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1. The lattice of submonoids of the uniform block permutations containing the symmetric group

   https://doi.org/10.1007/s00233-025-10505-6  

   Orellana, Rosa; Saliola, Franco; Schilling, Anne; Zabrocki, Mike (April 2025, Semigroup Forum)

   Abstract We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the$$\mathscr {J}$$ $J$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra. 

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2. Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

   https://doi.org/10.1002/cpa.22076  

   Humphries, Peter; Radziwiłł, Maksym (September 2022, Communications on Pure and Applied Mathematics)

   Abstract We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”, a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density 1. subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume with . We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weighting functions for annuli and for balls. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC. 

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3. Clustering with Neighborhoods is Hard for Squares

   Klimenko, Georgiy; Raichel, Benjamin (July 2023, Proceedings of the 35th Canadian Conference on Computational Geometry)

   Pankratov, Denis (Ed.)

   In the clustering with neighborhoods problem one is given a set S of disjoint convex objects in the plane and an integer parameter k ≥ 0, and the goal is to select a set C of k center points in the plane so as to minimize the maximum distance of an object in S to its nearest center in C. Previously [HKR21] showed that this problem cannot be approximated within any factor when S is a set of disjoint line segments, however, when S is a set of disjoint disks there is a roughly 8.46 approximation algorithm and a roughly 6.99 approximation lower bound. In this paper we investigate this significant discrepancy in hardness between these shapes. Specifically, we show that when S is a set of axis aligned squares of the same size, the problem again is hard to approximate within any factor. This surprising fact shows that the discrepancy is not due to the fatness of the object class, as one might otherwise naturally suspect. 

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4. The Emptiness Inside: Finding Gaps, Valleys, and Lacunae with Geometric Data Analysis

   https://doi.org/10.3847/1538-3881/ac961e  

   Contardo, Gabriella; Hogg, David W.; Hunt, Jason A.; Peek, Joshua E.; Chen, Yen-Chi (October 2022, The Astronomical Journal)

   Abstract Discoveries of gaps in data have been important in astrophysics. For example, there are kinematic gaps opened by resonances in dynamical systems, or exoplanets of a certain radius that are empirically rare. A gap in a data set is a kind of anomaly, but in an unusual sense: instead of being a single outlier data point, situated far from other data points, it is a region of the space, or a set of points, that is anomalous compared to its surroundings. Gaps are both interesting and hard to find and characterize, especially when they have nontrivial shapes. We present in this paper a statistic that can be used to estimate the (local) “gappiness” of a point in the data space. It uses the gradient and Hessian of the density estimate (and thus requires a twice-differentiable density estimator). This statistic can be computed at (almost) any point in the space and does not rely on optimization; it allows us to highlight underdense regions of any dimensionality and shape in a general and efficient way. We illustrate our method on the velocity distribution of nearby stars in the Milky Way disk plane, which exhibits gaps that could originate from different processes. Identifying and characterizing those gaps could help determine their origins. We provide in an appendix implementation notes and additional considerations for finding underdensities in data, using critical points and the properties of the Hessian of the density. 7 7 A Python implementation of t methods presented here is available at https://github.com/contardog/FindTheGap . 

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5. A Symmetric Neural Network to Compute Fractional Derivatives by Training with Integer Derivatives

   https://doi.org/10.1109/SII52469.2022.9708840  

   Chen, Tan; Goodwine, Bill (January 2022, 2022 IEEE/SICE International Symposium on System Integration (SII))

   Fractional calculus is an increasingly recognized important tool for modeling complicated dynamics in modern engineering systems. While, in some ways, fractional derivatives are a straight-forward generalization of integer-order derivatives that are ubiquitous in engineering modeling, in other ways the use of them requires quite a bit of mathematical expertise and familiarity with some mathematical concepts that are not in everyday use across the broad spectrum of engineering disciplines. In more colloquial terms, the learning curve is steep. While the authors recognize the need for fundamental competence in tools used in engineering, a computational tool that can provide an alternative means to compute fractional derivatives does have a useful role in engineering modeling. This paper presents the use of a symmetric neural network that is trained entirely on integer-order derivatives to provide a means to compute fractional derivatives. The training data does not contain any fractional-order derivatives at all, and is composed of only integer-order derivatives. The means by which a fractional derivative can be obtained is by requiring the neural network to be symmetric, that is, it is the composition of two identical sets of layers trained on integer-order derivatives. From that, the information contained in the nodes between the two sets of layers contains half-order derivative information 

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