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1. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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2. Block mapping class groups and their finiteness properties

   https://doi.org/10.1007/s10711-025-00985-9  

   Aramayona, J; Aroca, J; Cumplido, M; Skipper, R; Wu, X (April 2025, Geometriae Dedicata)

   Abstract Given$$g \in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$, let$$\Sigma _g$$ ${\Sigma }_g$denote the closed surface of genusgwith a Cantor set removed, if$$g<\infty $$ $g<\infty$; or the blooming Cantor tree, when$$g= \infty $$ $g=\infty$. We construct a family$$\mathfrak B(H)$$ $B\left(H\right)$of subgroups of$${{\,\textrm{Map}\,}}(\Sigma _g)$$ $\text{Map}\left({\Sigma }_g\right)$whose elements preserve ablock decompositionof$$\Sigma _g$$ ${\Sigma }_g$, andeventually like actlike an element ofH, whereHis a prescribed subgroup of the mapping class group of the block. The group$$\mathfrak B(H)$$ $B\left(H\right)$surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that$$\mathfrak B(H)$$ $B\left(H\right)$is of type$$F_n$$ $F_n$if and only ifHis. As a consequence, for every$$g\in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$and every$$n\ge 1$$ $n\ge 1$, we construct a subgroup$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$ $G<\text{Map}\left({\Sigma }_g\right)$that is of type$$F_n$$ $F_n$but not of type$$F_{n+1}$$ $F_{n+1}$, and which contains the mapping class group of every compact surface of genus$$\le g$$ $\le g$and with non-empty boundary. 

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3. A search for top-squark pair production, in final states containing a top quark, a charm quark and missing transverse momentum, using the 139 fb−1 of pp collision data collected by the ATLAS detector

   https://doi.org/10.1007/JHEP07(2024)250  

   Aad, G; Abbott, B; Abeling, K; Abicht, N J; Abidi, S H; Aboulhorma, A; Abramowicz, H; Abreu, H; Abulaiti, Y; Acharya, B S; et al (July 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> This paper presents a search for top-squark pair production in final states with a top quark, a charm quark and missing transverse momentum. The data were collected with the ATLAS detector during LHC Run 2 and correspond to an integrated luminosity of 139 fb⁻¹of proton-proton collisions at a centre-of-mass energy of$$ \sqrt{s} $$ $\sqrt{s}$= 13 TeV. The analysis is motivated by an extended Minimal Supersymmetric Standard Model featuring a non-minimal flavour violation in the second- and third-generation squark sector. The top squark in this model has two possible decay modes, either$$ {\tilde{t}}_1\to c{\overset{\sim }{\chi}}_1^0 $$ ${\tilde{t}}_1\to c{\tilde{\chi }}_1^0$or$$ {\tilde{t}}_1\to t{\overset{\sim }{\chi}}_1^0 $$ ${\tilde{t}}_1\to t{\tilde{\chi }}_1^0$, where the$$ {\overset{\sim }{\chi}}_1^0 $$ ${\tilde{\chi }}_1^0$is undetected. The analysis is optimised assuming that both of the decay modes are equally probable, leading to the most likely final state of$$ tc+{E}_T^{\textrm{miss}} $$ $\mathrm{tc}+E_T^{\text{miss}}$. Good agreement is found between the Standard Model expectation and the data in the search regions. Exclusion limits at 95% CL are obtained in the$$ m\left({\tilde{t}}_1\right) $$ $m\left({\tilde{t}}_1\right)$vs.$$ m\left({\overset{\sim }{\chi}}_1^0\right) $$ $m\left({\tilde{\chi }}_1^0\right)$plane and, in addition, limits on the branching ratio of the$$ {\tilde{t}}_1\to t{\overset{\sim }{\chi}}_1^0 $$ ${\tilde{t}}_1\to t{\tilde{\chi }}_1^0$decay as a function ofm($$ {\tilde{t}}_1 $$ ${\tilde{t}}_1$) are also produced. Top-squark masses of up to 800 GeV are excluded for scenarios with light neutralinos, and top-squark masses up to 600 GeV are excluded in scenarios where the neutralino and the top squark are almost mass degenerate. 

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4. Ordinary modules for vertex algebras of 𝔬𝔰𝔭 _1|2𝑛

   https://doi.org/10.1515/crelle-2024-0060  

   Creutzig, Thomas; Genra, Naoki; Linshaw, Andrew (August 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that the affine vertex superalgebra $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$V^{k}(\mathfrak{osp}_{1|2n})at generic level 𝑘 embeds in the equivariant 𝒲-algebra of $\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}times $4n$4nfree fermions.This has two corollaries:(1) it provides a new proof that, for generic 𝑘, the coset $\mathrm{Com}(V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right),V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right))$\operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n}))is isomorphic to ${\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})for $\mathrm{\ell }=-\left(n+1\right)+\left(k+n+1\right)/\left(2k+2n+1\right)$\ell=-(n+1)+(k+n+1)/(2k+2n+1), and(2) we obtain the decomposition of ordinary $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$V^{k}(\mathfrak{osp}_{1|2n})-modules into $V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})-modules.Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for $\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}, we show that the simple algebra $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$L_{k}(\mathfrak{osp}_{1|2n})is an extension of the simple subalgebra $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$L_{k}(\mathfrak{sp}_{2n})\otimes{\mathcal{W}}_{\ell}(\mathfrak{sp}_{2n}).Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$L_{k}(\mathfrak{osp}_{1|2n})-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects.It is equivalent to a certain subcategory of ${\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$\mathcal{W}_{\ell}(\mathfrak{sp}_{2n})-modules.A similar result also holds for the category of Ramond twisted modules.Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$L_{k}(\mathfrak{sp}_{2n})-modules are rigid. 

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5. Bounds on eigenfunctions of quantum cat maps

   https://doi.org/10.1088/1402-4896/ad0331  

   Kim, Elena; Koirala, Robert (October 2023, Physica Scripta)

   Abstract We studyℓ^∞norms ofℓ²-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000,Communications in Mathematical Physics,211, 659–686)) we show that there exists a sequence of eigenfunctionsuwith $\parallel u{\parallel }_{\infty }\gtrsim {\left(\log N\right)}^{-1/2}$. For general eigenfunctions we show the upper bound $\parallel u{\parallel }_{\infty }\lesssim {\left(\log N\right)}^{-1/2}$. Here the semiclassical parameter is $h={\left(2\pi N\right)}^{-1}$. Our upper bound is analogous to the one proved by Bérard in P Bérard (1977,Mathematische Zeitschrift,155, 249-276) for compact Riemannian manifolds without conjugate points. 

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