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1. 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

   https://doi.org/10.1007/JHEP01(2021)142  

   Gráf, Lukáš; Henning, Brian; Lu, Xiaochuan; Melia, Tom; Murayama, Hitoshi (January 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p 8 , as well as enumeration of CP -even, CP -odd, C -odd, and P -odd terms beginning from order p 6 . The method is extendable to very high orders, and we present results up to order p 16 . (The title sequence is the number of independent C -even and P -even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p 2 , p 4 , p 6 , …) 

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2. On the generalized Fitting height and insoluble length of finite groups

   https://doi.org/doi.org/10.1112/blms.12372  

   Guralnick, Robert; Tracey, Gareth (January 2020, The bulletin of the London Mathematical Society)

   null (Ed.)

   We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by‐product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism. 

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3. Words, Hausdorff dimension and randomly free groups

   https://doi.org/https://doi.org/10.1007/s00208-017-1635-y  

   Larsen, Michael; Shalev, Aner (July 2018, Mathematische Annalen)

   We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$ there exists $$\e > 0$$ such that if $$\Gamma$$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma^d \rightarrow \Gamma$$ have Hausdorff dimension at most $$d -\e$$. We conclude that profinite groups $$G := \hat\Gamma$$, $$\Gamma$$ as above, satisfy no probabilistic identity, and therefore they are \emph{randomly free}, namely, for any $$d \ge 1$$, the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$ freely generate a free subgroup (isomorphic to $$F_d$$) is $$1$$. This solves an open problem from \cite{DPSS}. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem. 

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4. Vehicle Miles Traveled and Environmental Impacts from On-Demand Delivery: A Literature Review

   https://doi.org/10.1061/9780784485521.004  

   Liu, Haishan; Hao, Peng; Tanvir, Shams; Pande, Anurag; Barth, Matthew (June 2024, American Society of Civil Engineers)

   The boom of e-commerce and the increasing demand for fast and reliable delivery services have led to the thriving of on-demand delivery (ODD), which provides delivery services to food takeout, grocery, pharmacy, and other light-weighted goods. The operational efficiency of ODD is subject to many factors—access to curbside, delays at the pick-up and drop-off locations, order dispatching mode, vehicle routing schedule, and vehicle refueling needs. The fast-growing delivery orders coupled with operational inefficiencies of ODD may lead to higher vehicle miles traveled (VMT) and pollutant emissions. Policymakers as well as practitioners need to evaluate the VMT and emissions impact of ODD, given the consumer behavior, operational paradigm, and business models. This paper conducted a systematic review of the existing literature to synthesize and summarize the impacts of ODD with a specific focus on VMT and emissions. The PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analysis) guideline was employed to systematically search for related studies in multiple databases and to crystallize the review scope. The impact evaluation was delved into three aspects: customer shopping behavior (online shopping vs. in-store shopping), ODD operational strategy (truck/van vs. green vehicles, professional delivery vs. crowdsourcing), and business models (home delivery vs. depot/collection point). Overall, this study found that online shopping with coordinated ODD can achieve significant VMT and emissions reduction compared with in-store shopping. The reduction extent depends on the customer trip chaining, travel mode choice, residential area type, and the ratio of product return. The use of zero-emissions vehicles in ODD, such as electric van/truck/vehicle, cargo-bike, UAV, provides relatively higher emissions reductions, but also brings new issues such as charging needs or capacity limits. Collection points (e.g., parcel locker, retailer store, postal service point) can reduce the VMT and emissions if they are optimally distributed, and customers visit them in zero-emissions modes or through trip chaining. 

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5. Fully discrete pointwise smoothing error estimates for measure valued initial data

   https://doi.org/10.1051/m2an/2023076  

   Leykekhman, Dmitriy; Vexler, Boris; Wagner, Jakob (September 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interiorL^∞error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interiorL^∞error estimates forL²initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements. 

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