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1. The immersion poset on partitions

   https://doi.org/10.1007/s10801-025-01380-z  

   Johnston, Lisa; Kenepp, David; Nguyen, Evuilynn; Paul, Digjoy; Schilling, Anne; Simone, Mary_Claire; Zhou, Regina (February 2025, Journal of Algebraic Combinatorics)

   Abstract We introduce the immersion poset$$({\mathcal {P}}(n), \leqslant _I)$$ $(P\left(n\right),{⩽}_I)$on partitions, defined by$$\lambda \leqslant _I \mu $$ $\lambda {⩽}_I\mu$if and only if$$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$ $s_{\mu }(x_1,\dots ,x_N)-s_{\lambda }(x_1,\dots ,x_N)$is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of$$GL_N({\mathbb {C}})$$ $G{L}_N\left(C\right)$form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections$$\textsf{SSYT}(\lambda , \nu ) \hookrightarrow \textsf{SSYT}(\mu , \nu )$$ $\mathrm{SSYT}(\lambda ,\nu )\hookrightarrow \mathrm{SSYT}(\mu ,\nu )$on semistandard Young tableaux given constraints on the shape of$$\lambda $$ $\lambda$, and present results on immersion relations among hook and two column partitions. The standard immersion poset$$({\mathcal {P}}(n), \leqslant _{std})$$ $(P\left(n\right),{⩽}_{\mathrm{std}})$is a refinement of the immersion poset, defined by$$\lambda \leqslant _{std} \mu $$ $\lambda {⩽}_{\mathrm{std}}\mu$if and only if$$\lambda \leqslant _D \mu $$ $\lambda {⩽}_D\mu$in dominance order and$$f^\lambda \leqslant f^\mu $$ $f^{\lambda }⩽f^{\mu }$, where$$f^\nu $$ $f^{\nu }$is the number of standard Young tableaux of shape$$\nu $$ $\nu$. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12]. 

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2. On a Bohr set analogue of Chowla’s conjecture

   https://doi.org/10.1007/s00209-025-03770-2  

   Teräväinen, Joni; Walker, Aled (May 2025, Mathematische Zeitschrift)

   Abstract Let$$\lambda $$ $\lambda$denote the Liouville function. We show that the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\lambda \left(\lfloor {\alpha }_{2}n\rfloor \right)$is 0 whenever$$\alpha _1,\alpha _2$$ ${\alpha }_1,{\alpha }_2$are positive reals with$$\alpha _1/\alpha _2$$ ${\alpha }_1/{\alpha }_2$irrational. We also show that for$$k\geqslant 3$$ $k⩾3$the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\cdots \lambda \left(\lfloor {\alpha }_{k}n\rfloor \right)$has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers$$\alpha _i.$$ ${\alpha }_i.$Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets. 

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3. Steady Radiating Gravity waves: An Exponential Asymptotics Approach

   https://doi.org/10.1007/s42286-023-00081-z  

   Kataoka, Takeshi; Akylas, T. R. (January 2024, Water Waves)

   Abstract The radiation of steady surface gravity waves by a uniform stream$$U_{0}$$ $U_0$over locally confined (width$$L$$ $L$) smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here, an asymptotic analysis is made for subcritical flow$$D/\lambda > 1$$ $D/\lambda >1$in the low-Froude-number ($$F^{2} \equiv \lambda /L \ll 1$$ $F^2\equiv \lambda /L\ll 1$) limit, where$$\lambda = U_{0}^{2} /g$$ $\lambda =U_0^2/g$is the lengthscale of radiating gravity waves and$$D$$ $D$is the uniform water depth. In this regime, the downstream wave amplitude, although formally exponentially small with respect to$$F$$ $F$, is determined by a fully nonlinear mechanism even for small topography amplitude. It is argued that this mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity. 

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4. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

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5. Observation of the $${{{\Lambda }} _{\text {b}}^{{0}}} \rightarrow {{\text {J}/\uppsi }} {{{\Xi }} ^{{-}}} {{\text {K}} ^{{+}}} $$ decay

   https://doi.org/10.1140/epjc/s10052-024-13114-9  

   Hayrapetyan, A; Tumasyan, A; Adam, W; Andrejkovic, J W; Bergauer, T; Chatterjee, S; Damanakis, K; Dragicevic, M; Valle, A_Escalante Del; Hussain, P S; et al (October 2024, The European Physical Journal C)

   Abstract Using proton–proton collision data corresponding to an integrated luminosity of$$140\hbox { fb}^{-1}$$ $140{\text{fb}}^{-1}$collected by the CMS experiment at$$\sqrt{s}= 13\,\text {Te}\hspace{-.08em}\text {V} $$ $\sqrt{s}=13\text{Te}\text{V}$, the$${{{\Lambda }} _{\text {b}}^{{0}}} \rightarrow {{\text {J}/\uppsi }} {{{\Xi }} ^{{-}}} {{\text {K}} ^{{+}}} $$ ${\Lambda }_{\text{b}}^0\to \text{J}/\psi {\Xi }^{-}{\text{K}}^{+}$decay is observed for the first time, with a statistical significance exceeding 5 standard deviations. The relative branching fraction, with respect to the$${{{\Lambda }} _{\text {b}}^{{0}}} \rightarrow {{{\uppsi }} ({2\textrm{S}})} {{\Lambda }} $$ ${\Lambda }_{\text{b}}^0\to \psi \left(2\text{S}\right)\Lambda$decay, is measured to be$$\mathcal {B}({{{\Lambda }} _{\text {b}}^{{0}}} \rightarrow {{\text {J}/\uppsi }} {{{\Xi }} ^{{-}}} {{\text {K}} ^{{+}}} )/\mathcal {B}({{{\Lambda }} _{\text {b}}^{{0}}} \rightarrow {{{\uppsi }} ({2\textrm{S}})} {{\Lambda }} ) = [3.38\pm 1.02\pm 0.61\pm 0.03]\%$$ $B({\Lambda }_{\text{b}}^0\to \text{J}/\psi {\Xi }^{-}{\text{K}}^{+})/B({\Lambda }_{\text{b}}^0\to \psi \left(2\text{S}\right)\Lambda )=[3.38\pm 1.02\pm 0.61\pm 0.03]\%$, where the first uncertainty is statistical, the second is systematic, and the third is related to the uncertainties in$$\mathcal {B}({{{\uppsi }} ({2\textrm{S}})} \rightarrow {{\text {J}/\uppsi }} {{{\uppi }} ^{{+}}} {{{\uppi }} ^{{-}}} )$$ $B(\psi \left(2\text{S}\right)\to \text{J}/\psi {\pi }^{+}{\pi }^{-})$and$$\mathcal {B}({{{\Xi }} ^{{-}}} \rightarrow {{\Lambda }} {{{\uppi }} ^{{-}}} )$$ $B({\Xi }^{-}\to \Lambda {\pi }^{-})$. 

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