Related papers: Torus Construction
The rotation curves of disk galaxies exhibit a number of striking regularities. The amplitude of the rotation is correlated with luminosity (Tully-Fisher), the shape of the rotation curve is well predicted by the luminous mass distribution,…
The non-resonant secular dynamics of compact planetary systems are modeled by a perturbing function which is usually expanded in eccentricity and absolute inclination with respect to the invariant plane. Here, the expressions are given in a…
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…
We already saw in [A1] that the space of dynamically marked rational maps can be identified to a subspace of the space of covers between trees of spheres on which there is a notion of convergence that makes it sequentially compact. In the…
An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}.…
The manner in which probability amplitudes of paths sum up to form wave functions of orbital angular momentum eigenstates is described. Using a generalization of stationary-phase analysis, distributions are derived that provide a measure of…
The slow deformation of terrestrial orbits in the medium range, subject to lunisolar resonances, is well approximated by a family of Hamiltonian flow with $2.5$ degree-of-freedom. The action variables of the system may experience chaotic…
We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic…
Planetary migration is the process by which a forming planet undergoes a drift of its semi-major axis caused by the tidal interaction with its parent protoplanetary disc. One of the key quantities to assess the migration of embedded planets…
A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori…
A common problem in the optimization of structures is the handling of uncertainties in the parameters. If the parameters appear in the constraints, the uncertainties can lead to an infinite number of constraints. Usually the constraints…
We apply optimal control theory to a model of a polar active fluid (the Toner-Tu model), with the objective of driving the system into particular emergent dynamical behaviors or programming switching between states on demand. We use the…
In this paper, we consider chaotic dynamics and variational structures of area-preserving maps. There is a lot of study on the dynamics of their maps and the works of Poincare and Birkhoff are well-known. To consider variational structures…
We give a geometric characterization of the elements of a TRO that can be represented as compact operators by a faithful representation of the TRO.
Trajectory prediction and planning in autonomous driving are highly challenging due to the complexity of predicting surrounding agents' movements and planning the ego agent's actions in dynamic environments. Existing methods encode map and…
We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which…
We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group…
Ring galaxies are amazing objects exemplified by the famous case of the Hoag's Object. Here the mass of the central galaxy may be comparable to the mass of the ring, making it a difficult case to model mechanically. In a previous paper, it…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
Functions which are covariant or invariant under the transformations of a compact linear group $G$ acting in a euclidean space $\real^n$, can be profitably studied as functions defined in the orbit space of the group. The orbit space is the…