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1. Schemes Supported on the Singular Locus of a Hyperplane Arrangement in ℙ n

   https://doi.org/10.1093/imrn/rnaa113  

   Migliore, Juan; Nagel, Uwe; Schenck, Henry (May 2020, International Mathematics Research Notices)

   Abstract A hyperplane arrangement in $$\mathbb P^n$$ is free if $R/J$ is Cohen–Macaulay (CM), where $$R = k[x_0,\dots ,x_n]$$ and $$J$$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $$ J^{un}$$, the intersection of height two primary components, and $$\sqrt{J}$$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $$\mathbb P^3$$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $$r$$, there is an arrangement for which $$R/J^{un}$$ (resp. $$R/\sqrt{J}$$) fails to be CM in only one degree, and this failure is by $$r$$. We draw consequences for the even liaison class of $$J^{un}$$ or $$\sqrt{J}$$. 

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2. J ‐modulated ¹⁹ F‐ and ¹ H‐detected dual‐optimized inverted ¹ J _CC 1,n‐ADEQUATE: A universal ADEQUATE experiment

   https://doi.org/10.1002/mrc.5324  

   Crouch, Ronald C; Pelmuş, Marius; Raab, Jeffrey G; Tischenko, Evgeny; Frey, Michael; Wang, Yunyi; Reibarkh, Mikhail; Williamson, R Thomas; Martin, Gary E (March 2023, Magnetic Resonance in Chemistry)

   Abstract The recently reported¹⁹F‐detected dual‐optimized inverted¹J_CC1,n‐ADEQUATE experiment and the previously reported¹H‐detected version have been modified to incorporateJ‐modulation, making it feasible to acquire all 1,1‐ and 1,n‐ADEQUATE correlations as well as¹J_CCandⁿJ_CChomonuclear scalar couplings in a single experiment. The experiments are demonstrated usingN,N‐dimethylamino‐2,5,6‐trifluoro‐3,4‐phthalonitrile andN,N‐dimethylamino‐3,4‐phthalonitrile. 

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3. Paucity problems and some relatives of Vinogradov’s mean value theorem

   https://doi.org/10.1017/S0305004123000166  

   WOOLEY, TREVOR D. (September 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract When$$k\geqslant 4$$and$$0\leqslant d\leqslant (k-2)/4$$, we consider the system of Diophantine equations\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when$$d=o\!\left(k^{1/4}\right)$$. 

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4. DFT investigation of coupling constant anomalies in substituted β‐lactams

   https://doi.org/10.1002/mrc.5444  

   Crull, Emily_B; Buevich, Alexei_V; Martin, Gary_E; Mahar, Rohit; Qu, Bo; Senanayake, Chris_H; Molinski, Tadeusz_F; Williamson, R_Thomas (March 2024, Magnetic Resonance in Chemistry)

   Abstract β‐lactams are a chemically diverse group of molecules with a wide range of biological activities. Having recently observed curious trends in²J_HHcoupling values in studies on this structural class, we sought to obtain a more comprehensive understanding of these diagnostic NMR parameters, specifically interrogating¹J_CH,²J_CH, and²J_HH, to differentiate 3‐ and 4‐monosubstituted β‐lactams. Further investigation using computational chemistry methods was employed to explore the geometric and electronic origins for the observed and calculated differences between the two substitution patterns. 

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5. Quadrupling the depairing current density in the iron-based superconductor SmFeAsO1–xHx

   https://doi.org/10.1038/s41563-024-01952-7  

   Miura, Masashi; Eley, Serena; Iida, Kazumasa; Hanzawa, Kota; Matsumoto, Jumpei; Hiramatsu, Hidenori; Ogimoto, Yuki; Suzuki, Takumi; Kobayashi, Tomoki; Ozaki, Toshinori; et al (July 2024, Nature Materials)

   Abstract Iron-based 1111-type superconductors display high critical temperatures and relatively high critical current densitiesJ_c. The typical approach to increasingJ_cis to introduce defects to control dissipative vortex motion. However, when optimized, this approach is theoretically predicted to be limited to achieving a maximumJ_cof only ∼30% of the depairing current densityJ_d, which depends on the coherence length and the penetration depth. Here we dramatically boostJ_cin SmFeAsO_1–xH_xfilms using a thermodynamic approach aimed at increasingJ_dand incorporating vortex pinning centres. Specifically, we reduce the penetration depth, coherence length and critical field anisotropy by increasing the carrier density through high electron doping using H substitution. Remarkably, the quadrupledJ_dreaches 415 MA cm^–2, a value comparable to cuprates. Finally, by introducing defects using proton irradiation, we obtain highJ_cvalues in fields up to 25 T. We apply this method to other iron-based superconductors and achieve a similar enhancement of current densities. 

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