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   https://doi.org/10.6084/m9.figshare.28200776  

   He, Qinsi (January 2025, figshare)

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2. 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. 

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3. QTL × environment interactions underlie ionome divergence in switchgrass

   https://doi.org/10.1093/g3journal/jkab144  

   Zhang, Li; MacQueen, Alice; Bonnette, Jason; Fritschi, Felix B; Lowry, David B; Juenger, Thomas E (April 2021, G3 Genes|Genomes|Genetics)

   Wisser, R J (Ed.)

   Abstract Ionomics measures elemental concentrations in biological organisms and provides a snapshot of physiology under different conditions. In this study, we evaluate genetic variation of the ionome in outbred, perennial switchgrass in three environments across the species’ native range, and explore patterns of genotype-by-environment interactions. We grew 725 clonally replicated genotypes of a large full sib family from a four-way linkage mapping population, created from deeply diverged upland and lowland switchgrass ecotypes, at three common gardens. Concentrations of 18 mineral elements were determined in whole post-anthesis tillers using ion coupled plasma mass spectrometry (ICP-MS). These measurements were used to identify quantitative trait loci (QTL) with and without QTL-by-environment interactions (QTLxE) using a multi-environment QTL mapping approach. We found that element concentrations varied significantly both within and between switchgrass ecotypes, and GxE was present at both the trait and QTL level. Concentrations of 14 of the 18 elements were under some genetic control, and 77 QTL were detected for these elements. Seventy-four percent of QTL colocalized multiple elements, half of QTL exhibited significant QTLxE, and roughly equal numbers of QTL had significant differences in magnitude and sign of their effects across environments. The switchgrass ionome is under moderate genetic control and by loci with highly variable effects across environments. 

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4. Learning Geometry with Dynamic Geometry Environments: A Systematic Review of 30 Years of Research

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   Hoyes, Michael; Diaz, Brayan; Hollebrands, Karen (September 2026, International Journal of Education in Mathematics, Science and Technology)

   This systematic review synthesizes 233 empirical journal articles published between 1991 and 2023 examining research on dynamic geometry environments (DGEs) in the context of learning and teaching geometry. Following PRISMA procedures, we combined structural topic modeling with interpretive thematic analysis to identify major research trends and their development over time. Five overarching themes were identified: mathematical reasoning and proof, instructional practices and teacher orchestration, conceptual understanding and learning outcomes, learner characteristics and diversity, and affective dimensions of engagement. Temporal analyses indicate a shift from early emphases on conjecture generation and proof toward increased attention to teacher orchestration, learner diversity, and affective dimensions of geometric learning. The findings provide a longitudinal account of how research priorities in DGE-supported geometry learning have evolved and highlight emerging directions for supporting equitable and conceptually oriented geometry instruction. 

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5. Measurement of the energy dependence of the e+e− → $$ B\overline{B} $$, $$ B{\overline{B}}^{\ast } $$ and $$ {B}^{\ast }{\overline{B}}^{\ast } $$ exclusive cross sections

   https://doi.org/10.1007/JHEP06(2021)137  

   Mizuk, R.; Bondar, A.; Adachi, I.; Aihara, H.; Al Said, S.; Asner, D. M.; Atmacan, H.; Aulchenko, V.; Aushev, T.; Ayad, R.; et al (June 2021, Journal of High Energy Physics)

   A bstract We report the first measurement of the exclusive cross sections e + e − → $$ B\overline{B} $$ B B ¯ , e + e − → $$ B{\overline{B}}^{\ast } $$ B B ¯ ∗ , and e + e − → $$ {B}^{\ast }{\overline{B}}^{\ast } $$ B ∗ B ¯ ∗ in the energy range from 10 . 63 GeV to 11 . 02 GeV. The B mesons are fully reconstructed in a large number of hadronic final states and the three channels are identified using a beam-constrained-mass variable. The shapes of the exclusive cross sections show oscillatory behavior with several maxima and minima. The results are obtained using data collected by the Belle experiment at the KEKB asymmetric-energy e + e − collider. 

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