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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. 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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5. The fields of values of characters of degree not divisible by p

   https://doi.org/10.1017/fmp.2021.1  

   Navarro, Gabriel; Tiep, Pham Huu (January 2021, Forum of Mathematics, Pi)

   Abstract We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p . In the case where $p=2$ , we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups. 

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