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1. Experimental study of the $$ \gamma p \rightarrow K^{0}\Sigma^{+}$$, $$ \gamma n \rightarrow K^{0} \Lambda$$, and $$ \gamma n \rightarrow K^{0} \Sigma^{0}$$ reactions at the Mainz Microtron

   https://doi.org/10.1140/epja/i2019-12924-x  

   Akondi, C. S.; Bantawa, K.; Manley, D. M.; Abt, S.; Achenbach, P.; Afzal, F.; Aguar-Bartolomé, P.; Ahmed, Z.; Annand, J. R.; Arends, H. J.; et al (November 2019, The European Physical Journal A)

   null (Ed.)

   Abstract. This work measured $$ \mathrm{d}\sigma/\mathrm{d}\Omega$$ d σ / d Ω for neutral kaon photoproduction reactions from threshold up to a c.m. energy of 1855MeV, focussing specifically on the $$ \gamma p\rightarrow K^0\Sigma^+$$ γ p → K 0 Σ + , $$ \gamma n\rightarrow K^0\Lambda$$ γ n → K 0 Λ , and $$ \gamma n\rightarrow K^0 \Sigma^0$$ γ n → K 0 Σ 0 reactions. Our results for $$ \gamma n\rightarrow K^0 \Sigma^0$$ γ n → K 0 Σ 0 are the first-ever measurements for that reaction. These data will provide insight into the properties of $$ N^{\ast}$$ N * resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the $$ \pi N$$ π N channel. Integrated cross sections were extracted by fitting the differential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions. 

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2. Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

   https://doi.org/10.1017/fms.2020.60  

   Chirvasitu, Alex; Kanda, Ryo; Smith, S. Paul (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract The elliptic algebras in the title are connected graded $$\mathbb {C}$$ -algebras, denoted $$Q_{n,k}(E,\tau )$$ , depending on a pair of relatively prime integers $$n>k\ge 1$$ , an elliptic curve E and a point $$\tau \in E$$ . This paper examines a canonical homomorphism from $$Q_{n,k}(E,\tau )$$ to the twisted homogeneous coordinate ring $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ on the characteristic variety $$X_{n/k}$$ for $$Q_{n,k}(E,\tau )$$ . When $$X_{n/k}$$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ is surjective, the relations for $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ are generated in degrees $$\le 3$$ and the noncommutative scheme $$\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $$X_{n/k}=E^g$$ and $$\tau =0$$ , the results about $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ show that the morphism $$\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces. 

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3. SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

   https://doi.org/10.1017/s1446788718000198  

   LIU, WENCAI (August 2019, Journal of the Australian Mathematical Society)

   In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence $${\mathcal{D}}$$ , we obtain a sharp criterion such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ for a certain one-parameter family of $$\unicode[STIX]{x1D713}$$ . Also, under a minor condition on pseudo-absolute-value sequences $${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$$ , we obtain a sharp criterion on a general sequence $$\unicode[STIX]{x1D713}(n)$$ such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ . 

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4. Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

   https://doi.org/10.1017/S096354831900021X  

   Gao, Wei; Han, Jie; Zhao, Yi (November 2019, Combinatorics, Probability and Computing)

   Abstract Given two k -graphs ( k -uniform hypergraphs) F and H , a perfect F -tiling (or F -factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H . For all complete k -partite k -graphs K , Mycroft proved a minimum codegree condition that guarantees a K -factor in an n -vertex k -graph, which is tight up to an error term o ( n ). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K (k) (1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively. 

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5. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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