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1. Semiclassical resolvent bounds for short range L ∞ potentials with singularities at the origin

   https://doi.org/10.3233/ASY-231872  

   Shapiro, Jacob (February 2024, Asymptotic Analysis)

   We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity. 

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2. A Buchsbaum theory for tight closure

   https://doi.org/10.1090/tran/8762  

   Ma, Linquan; Quy, Pham (November 2022, Transactions of the American Mathematical Society)

   A Noetherian local ring ( R , m ) (R,\frak {m}) is called Buchsbaum if the difference ℓ ( R / q ) − e ( q , R ) \ell (R/\mathfrak {q})-e(\mathfrak {q}, R) , where q \mathfrak {q} is an ideal generated by a system of parameters, is a constant independent of q \mathfrak {q} . In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring ( R , m ) (R,\frak {m}) of prime characteristic p > 0 p>0 and dimension at least one, the difference e ( q , R ) − ℓ ( R / q ∗ ) e(\mathfrak {q}, R)-\ell (R/\mathfrak {q}^*) is independent of q \mathfrak {q} if and only if the parameter test ideal τ p a r ( R ) \tau _{\mathrm {par}}(R) contains m \frak {m} . We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings. 

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3. Generalized cross validation for ℓp-ℓq minimization

   https://doi.org/10.1007/s11075-021-01087-9  

   Buccini, Alessandro; Reichel, Lothar (March 2021, Numerical Algorithms)

   null (Ed.)

   Abstract Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p -norm, and a regularization term, that is defined in terms of a q -norm. We allow 0 < p , q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed. 

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4. Measurement of two-photon decay width of χc2(1P) in γγ → χc2(1P) → J/ψγ at Belle

   https://doi.org/10.1007/JHEP01(2023)160  

   Seino, Y.; Hayasaka, K.; Uehara, S.; Adachi, I.; Aihara, H.; Al Said, S.; Asner, D. M.; Atmacan, H.; Aushev, T.; Ayad, R.; et al (January 2023, Journal of High Energy Physics)

   A bstract We report the measurement of the two-photon decay width of χ c 2 (1 P ) in two-photon processes at the Belle experiment. We analyze the process γγ → χ c 2 (1 P ) → J/ψγ , J/ψ → ℓ + ℓ − ( ℓ = e or μ ) using a data sample of 971 fb − 1 collected with the Belle detector at the KEKB e + e − collider. In this analysis, the product of the two-photon decay width of χ c 2 (1 P ) and the branching fraction is determined to be $$ {\Gamma}_{\gamma \gamma}\left({\chi}_{c2}(1P)\right)\mathcal{B}\left({\chi}_{c2}(1P)\to J/\psi \gamma \right)\mathcal{B}\left(J/\psi \to {\ell}^{+}{\ell}^{-}\right)=14.8\pm 0.3\left(\textrm{stat}.\right)\pm 0.7\left(\textrm{syst}.\right) $$ Γ γγ χ c 2 1 P B χ c 2 1 P → J / ψγ B J / ψ → ℓ + ℓ − = 14.8 ± 0.3 stat . ± 0.7 syst . eV, which corresponds to Γ γγ ( χ c 2 (1 P )) = 653 ± 13(stat.) ± 31(syst.) ± 17(B.R.) eV, where the third uncertainty is from $$ \mathcal{B} $$ B ( χ c 2 (1 P ) → J/ψγ ) and $$ \mathcal{B} $$ B ( J/ψ → ℓ + ℓ − ). 

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5. Search for Rare $b\to d{\ell }^{+}{\ell }^{-}$ Transitions at Belle

   https://doi.org/10.1103/PhysRevLett.133.101804  

   Adachi, I; Aggarwal, L; Aihara, H; Akopov, N; Aloisio, A; Al_Said, S; Asner, D M; Atmacan, H; Aushev, V; Aversano, M; et al (September 2024, Physical Review Letters)

   We present the results of a search for the $b\to d{\ell }^{+}{\ell }^{-}$flavor-changing neutral-current rare decays $B^{+,0}\to (\eta ,\omega ,{\pi }^{+,0},{\rho }^{+,0})e^{+}{e}^{-}$and $B^{+,0}\to (\eta ,\omega ,{\pi }^0,{\rho }^{+}){\mu }^{+}{\mu }^{-}$using a $711{\mathrm{fb}}^{-1}$data sample that contains $772\times {10}^{6}B\overline{B}$events. The data were collected at the $\mathrm{\Upsilon }\left(4S\right)$resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$collider. We find no evidence for signal and set upper limits on branching fractions at the 90% confidence level in the range $\left(3.8–47\right)\times {10}^{-8}$depending on the decay channel. The obtained limits are the world’s best results. This is the first search for the channels $B^{+,0}\to (\omega ,{\rho }^{+,0})e^{+}{e}^{-}$and $B^{+,0}\to (\omega ,{\rho }^{+}){\mu }^{+}{\mu }^{-}$. Published by the American Physical Society2024 

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