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1. Partitioned matching games for international kidney exchange

   https://doi.org/10.1007/s10107-025-02200-9  

   Benedek, Márton; Biró, Péter; Kern, Walter; Pálvölgyi, Dömötör; Paulusma, Daniel (February 2025, Mathematical Programming)

   Abstract We introduce partitioned matching games as a suitable model for international kidney exchange programmes, where in each round the total number of available kidney transplants needs to be distributed amongst the participating countries in a “fair” way. A partitioned matching game (N, v) is defined on a graph$$G=(V,E)$$ $G=(V,E)$with an edge weightingwand a partition$$V=V_1 \cup \dots \cup V_n$$ $V=V_1\cup \cdots \cup V_n$. The player set is$$N = \{ 1, \dots , n\}$$ $N=\{1,\cdots ,n\}$, and player$$p \in N$$ $p\in N$owns the vertices in$$V_p$$ $V_p$. The valuev(S) of a coalition $$S \subseteq N$$ $S\subseteq N$is the maximum weight of a matching in the subgraph ofGinduced by the vertices owned by the players in S. If$$|V_p|=1$$ $|V_p|=1$for all$$p\in N$$ $p\in N$, then we obtain the classical matching game. Let$$c=\max \{|V_p| \; |\; 1\le p\le n\}$$ $c=max\left\{\right|V_p\left|\right|1\le p\le n\}$be the width of (N, v). We prove that checking core non-emptiness is polynomial-time solvable if$$c\le 2$$ $c\le 2$but co--hard if$$c\le 3$$ $c\le 3$. We do this via pinpointing a relationship with the known class ofb-matching games and completing the complexity classification on testing core non-emptiness forb-matching games. With respect to our application, we prove a number of complexity results on choosing, out of possibly many optimal solutions, one that leads to a kidney transplant distribution that is as close as possible to some prescribed fair distribution. 

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2. More accurate $$ \sigma \left(\mathcal{GG}\to h\right),\Gamma \left(h\to \mathcal{GG},\mathcal{AA},\overline{\Psi}\Psi \right) $$ and Higgs width results via the geoSMEFT

   https://doi.org/10.1007/JHEP01(2024)170  

   Martin, Adam; Trott, Michael (January 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We develop Standard Model Effective Field Theory (SMEFT) predictions ofσ($$ \mathcal{GG} $$ $\mathrm{GG}$→h), Γ(h→$$ \mathcal{GG} $$ $\mathrm{GG}$), Γ(h→$$ \mathcal{AA} $$ $\mathrm{AA}$) to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent Γ(h→$$ \overline{\Psi}\Psi $$ $\overline{\Psi }\Psi$) results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes Γ, σto a full set of corrections at$$ \mathcal{O}\left({\overline{v}}_T^2/{\varLambda}^2{\left(16{\pi}^2\right)}^2\right) $$ $O\left({\overline{v}}_T^2/{\Lambda }^2{\left(16{\pi }^2\right)}^2\right)$and$$ \mathcal{O}\left({\overline{v}}_T^4/{\Lambda}^4\right) $$ $O\left({\overline{v}}_T^4/{\Lambda }^4\right)$, where$$ {\overline{v}}_T $$ ${\overline{v}}_T$is the electroweak scale vacuum expectation value and Λ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For Γ(h→$$ \overline{\Psi}\Psi $$ $\overline{\Psi }\Psi$), we include results at$$ \mathcal{O}\left({\overline{v}}_T^2/{\Lambda}^2\left(16{\pi}^2\right)\right) $$ $O\left({\overline{v}}_T^2/{\Lambda }^2\left(16{\pi }^2\right)\right)$in the limit where subleadingm_Ψ→ 0 corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects. 

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3. Student Interpretations of Eigenequations in Linear Algebra and Quantum Mechanics

   https://doi.org/10.1007/s40753-024-00241-7  

   Wawro, Megan; Pina, Andi; Thompson, John_R; Topdemir, Zeynep; Watson, Kevin (June 2024, International Journal of Research in Undergraduate Mathematics Education)

   Abstract This work investigates how students interpret various eigenequations in different contexts for$$2 \times 2$$ $2\times 2$matrices:$$A\vec {x}=\lambda \vec {x}$$ $A\overrightarrow{x}=\lambda \overrightarrow{x}$in mathematics and either$$\hat{S}_x| + \rangle _x=\frac{\hbar }{2}| + \rangle _x$$ ${\hat{S}}_x{|+⟩}_x=\frac{ħ}{2}{|+⟩}_x$or$$\hat{S}_z| + \rangle =\frac{\hbar }{2}| + \rangle$$ ${\hat{S}}_z|+⟩=\frac{ħ}{2}|+⟩$in quantum mechanics. Data were collected from two sources in a senior-level quantum mechanics course; one is video, transcript and written work of individual, semi-structured interviews; the second is written work from the same course three years later. We found two principal ways in which students reasoned about the equal sign within the mathematics eigenequation and at times within the quantum mechanical eigenequations: with a functional interpretation and/or a relational interpretation. Second, we found three distinct ways in which students explained how they made sense of the physical meaning conveyed by the quantum mechanical eigenequations: via a measurement interpretation, potential measurement interpretation, or correspondence interpretation of the equation. Finally, we present two themes that emerged in the ways that students compared the different eigenequations: attention to form and attention to conceptual (in)compatibility. These findings are discussed in relation to relevant literature, and their instructional implications are also explored. 

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4. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (March 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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5. 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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