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1. Four-body bound states in momentum space: the Yakubovsky approach without two-body t − matrices

   https://doi.org/10.3389/fphy.2023.1232691  

   Mohammadzadeh, M; Radin, M; Mohseni, K; Hadizadeh, M R (December 2023, Frontiers in Physics)

   This study presents a solution to the Yakubovsky equations for four-body bound states in momentum space, bypassing the common use of two-bodyt− matrices. Typically, such solutions are dependent on the fully-off-shell two-bodyt− matrices, which are obtained from the Lippmann-Schwinger integral equation for two-body subsystem energies controlled by the second and third Jacobi momenta. Instead, we use a version of the Yakubovsky equations that does not requiret− matrices, facilitating the direct use of two-body interactions. This approach streamlines the programming and reduces computational time. Numerically, we found that this direct approach to the Yakubovsky equations, using 2B interactions, produces four-body binding energy results consistent with those obtained from the conventionalt− matrix dependent Yakubovsky equations, for both separable (Yamaguchi and Gaussian) and non-separable (Malfliet-Tjon) interactions. 

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2. A note on the accuracy of the generalized‐α scheme for the incompressible Navier‐Stokes equations

   https://doi.org/10.1002/nme.6550  

   Liu, Ju; Lan, Ingrid S.; Tikenogullari, Oguz Z.; Marsden, Alison L. (October 2020, International Journal for Numerical Methods in Engineering)

   Abstract We investigate the temporal accuracy of two generalized‐ schemes for the incompressible Navier‐Stokes equations. In a widely‐adopted approach, the pressure is collocated at the time stept_n + 1while the remainder of the Navier‐Stokes equations is discretized following the generalized‐ scheme. That scheme has been claimed to besecond‐order accurate in time. We developed a suite of numerical code using inf‐sup stable higher‐order non‐uniform rational B‐spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve high spatial accuracy and to investigate asymptotic temporal convergence behavior. Numerical evidence suggests that onlyfirst‐order accuracyis achieved, at least for the pressure, in this aforesaid temporal discretization approach. On the other hand, evaluating the pressure at the intermediate time step recovers second‐order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized‐ scheme of choice when integrating the incompressible Navier‐Stokes equations. 

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3. A Universal Equation to Predict Ω _m from Halo and Galaxy Catalogs

   https://doi.org/10.3847/1538-4357/acee6f  

   Shao, Helen; de_Santi, Natalí_S M; Villaescusa-Navarro, Francisco; Teyssier, Romain; Ni, Yueying; Anglés-Alcázar, Daniel; Genel, Shy; Steinwandel, Ulrich P; Hernández-Martínez, Elena; Dolag, Klaus; et al (October 2023, The Astrophysical Journal)

   Abstract We discover analytic equations that can infer the value of Ω_mfrom the positions and velocity moduli of halo and galaxy catalogs. The equations are derived by combining a tailored graph neural network (GNN) architecture with symbolic regression. We first train the GNN on dark matter halos from GadgetN-body simulations to perform field-level likelihood-free inference, and show that our model can infer Ω_mwith ∼6% accuracy from halo catalogs of thousands ofN-body simulations run with six different codes: Abacus, CUBEP³M, Gadget, Enzo, PKDGrav3, and Ramses. By applying symbolic regression to the different parts comprising the GNN, we derive equations that can predict Ω_mfrom halo catalogs of simulations run with all of the above codes with accuracies similar to those of the GNN. We show that, by tuning a single free parameter, our equations can also infer the value of Ω_mfrom galaxy catalogs of thousands of state-of-the-art hydrodynamic simulations of the CAMELS project, each with a different astrophysics model, run with five distinct codes that employ different subgrid physics: IllustrisTNG, SIMBA, Astrid, Magneticum, SWIFT-EAGLE. Furthermore, the equations also perform well when tested on galaxy catalogs from simulations covering a vast region in parameter space that samples variations in 5 cosmological and 23 astrophysical parameters. We speculate that the equations may reflect the existence of a fundamental physics relation between the phase-space distribution of generic tracers and Ω_m, one that is not affected by galaxy formation physics down to scales as small as 10h⁻¹kpc. 

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4. Singularity formation in 3D Euler equations with smooth initial data and boundary

   https://doi.org/10.1073/pnas.2500940122  

   Chen, Jiajie; Hou, Thomas_Y (June 2025, Proceedings of the National Academy of Sciences)

   A long-standing fundamental open problem in mathematical fluid dynamics and nonlinear partial differential equations is to determine whether solutions of the 3D incompressible Euler equations can develop a finite-time singularity from smooth, finite-energy initial data. Leonhard Euler introduced these equations in 1757 [L. Euler,Mémoires de l’Académie des Sci. de Berlin11, 274–315 (1757).], and they are closely linked to the Navier–Stokes equations and turbulence. While the general singularity formation problem remains unresolved, we review a recent computer-assisted proof of finite-time, nearly self-similar blowup for the 2D Boussinesq and 3D axisymmetric Euler equations in a smooth bounded domain with smooth initial data. The proof introduces a framework for (nearly) self-similar blowup, demonstrating the nonlinear stability of an approximate self-similar profile constructed numerically via the dynamical rescaling formulation. 

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5. Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

   https://doi.org/10.1090/bull/1838  

   Xin, Jack; Yu, Yifeng; Ronney, Paul (July 2024, Bulletin of the American Mathematical Society)

   G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. Thelack of coercivityof Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomeshighly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection. 

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