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1. On the stationarity for nonlinear optimization problems with polyhedral constraints

   https://doi.org/10.1007/s10107-023-01979-9  

   di Serafino, Daniela; Hager, William W.; Toraldo, Gerardo; Viola, Marco (May 2023, Mathematical Programming)

   Abstract For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$ $x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$ ${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$ $\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$ ${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$ $\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$ $\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$ $\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$ $\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$ $x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$ $\phi \left(x\right)$only vanishes when$$\textbf{x}$$ $x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$ $\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$ $\Vert \phi \left(x\right)\Vert$. 

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2. Azimuthal correlations of heavy-flavor hadron decay electrons with charged particles in pp and p–Pb collisions at $$\pmb {\sqrt{s_{\mathrm{{NN}}}}}$$ = 5.02 TeV

   https://doi.org/10.1140/epjc/s10052-023-11835-x  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (August 2023, The European Physical Journal C)

   Abstract The azimuthal ($$\Delta \varphi $$ $\Delta \phi$) correlation distributions between heavy-flavor decay electrons and associated charged particles are measured in pp and p–Pb collisions at$$\sqrt{s_{\mathrm{{NN}}}} = 5.02$$ $\sqrt{s_{\mathrm{NN}}}=5.02$TeV. Results are reported for electrons with transverse momentum$$4<16$$ $4<p_{\text{T}}<16$$$\textrm{GeV}/c$$ $\text{GeV}/c$ and pseudorapidity$$|\eta |<0.6$$ $\left|\eta \right|<0.6$. The associated charged particles are selected with transverse momentum$$1<7$$ $1<p_{\text{T}}<7$$$\textrm{GeV}/c$$ $\text{GeV}/c$, and relative pseudorapidity separation with the leading electron$$|\Delta \eta | < 1$$ $\left|\Delta \eta \right|<1$. The correlation measurements are performed to study and characterize the fragmentation and hadronization of heavy quarks. The correlation structures are fitted with a constant and two von Mises functions to obtain the baseline and the near- and away-side peaks, respectively. The results from p–Pb collisions are compared with those from pp collisions to study the effects of cold nuclear matter. In the measured trigger electron and associated particle kinematic regions, the two collision systems give consistent results. The$$\Delta \varphi $$ $\Delta \phi$distribution and the peak observables in pp and p–Pb collisions are compared with calculations from various Monte Carlo event generators. 

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3. 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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4. Linking Numbers in Three-Manifolds

   https://doi.org/10.1007/s00454-021-00287-3  

   Cahn, Patricia; Kjuchukova, Alexandra (July 2021, Discrete & Computational Geometry)

   Abstract LetMbe a connected, closed, oriented three-manifold andK, Ltwo rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number betweenKandLin terms of a presentation of Mas an irregular dihedral three-fold cover of $$S^3$$ $S^3$branched along a knot$$\alpha \subset S^3$$ $\alpha \subset S^3$. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ $\alpha$can be derived from dihedral covers of $$\alpha $$ $\alpha$. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. 

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5. Advances in tabulating Carmichael numbers

   https://doi.org/10.1007/s40993-024-00598-3  

   Shallue, Andrew; Webster, Jonathan (March 2025, Research in Number Theory)

   Abstract We report that there are 49679870 Carmichael numbers less than$$10^{22}$$ ${10}^{22}$which is an order of magnitude improvement on Richard Pinch’s prior work. We find Carmichael numbers of the form$$n = Pqr$$ $n=Pqr$using an algorithm bifurcated by the size ofPwith respect to the tabulation boundB. For$$P < 7 \times 10^7$$ $P<7\times {10}^7$, we found 35985331 Carmichael numbers and 1202914 of them were less than$$10^{22}$$ ${10}^{22}$. When$$P > 7 \times 10^7$$ $P>7\times {10}^7$, we found 48476956 Carmichael numbers less than$$10^{22}$$ ${10}^{22}$. We provide a comprehensive overview of both cases of the algorithm. For the large case, we show and implement asymptotically faster ways to tabulate compared to the prior tabulation. We also provide an asymptotic estimate of the cost of this algorithm. It is interesting that Carmichael numbers are worst case inputs to this algorithm. So, providing a more robust asymptotic analysis of the cost of the algorithm would likely require resolution of long-standing open questions regarding the asymptotic density of Carmichael numbers. 

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