Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies

In this paper, we study the existence and stability of collinear and noncollinear equilibrium points within the frame of the perturbed restricted problem of 2 + 2 bodies by a planetesimal belt. We compare and investigate the corresponding results of the perturbed and unperturbed models. The impact of the planetesimal belt is observed on collinear and noncollinear equilibrium points. We demonstrate that all equilibrium points are unstable, and we numerically investigate the noncollinear equilibrium points. Finally, we emphasize that the proposed problem is a credible model for describing the capture of small bodies by a planet.

Contrastive Explanations of Plans through Model Restrictions

In automated planning, the need for explanations arises when there is a mismatch between a proposed plan and the user’s expectation. We frame Explainable AI Planning as an iterative plan exploration process, in which the user asks a succession of contrastive questions that lead to the generation and solution of hypothetical planning problems that are restrictions of the original problem. The object of the exploration is for the user to understand the constraints that govern the original plan and, ultimately, to arrive at a satisfactory plan. We present the results of a user study that demonstrates that when users ask questions about plans, those questions are usually contrastive, i.e. “why A rather than B?”. We use the data from this study to construct a taxonomy of user questions that often arise during plan exploration. Our approach to iterative plan exploration is a process of successive model restriction. Each contrastive user question imposes a set of constraints on the planning problem, leading to the construction of a new hypothetical planning problem as a restriction of the original. Solving this restricted problem results in a plan that can be compared with the original plan, admitting a contrastive explanation. We formally define model-based compilations in PDDL2.1 for each type of constraint derived from a contrastive user question in the taxonomy, and empirically evaluate the compilations in terms of computational complexity. The compilations were implemented as part of an explanation framework supporting iterative model restriction. We demonstrate its benefits in a second user study.

Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

In this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.

The method of envelopes to concisely calculate semiparametric efficient scores under parametric restrictions

When addressing semiparametric problems with parametric restrictions (assumptions on the distribution), the efficient score (ES) of a parameter is often important for generating useful estimates. However, usual derivation of ES, although conceptually simple, is often lengthy and with many steps that do not help in understanding why its final form arises. This drawback often casts onto semiparametric estimation a mantle that can turn away otherwise able doctoral students or researchers. Here we show that many ESs can be obtained as a one-step derivation after we characterize those features (envelopes) of the unrestricted problem that are constrained in the restricted problem. We demonstrate our arguments in three problems with known ES but whose usual derivations are lengthy. We show that the envelope-based derivation is dramatically explanatory and compact, needing essentially two lines where the standard approach needs 10 or more pages. This suggests that the envelope method can add useful intuition and exegesis to both teaching and research of semiparametric estimation.

Geometrical characteristics of the stability domain in the restricted problem of eight bodies

The eight-body Newtonian problem is studied. Applying the symbolic calculation system Mathematica the stationary solutions, their stability in numerical form and the geometric characteristics of the stability domain are studied.