More Like this

1. Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)

   https://doi.org/10.1093/imrn/rnaa218  

   Ford, Kevin (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show that for some even $$k\leqslant 3570$$ and all  $$k$$ with $442720643463713815200|k$, the equation $$\phi (n)=\phi (n+k)$$ has infinitely many solutions $$n$$, where $$\phi $$ is Euler’s totient function. We also show that for a positive proportion of all $$k$$, the equation $$\sigma (n)=\sigma (n+k)$$ has infinitely many solutions $$n$$. The proofs rely on recent progress on the prime $$k$$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath. 

   more » « less
2. Rationality of forms of M¯0,n$$\overline{{\mathcal {M}}}_{0,n}$$

   https://doi.org/10.1112/jlms.70097  

   Hassett, Brendan; Tschinkel, Yuri; Zhang, Zhijia (February 2025, Journal of the London Mathematical Society)

   Abstract We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with permutations of the points. 

   more » « less
3. Preparation and Characterization of N , N′ ‐Dialkyl‐1,3‐propanedialdiminium Chlorides, N , N′ ‐Dialkyl‐1,3‐propanedialdimines, and Lithium N , N′ ‐Dialkyl‐1,3‐propanedialdiminates

   https://doi.org/10.1002/hlca.202200152  

   Schmit, Claire E.; Girolami, Gregory S. (May 2023, Helvetica Chimica Acta)

   Abstract We describe convenient preparations ofN,N′‐dialkyl‐1,3‐propanedialdiminium chlorides,N,N′‐dialkyl‐1,3‐propanedialdimines, and lithiumN,N′‐dialkyl‐1,3‐propanedialdiminates in which the alkyl groups are methyl, ethyl, isopropyl, ortert‐butyl. For the dialdiminium salts, the N₂C₃backbone is always in thetrans‐s‐transconfiguration. Three isomers are present in solution except for thetert‐butyl compound, for which only two isomers are present; increasing the steric bulk of theN‐alkyl substituents shifts the equilibrium away from the (Z,Z) isomer in favor of the (E,Z), and (E,E) isomers. For the neutral dialdimines, crystal structures show that the methyl and isopropyl compounds adopt the (E,Z) form, whereas thetert‐butyl compound is in the (E,E) form. In aprotic solvents all four dialdimines (as well as the lithium dialdiminate salts) adoptcis‐s‐cisconformations in which there presumably is either an intramolecular hydrogen bond (or a lithium cation) between the two nitrogen atoms. 

   more » « less
4. Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

   https://doi.org/10.1215/21562261-2024-0009  

   Fisher, David; Melnick, Karin (November 2024, Kyoto Journal of Mathematics)
5. Beyond N-Alkylation: Synthesis, Structure, and Function of N-Amino Peptides

   https://doi.org/10.1021/acs.accounts.4c00024  

   Angera, Isaac J; Wright, Madison M; Del_Valle, Juan R (May 2024, Accounts of Chemical Research)