ed.)

   Abstract: Morgan and Parker proved that if G is a group with Z(G)=1, then the connected components of the commuting graph of G have diameter at most 10. Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a 2-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most 8. We prove that the hypothesis Z (G) = 1 in these results can be replaced with G' \cap Z(G)=1. We also prove that if G is solvable and G/Z(G) is either a Frobenius group or a 2-Frobenius group, then the commuting graph of G is disconnected. 

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