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1. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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2. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo; Wang, Jinhuan (June 2024, Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

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3. Spectroscopic study of the 4 f ⁷ 6 s ^{2 8} S 7/2∘−4 f ⁷ ( ⁸ S ^∘ )6 s 6 p ( ¹ P ^∘ ) ⁸ P _9/2 transition in neutral europium-151 and europium-153: absolute frequency and hyperfine structure

   https://doi.org/10.1364/JOSAB.467968  

   Herd, M_T; Maruko, C.; Herzog, M_M; Brand, A.; Cannon, G.; Duah, B.; Hollin, N.; Karani, T.; Wallace, A.; Whitmore, M.; et al (September 2022, Journal of the Optical Society of America B)

   We report on spectroscopic measurements on the $4f^{7}6s^2{}^8{S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4f^7{(}^8{S}^{\circ <\#comment/>}\left)6s6p{(}^1{P}^{\circ <\#comment/>}\right){}^8{P}_{9/2}$transition in neutral europium-151 and europium-153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the $6s6p{(}^1{P}^{\circ <\#comment/>}){}^8{P}_{9/2}$state were found to be $\mathrm{A}\left(151\right)=-<\#comment/>228.84\left(2\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(151\right)=226.9\left(5\right)\mathrm{M}\mathrm{H}\mathrm{z}$and $\mathrm{A}\left(153\right)=-<\#comment/>101.87\left(6\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(153\right)=575.4\left(1.5\right)\mathrm{M}\mathrm{H}\mathrm{z}$, which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results. 

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4. Demonstration of the DC-Kerr effect in silicon-rich nitride

   https://doi.org/10.1364/OL.432359  

   Friedman, Alex; Nejadriahi, Hani; Sharma, Rajat; Fainman, Yeshaiahu (August 2021, Optics Letters)

   We demonstrate the DC-Kerr effect in plasma enhanced chemical vapor deposition (PECVD) silicon-rich nitride (SRN) and use it to demonstrate a third order nonlinear susceptibility, ${\mathrm{\chi <\#comment/>}}^{\left(3\right)}$, as high as $(6\pm <\#comment/>0.58)\times <\#comment/>{10}^{-<\#comment/>19}{\mathrm{m}}^2/{\mathrm{V}}^2$. We employ spectral shift versus applied voltage measurements in a racetrack resonator as a tool to characterize the nonlinear susceptibilities of these films. In doing so, we demonstrate a ${\mathrm{\chi <\#comment/>}}^{\left(3\right)}$larger than that of silicon and argue that PECVD SRN can provide a versatile platform for employing optical phase shifters while maintaining a low thermal budget using a deposition technique readily available in CMOS process flows. 

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5. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel (January 2023, Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. The linear representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$that are subrepresentations of a symmetric one. 

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