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1. 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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2. The stable Galois correspondence for real closed fields

   https://doi.org/10.1090/conm/707/14250  

   Heller, J.; Ormsby, K. (January 2018, Contemporary mathematics - American Mathematical Society)

   In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a finite Galois extension of fields with Galois group G, there is a functor c∗L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗L/k(G/H+) = Spec(LH)+. The main theorem of [7] says that when k is a real closed field and L = k[i], the restriction of c∗L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c∗L/k itself. Our main tools are Bachmann’s theorem on the (2,η)- periodic stable motivic homotopy category and an isomorphism range for the map πRS → πC2 S induced by C2-equivariant Betti realization. 

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3. Unnatural Lysines with Reduced Sidechain N ‐Basicity: Synthesis of N ‐trifluoroethyl Substituted Lysine and Homologs

   https://doi.org/10.1002/slct.202203132  

   Jameson, Brian; Glaser, Rainer (December 2022, ChemistrySelect)

   Abstract The syntheses are reported of N^ϵ‐(2,2,2‐trifluoroethyl)‐D,L‐lysine (^tFK) and N^ζ‐(2,2,2‐trifluoroethyl)‐D,L‐homolysine (^tFK₊₁) from amino alcohols HO−(CH₂)_n−NH₂. The syntheses involve reductive amination, Appel bromination, and the stereoselective bond formation between C_α of the amino acid and the fluorinated alkyl chain in the Schöllkopf bislactim amino acid synthesis. The methyl esters of the fluorinated amino acids are the relevant substrates for oligopeptide synthesis. With theR‐Schöllkopf reagent, we stereoselectively generated methyl N^ϵ‐boc‐N^ϵ‐(2,2,2‐trifluoroethyl)‐L‐lysinate and methyl N^ζ‐boc‐N^ζ‐(2,2,2‐trifluoroethyl)‐L‐homolysinate. Products and intermediates were characterized by ¹H NMR, ¹³C NMR, COSY, HSQC, and LCMS. A variety of N‐functionality may be introduced by reacting hemiacetals with different appendages. This fluorine modification reduces the sidechain N‐basicity by combined ‐I effect of the three fluorines. This effect increases the [amine]/[ammonium ion] ratio of the sidechain amine in lysine to facilitate carbamylation at lower pH conditions. 

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4. Glass transition temperature from the chemical structure of conjugated polymers

   https://doi.org/10.1038/s41467-020-14656-8  

   Xie, Renxuan; Weisen, Albree_R; Lee, Youngmin; Aplan, Melissa_A; Fenton, Abigail_M; Masucci, Ashley_E; Kempe, Fabian; Sommer, Michael; Pester, Christian_W; Colby, Ralph_H; et al (February 2020, Nature Communications)

   Abstract The glass transition temperature (T_g) is a key property that dictates the applicability of conjugated polymers. TheT_gdemarks the transition into a brittle glassy state, making its accurate prediction for conjugated polymers crucial for the design of soft, stretchable, or flexible electronics. Here we show that a single adjustable parameter can be used to build a relationship between theT_gand the molecular structure of 32 semiflexible (mostly conjugated) polymers that differ drastically in aromatic backbone and alkyl side chain chemistry. An effective mobility value,ζ, is calculated using an assigned atomic mobility value within each repeat unit. The only adjustable parameter in the calculation ofζis the ratio of mobility between conjugated and non-conjugated atoms. We show thatζcorrelates strongly to theT_g, and that this simple method predicts theT_gwith a root-mean-square error of 13 °C for conjugated polymers with alkyl side chains. 

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5. Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

   https://doi.org/10.1007/s11856-024-2613-1  

   Hung, Nguyen Ngoc; Sambale, Benjamin; Tiep, Pham Huu (September 2024, Israel Journal of Mathematics)

   Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ $G/O_{p^{\prime }}\left(G\right)\cong C_{11}^2⋊\text{SL}(2,5)$. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ $2\sqrt{p-1}+1-k_p\left(G\right)$, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 

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