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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. Quasiconformal surgery and linear differential equations

   https://doi.org/10.1007/s11854-019-0007-9  

   Bergweiler, Walter ; Eremenko, Alexandre ( August 2019 , Journal d'Analyse Mathématique)

   We describe a new method of constructing transcendental entire functions A such that the differential equation w″ + Aw = 0 has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functions A of any prescribed order greater than 1/2 such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the estimate obtained is best possible. Our method is based on gluing solutions of the Schwarzian differential equation S(F) = 2A for infinitely many coefficients A. 

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