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1. Geometric Separation in $$\mathbb {R}^3$$ R 3

   https://doi.org/10.1007/s00041-017-9569-z  

   Guo, Kanghui; Labate, Demetrio (January 2018, Journal of Fourier Analysis and Applications)
2. Wide‐Bandgap R Ba ₃ (B ₃ O ₆ ) ₃ ( R = Nd, Sm, Tb, Dy, and Er) Single Crystals for Ultraviolet Nonlinear Optics

   https://doi.org/10.1002/adom.202401437  

   Sarker, Saugata; Wang, Yu; Benyacko, Caeli; Guan, Yingdong; Yoshida, Suguru; Yennawar, Hemant; He, Jingyang; Mao, Zhiqiang; Gopalan, Venkatraman (August 2024, Advanced Optical Materials)

   Abstract Discovery of new materials with enhanced optical properties in the visible and UV‐C range can impact applications in lasers, nonlinear optics, and quantum optics. Here, the optical floating zone growth of a family of rare earth borates,RBa₃(B₃O₆)₃(R= Nd, Sm, Tb, Dy, and Er), with promising linear and nonlinear optical (NLO) properties is reported. Although previously identified to be centrosymmetric, the X‐ray analysis combined with optical second harmonic generation (SHG) assigns the noncentrosymmetricPspace group to these crystals. Characterization of linear optical properties reveals a direct bandgap of ≈5.61–5.72 eV and strong photoluminescence in both the visible and mid‐IR regions. Anisotropic linear and nonlinear optical characterization reveals both Type‐I and Type‐II SHG phase matchability, with the highest effective phase‐matched SHG coefficient of 1.2 pm V⁻¹at 800‐nm fundamental wavelength (for DyBa₃(B₃O₆)₃), comparable to β‐BaB₂O₄(phase‐matchedd₂₂≈ 1.9 pm V⁻¹). Laser‐induced surface damage threshold for these environmentally stable crystals is 650–900 GW cm⁻², which is four to five times higher than that of β‐BaB₂O₄, thus providing an opportunity to pump with significantly higher power to generate about six to seven times stronger SHG light. Since the SHG arises from disorder on the Ba‐site, significantly larger SHG coefficients may be realized by “poling” the crystals to align the Ba displacements. These properties motivate further development of this crystal family for laser and wide bandgap NLO applications. 

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3. The R -Process Alliance: Analysis of limited- r stars

   https://doi.org/10.1051/0004-6361/202449376  

   Xylakis-Dornbusch, T; Hansen, T T; Beers, T C; Christlieb, N; Ezzeddine, R; Frebel, A; Holmbeck, E; Placco, V M; Roederer, I U; Sakari, C M; et al (August 2024, Astronomy & Astrophysics)

   Context.In recent years, theR-Process Alliance (RPA) has conducted a successful search for stars that are enhanced in elements produced by the rapid neutron-capture (r-)process. In particular, the RPA has uncovered a number of stars that are strongly enriched in lightr-process elements, such as Sr, Y, and Zr. These so-called limited-rstars were investigated to explore the astrophysical production site(s) of these elements. Aims.We investigate the possible formation sites for light neutron-capture elements by deriving detailed abundances for neutron-capture elements from high-resolution spectra with a high signal-to-noise ratio of three limited-rstars. Methods.We conducted a kinematic analysis and a 1D local thermodynamic equilibrium spectroscopic abundance analysis of three stars. Furthermore, we calculated the lanthanide mass fraction (X_La) of our stars and of limited-rstars from the literature. Results.We found that the abundance pattern of neutron-capture elements of limited-rstars behaves differently depending on their [Ba/Eu] ratios, and we suggest that this should be taken into account in future investigations of their abundances. Furthermore, we found that theX_Laof limited-rstars is lower than that of the kilonova AT2017gfo. The latter seems to be in the transition zone between limited-rX_Laand that ofr-I andr-II stars. Finally, we found that unliker-I andr-II stars, the current sample of limited-rstars is largely born in the Galaxy and is not accreted. 

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4. Making K _r+1 -free graphs r -partite

   https://doi.org/10.1017/S0963548320000590  

   Balogh, József; Clemen, Felix Christian; Lavrov, Mikhail; Lidický, Bernard; Pfender, Florian (July 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+ 1 -free graph on n vertices with e ( G ) > ex( n , K r +1 )– α n 2 , then one can remove εn 2 edges from G to obtain an r -partite graph. Füredi gave a short proof that one can choose α = ε . We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0. 

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5. On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)∖SL(3,R)/SO(3,R)

   https://doi.org/10.1007/s40993-019-0168-8  

   Miller, Stephen D.; Trinh, Tien D. (October 2019, Research in number theory)

   It is well-known that there are automorphic eigenfunctions on SL(2,Z)∖SL(2,R)/SO(2,R)—such as the classical j-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of q=e2πiz, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3,Z)∖SL(3,R)/SO(3,R) and eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confirms part of a conjecture of Miatello and Wallach, who assert all automorphic eigenfunctions on this quotient (among other rank ≥2 examples) always have moderate growth. We additionally confirm their conjecture under certain natural hypotheses, such as the absolute convergence of the eigenfunction’s Fourier expansion. 

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