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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. 

Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings. 

Permanent magnet manipulators provide a unique potential for safe operation of magnetically driven medical tools inside the human body for noninvasive surgical, imaging, and drug targeting procedures. These systems manipulate magnetic objects from a distance without direct contact, by generating a magnetic field using strong permanent magnets, and controlling the shape of this field properly. Control over the magnetic field is gained by displacement of the magnets using independent mechanical actuators for each magnet. However, interactions between the magnets result in a coupling between the actuators, which prevents them from independent and precise operation. This paper develops a multivariate, nonlinear feedback control to cancel the magnetic coupling between the actuators in effect. This feedback control incorporates a complex mathematical model of the magnetic interactions between the actuators, which does not admit a simple analytical form. Instead, this model is constructed numerically using the finite element method. The decoupling performance of the proposed feedback control is verified by numerical simulations.