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1. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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2. A Duality Principle for Groups II: Multi-frames Meet Super-Frames

   https://doi.org/10.1007/s00041-020-09792-0  

   Balan, R.; Dutkay, D.; Han, D.; Larson, D.; Luef, F. (December 2020, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

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3. Evidence for the singly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ and search for $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ and Ω−K+ decays at Belle

   https://doi.org/10.1007/JHEP01(2023)055  

   Han, X.; Jia, S.; Yuan, L.; Shen, C. P.; Adachi, I.; Ahn, J. K.; Aihara, H.; Asner, D. M.; Aushev, T.; Ayad, R.; et al (January 2023, Journal of High Energy Physics)

   A bstract Using a data sample of 980 fb − 1 collected with the Belle detector at the KEKB asymmetric-energy e + e − collider, we study for the first time the singly Cabibbo-suppressed decays $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ Ω c 0 → Ξ − π + and Ω − K + and the doubly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + . Evidence for an $$ {\Omega}_c^0 $$ Ω c 0 signal in the $$ {\Omega}_c^0 $$ Ω c 0 → Ξ − π + mode is reported with a significance of 4 . 5 σ including systematic uncertainties. The ratio of branching fractions to the normalization mode $$ {\Omega}_c^0 $$ Ω c 0 → Ω − π + is measured to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{\pi}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)=0.253\pm 0.052\left(\textrm{stat}.\right)\pm 0.030\left(\textrm{syst}.\right). $$ B Ω c 0 → Ξ − π + / B Ω c 0 → Ω − π + = 0.253 ± 0.052 stat . ± 0.030 syst . . No significant signals of $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + and Ω − K + modes are found. The upper limits at 90% confidence level on ratios of branching fractions are determined to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.070 $$ B Ω c 0 → Ξ − K + / B Ω c 0 → Ω − π + < 0.070 and $$ \mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.29. $$ B Ω c 0 → Ω − K + / B Ω c 0 → Ω − π + < 0.29 . 

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4. Inequalities for the Radon Transform on Convex Sets

   https://doi.org/10.1093/imrn/rnab122  

   Giannopoulos, Apostolos; Koldobsky, Alexander; Zvavitch, Artem (May 2021, International Mathematics Research Notices)

   Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 

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5. Light-flavor particle production in high-multiplicity pp collisions at $$ \sqrt{\textrm{s}} $$ = 13 TeV as a function of transverse spherocity

   https://doi.org/10.1007/JHEP05(2024)184  

   Acharya, S; Adamová, D; Aglieri_Rinella, G; Agnello, M; Agrawal, N; Ahammed, Z; Ahmad, S; Ahn, S U; Ahuja, I; Akindinov, A; et al (May 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> Results on the transverse spherocity dependence of light-flavor particle production (π, K, p,ϕ, K^*0,$$ {\textrm{K}}_{\textrm{S}}^0 $$ $K_S^0$, Λ, Ξ) at midrapidity in high-multiplicity pp collisions at$$ \sqrt{s} $$ $\sqrt{s}$= 13 TeV were obtained with the ALICE apparatus. The transverse spherocity estimator$$ \left({S}_{\textrm{O}}^{p_{\textrm{T}}=1}\right) $$ $\left(S_O^{p_T=1}\right)$categorizes events by their azimuthal topology. Utilizing narrow selections on$$ {S}_{\textrm{O}}^{p_{\textrm{T}}=1} $$ $S_O^{p_T=1}$, it is possible to contrast particle production in collisions dominated by many soft initial interactions with that observed in collisions dominated by one or more hard scatterings. Results are reported for two multiplicity estimators covering different pseudorapidity regions. The$$ {S}_{\textrm{O}}^{p_{\textrm{T}}=1} $$ $S_O^{p_T=1}$estimator is found to effectively constrain the hardness of the events when the midrapidity (|η| < 0.8) estimator is used. The production rates of strange particles are found to be slightly higher for soft isotropic topologies, and severely suppressed in hard jet-like topologies. These effects are more pronounced for hadrons with larger mass and strangeness content, and observed when the topological selection is done within a narrow multiplicity interval. This demonstrates that an important aspect of the universal scaling of strangeness enhancement with final-state multiplicity is that high-multiplicity collisions are dominated by soft, isotropic processes. On the contrary, strangeness production in events with jet-like processes is significantly reduced. The results presented in this article are compared with several QCD-inspired Monte Carlo event generators. Models that incorporate a two-component phenomenology, either through mechanisms accounting for string density, or thermal production, are able to describe the observed strangeness enhancement as a function of$$ {S}_{\textrm{O}}^{p_{\textrm{T}}=1} $$ $S_O^{p_T=1}$. 

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