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1. Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

   https://doi.org/10.1090/tran/8903  

   Bañuelos, Rodrigo; Mariano, Phanuel; Wang, Jing (August 2023, Transactions of the American Mathematical Society)

   For domains in R d \mathbb {R}^d , d ≥ 2 d\geq 2 , we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power p > 0 p>0 and the supremum over all starting points of the p p -moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of p p and that for p ≥ 1 p \geq 1 , the upper bound is asymptotically sharp as d → ∞ d\to \infty . For all p > 0 p>0 , we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal. 

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2. MBSE Online Professional Development Learning Modules Data

   https://doi.org/10.4231/3DR2-VY15  

   Douglas, Kerrie; Huang, Wanju; Li, Tiantian; Marcos, Leonardo; Huang, Wanju (January 2025, Purdue University Research Repository)

   <p>The data was downloaded and captured through MBSE online learning modules. Deidentified learners&#39; activities within the modules, such as clickstreams and assignments, were captured in the data/</p> <p>&bull; All files here are student submissions to one or more of the modules in the MBSE program.</p> <p>&bull; All user data has either been removed or redacted from the submission.</p> <p>&bull; &ldquo;Andrew Hurt&rdquo; is not a student and none of these files came from him. He is the person who did the redaction.</p> <p>&bull; The naming structure of the files is as follows: [Module number]-[Module Offering Date]-[Submission Number]-[Part Number of Submission]. Example: M5-040422-S1-Part3. This file is from Module 5, which was offered on April 4, 2022, it is submission 1, and part 3 of submission 1.</p> <p>&bull; Note that submissions within or between modules are not necessarily connected to specific students. So &ldquo;Submission 1&rdquo; from module 5 is not the same user as &ldquo;Submission 1&rdquo; from module 6.</p> <p>&bull; Not all submissions have multiple parts.</p> <p>&bull; No .mdzip files (proprietary MagicDraw software files) have been included in this list.</p> <p>&bull; If a module or folder in the module is missing content from a particular offering, it is because either no one submitted anything or because the file was a .mdzip file and was not downloaded.</p> <p>&nbsp;</p> 

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3. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces

   https://doi.org/10.54330/afm.127419  

   Alvarado, Ryan; Hajłasz, Piotr; Malý, Lukáš (October 2022, Annales Fennici Mathematici)

   We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space \(X\) supports a \(p\)-Poincaré inequality, then the \(N^{1,p}(X)\) Sobolev space is reflexive and separable whenever \(p\in (1,\infty)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^{1,p}(X)\), \(p\in [1,\infty)\), that is uniformly convex when \(p\in (1,\infty)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p\in (1,\infty)\). 

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4. ChemML : A machine learning and informatics program package for the analysis, mining, and modeling of chemical and materials data

   https://doi.org/10.1002/wcms.1458  

   Haghighatlari, Mojtaba; Vishwakarma, Gaurav; Altarawy, Doaa; Subramanian, Ramachandran; Kota, Bhargava U.; Sonpal, Aditya; Setlur, Srirangaraj; Hachmann, Johannes (January 2020, WIREs Computational Molecular Science)

   Abstract ChemMLis an open machine learning (ML) and informatics program suite that is designed to support and advance the data‐driven research paradigm that is currently emerging in the chemical and materials domain.ChemMLallows its users to perform various data science tasks and execute ML workflows that are adapted specifically for the chemical and materials context. Key features are automation, general‐purpose utility, versatility, and user‐friendliness in order to make the application of modern data science a viable and widely accessible proposition in the broader chemistry and materials community.ChemMLis also designed to facilitate methodological innovation, and it is one of the cornerstones of the software ecosystem for data‐driven in silico research. This article is categorized under:Software > Simulation MethodsComputer and Information Science > ChemoinformaticsStructure and Mechanism > Computational Materials ScienceSoftware > Molecular Modeling 

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5. The pentagram map, Poncelet polygons, and commuting difference operators

   https://doi.org/10.1112/S0010437X22007345  

   Izosimov, Anton (May 2022, Compositio Mathematica)

   The pentagram map takes a planar polygon $$P$$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $$P$$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $$P$$ is a Poncelet polygon, then the image of $$P$$ under the pentagram map is projectively equivalent to $$P$$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions. 

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