The construction of higher order uniformly convergent methods for the numerical resolution of any type of singularly perturbed problems, is an interesting task in applied mathematics. The main reason is that those methods permit obtaining good and efficient numerical approximations without increasing the computational cost of the numerical algorithm. Here, we study 2D elliptic weakly coupled systems of convection diffusion type, for which small positive parameters appear in both the diffusion and the convection terms. Moreover, we assume that the diffusion parameters at each equation of the system are different and the convection term has only a non zero component in one of the two spatial directions. Then, the exact solution has parabolic and regular layers in the boundary of the domain, which depend on the value and the ratio between the three small parameters. To solve the continuous problem, an hybrid finite difference scheme is used, which is constructed on an adequate Bakhvalov-Shishkin mesh; then, the scheme is a second order uniformly convergent method; this result is better than the obtained by previous methods existing in the literature to solve the same type of systems. We show the numerical results that the computational algorithm gives for a test problem; from them, we clearly observe the order of uniform convergence of the method and also its efficiency due the low computational cost needed to achieve the numerical results.
This work focusses on developing and analyzing a uniformly convergent numerical method to solve a linear two-dimensional parabolic singularly perturbed reaction diffusion problem, with a small positive diffusion parameter multiplying the highest derivative term in the differential equation. Moreover, it is supposed that the source term of the continuous problem exhibits discontinuous behavior along axis-aligned lines of the spatial domain, known as interface lines. In cases where the diffusion parameter is adequately small, in general, parabolic boundary layers appear in the exact solution, on all sides of the spatial boundary; moreover the discontinuities in the source function induce the formation of interior layers within the neighbourhoods adjacent to the interfaces. To solve efficiently this problem, we construct a numerical technique by developing a second order difference scheme, that has been implemented on a mesh in the spatial direction adapted to the layer behavior of the solution, combined with an alternating direction implicit scheme, defined on a uniformly distributed grid, to discretize in the temporal direction. The proposed numerical method is proven to be uniformly convergent while maintaining accuracy of first order in time and almost second order in space. Numerical computations for various test examples are carried out to reinforce the theoretical framework.
Deformable linear objects (DLOs) are widely encountered in everyday life, taking forms such as plastic tubes, wires, ropes, and cables. They are prevalent across diverse settings, including industrial, domestic, and medical environments, as well as in outdoor applications like electric power lines, subaquatic cables, and aerial transport systems. These objects are termed deformable due to their ability to undergo significant shape changes under external forces, and linear because their length vastly exceeds their cross-sectional dimensions. Despite their importance and widespread presence, developing robotic systems capable of interacting with DLOs poses numerous challenges. This survey presents a comprehensive review of the state-of-the-art methods developed over the past decade to address these challenges. It covers key areas including physical and data-driven modeling techniques, simulation environments, perception approaches based on vision and tactile sensing, as well as strategies for estimation, planning, and control. It also reviews common manipulation tasks such as grasping, shaping, routing, knotting, suturing, and transport. The survey concludes with a critical discussion of current limitations and outlines promising directions for future research.
This paper continues from the discussion of Florax et al. (Florax, R., H. Folmer and S. Rey, 2003. Specification searches in spatial econometrics: the relevance of Hendry's methodology. Regional Science and Urban Economics, 33, 557–579.), regarding the properties of various specification strategies for spatial econometric models. Habitual practise has popularised a technique based on the well-known Lagrange Multipliers, characterized as a Specific-to-General approach, and which seems to give good results. In our work, we contemplate other alternatives, some of which may be seen as slight variations of this proposal, including the selection tests of Vuong (Vuong, Q., 1989. Likelihood ratio-tests for model selection and non-nested hypotheses. Econometrica, 57, 307–333.) and of Clarke (Clarke, K., 2003. Nonparametric model discrimination in international relations. Journal of Conflict Resolutions, 47, 72–93.). We also examine an approach of the General-to-Specific type, as clearly opposite to the others. The comparison of the two strategies is carried out through a Monte Carlo experiment, the results of which are quite diffuse, in the sense that we do not find conclusive evidence in favour of either of these two approaches. However, it should be recognized that the General-to-Specific strategy seems to be more robust to the existence of anomalies in the Data Generating Process.
Multi-lanthanide metal-organic frameworks (MOFs) offer a flexible route for designing multifunctional materials. Here we report a carborane-based isostructural series of MOFs of formula {[(NdyYb1-y)3(mCB-L)4(NO3)(DMF)x]n & centerdot;Solv}, including the homometallic Nd (y = 1) and Yb (y = 0) compounds and a mixed Nd/Yb analogue (y = 0.44). Magnetic, magnetocaloric and near-infrared (NIR) optical properties were investigated by dc/ac magnetometry, X-ray absorption spectroscopy (XAS), X-ray magnetic circular dichroism (XMCD), and photoluminescence. Nd3+ and Yb3+ yield MOFs combining slow relaxation of the magnetization (U/kB similar to 19 K), cryogenic magnetocaloric response (-Delta Sm similar to 1.6 R at 5 T, 1.8 K) and ion-centered NIR luminescence. Notably, the mixed Nd/Yb MOF further extends this multifunctionality by exhibiting dual NIR emission at 998 nm and 1060 nm arising from partial Nd -> Yb energy transfer. These results underscore carborane ligands as effective blocks for engineering multi-lanthanide frameworks, and highlight {Nd/Yb} MOFs as multifunctional materials for quantum technologies, optical communication, and cryogenic cooling.
