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1. Measurement of the mass dependence of the transverse momentum of lepton pairs in Drell–Yan production in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te\hspace{-.08em}V} $$

   https://doi.org/10.1140/epjc/s10052-023-11631-7  

   Tumasyan, A.; Adam, W.; Andrejkovic, J. W.; Bergauer, T.; Chatterjee, S.; Dragicevic, M.; Valle, A. Escalante; Frühwirth, R.; Jeitler, M.; Krammer, N.; et al (July 2023, The European Physical Journal C)

   Abstract The double differential cross sections of the Drell–Yan lepton pair ($$\ell ^+\ell ^-$$ ${\ell }^{+}{\ell }^{-}$, dielectron or dimuon) production are measured as functions of the invariant mass$$m_{\ell \ell }$$ $m_{\ell \ell }$, transverse momentum$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$, and$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$. The$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$observable, derived from angular measurements of the leptons and highly correlated with$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$, is used to probe the low-$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$region in a complementary way. Dilepton masses up to 1$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$are investigated. Additionally, a measurement is performed requiring at least one jet in the final state. To benefit from partial cancellation of the systematic uncertainty, the ratios of the differential cross sections for various$$m_{\ell \ell }$$ $m_{\ell \ell }$ranges to those in the Z mass peak interval are presented. The collected data correspond to an integrated luminosity of 36.3$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$of proton–proton collisions recorded with the CMS detector at the LHC at a centre-of-mass energy of 13$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$. Measurements are compared with predictions based on perturbative quantum chromodynamics, including soft-gluon resummation. 

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2. Search for heavy resonances decaying into a Z or W boson and a Higgs boson in final states with leptons and b-jets in 139 fb−1 of pp collisions at $$ \sqrt{s} $$ = 13 TeV with the ATLAS detector

   https://doi.org/10.1007/JHEP06(2023)016  

   Aad, G.; Abbott, B.; Abbott, D. C.; Abeling, K.; Abidi, S. H.; Aboulhorma, A.; Abramowicz, H.; Abreu, H.; Abulaiti, Y.; Abusleme Hoffman, A. C.; et al (June 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> This article presents a search for new resonances decaying into aZorWboson and a 125 GeV Higgs bosonh, and it targets the$$ \nu \overline{\nu}b\overline{b} $$ $\nu \overline{\nu }b\overline{b}$,$$ {\ell}^{+}{\ell}^{-}b\overline{b} $$ ${\ell }^{+}{\ell }^{-}b\overline{b}$, or$$ {\ell}^{\pm}\nu b\overline{b} $$ ${\ell }^{\pm }\mathrm{\nu b}\overline{b}$final states, whereℓ=eorμ, in proton-proton collisions at$$ \sqrt{s} $$ $\sqrt{s}$= 13 TeV. The data used correspond to a total integrated luminosity of 139 fb⁻¹collected by the ATLAS detector during Run 2 of the LHC at CERN. The search is conducted by examining the reconstructed invariant or transverse mass distributions ofZhorWhcandidates for evidence of a localised excess in the mass range from 220 GeV to 5 TeV. No significant excess is observed and 95% confidence-level upper limits between 1.3 pb and 0.3 fb are placed on the production cross section times branching fraction of neutral and charged spin-1 resonances and CP-odd scalar bosons. These limits are converted into constraints on the parameter space of the Heavy Vector Triplet model and the two-Higgs-doublet model. 

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3. Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

   https://doi.org/10.1007/s10107-023-01966-0  

   He, Ziyu; Han, Shaoning; Gómez, Andrés; Cui, Ying; Pang, Jong-Shi (May 2023, Mathematical Programming)

   Abstract This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “$$\ell _0$$ ${\ell }_0$-path” where the discontinuous$$\ell _0$$ ${\ell }_0$-function provides the exact sparsity count; the “$$\ell _1$$ ${\ell }_1$-path” where the$$\ell _1$$ ${\ell }_1$-function provides a convex surrogate of sparsity count; and the “capped$$\ell _1$$ ${\ell }_1$-path” where the nonconvex nondifferentiable capped$$\ell _1$$ ${\ell }_1$-function aims to enhance the$$\ell _1$$ ${\ell }_1$-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped$$\ell _1$$ ${\ell }_1$-path. Our study of the capped$$\ell _1$$ ${\ell }_1$-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped$$\ell _1$$ ${\ell }_1$-regularized problem offers interesting theoretical properties and practical compromise between the$$\ell _0$$ ${\ell }_0$-path and the$$\ell _1$$ ${\ell }_1$-path. Indeed, while the$$\ell _0$$ ${\ell }_0$-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of$$\ell _1$$ ${\ell }_1$-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped$$\ell _1$$ ${\ell }_1$-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function. 

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4. Higher connectivity of tropicalizations

   https://doi.org/10.1007/s00208-021-02281-9  

   Maclagan, Diane; Yu, Josephine (October 2021, Mathematische Annalen)

   Abstract We show that the tropicalization of an irreducibled-dimensional variety over a field of characteristic 0 is$$(d-\ell )$$ $(d-\ell )$-connected through codimension one, where$$\ell $$ $\ell$is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety. 

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5. Inexact Fixed-Point Proximity Algorithm for the $$\ell _0$$ Sparse Regularization Problem

   https://doi.org/10.1007/s10915-024-02600-7  

   Fang, Ronglong; Xu, Yuesheng; Yan, Mingsong (July 2024, Journal of Scientific Computing)

   Abstract We studyinexactfixed-point proximity algorithms for solving a class of sparse regularization problems involving the$$\ell _0$$ ${\ell }_0$norm. Specifically, the$$\ell _0$$ ${\ell }_0$model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the$$\ell _0$$ ${\ell }_0$norm regularization term. Such an$$\ell _0$$ ${\ell }_0$model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the$$\ell _0$$ ${\ell }_0$models found by the proposed inexact algorithm outperform global minimizers of the corresponding$$\ell _1$$ ${\ell }_1$models, in terms of approximation accuracy and sparsity of the solutions. 

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