Cusp Algebras

We consider simple cusp algebras, that is certain subalgebras of the algebra of holomorphic functions on a disk that are annihilated by some distributions living on a singleton. We determine when these algebras can be holized in two dimensions, and when these holizations are globally biholomorphic

Cluster observations of the midaltitude cusp under strong northward interplanetary magnetic field

We report on a multispacecraft cusp observation lasting more than 100 min. We determine the cusp boundary motion and reveal the effect on the cusp size of the interplanetary magnetic field (IMF) changing from southward to northward. The cusp shrinks at the beginning of the IMF rotation and it reexpands at the rate of 0.40° invariant latitude per hour under stable northward IMF. On the basis of plasma signatures inside the cusp, such as counterstreaming electrons with balanced fluxes, we propose that pulsed dual lobe reconnection operates during the time of interest. SC1 and SC4 observations suggest a long-term regular periodicity of the pulsed dual reconnection, which we estimate to be ~1–5 min. Further, the distances from the spacecraft to the reconnection site are estimated on the basis of observations from three satellites. The distance determined using SC1 and SC4 observations is ~15 RE and that determined from SC3 data is ~8 RE. The large-scale speed of the reconnection site sunward motion is ~16 km s-1. We observe also a fast motion of the reconnection site by SC1, which provides new information about the transitional phase after the IMF rotation. Finally, a statistical study of the dependency of plasma convection inside the cusp on the IMF clock angle is performed. The relationship between the cusp stagnation, the dual lobe reconnection process, and the IMF clock angle is discussed

Can a Drinfeld module be modular?

Let $k$ be a global function field with field of constants \Fr and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard B\"ockle attaches abelian Galois representations to characteristic $p$ valued cusp eigenforms and double cusp eigenforms \cite{go1} such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where k=\Fr(T) and $\infty$ corresponds to the pole of $T$, it then becomes reasonable to ask whether rank 1 Drinfeld modules over $k$ are themselves ``modular'' in that their Galois representations arise from a cusp or double cusp form. This paper gives an introduction to \cite{boc2} with an emphasis on modularity and closes with some specific questions raised by B\"ockle's work.Comment: Final corrected versio

An Algorithm for Computing Cusp Points in the Joint Space of 3-RPR Parallel Manipulators

This paper presents an algorithm for detecting and computing the cusp points in the joint space of 3-RPR planar parallel manipulators. In manipulator kinematics, cusp points are special points, which appear on the singular curves of the manipulators. The nonsingular change of assembly mode of 3-RPR parallel manipulators was shown to be associated with the existence of cusp points. At each of these points, three direct kinematic solutions coincide. In the literature, a condition for the existence of three coincident direct kinematic solutions was established, but has never been exploited, because the algebra involved was too complicated to be solved. The algorithm presented in this paper solves this equation and detects all the cusp points in the joint space of these manipulators

Are the Ultra-Faint Dwarf Galaxies Just Cusps?

We develop a technique to investigate the possibility that some of the recently discovered ultra-faint dwarf satellites of the Milky Way might be cusp caustics rather than gravitationally self-bound systems. Such cusps can form when a stream of stars folds, creating a region where the projected 2-D surface density is enhanced. In this work, we construct a Poisson maximum likelihood test to compare the cusp and exponential models of any substructure on an equal footing. We apply the test to the Hercules dwarf (d ~ 113 kpc, M_V ~ -6.2, e ~ 0.67). The flattened exponential model is strongly favored over the cusp model in the case of Hercules, ruling out at high confidence that Hercules is a cusp catastrophe. This test can be applied to any of the Milky Way dwarfs, and more generally to the entire stellar halo population, to search for the cusp catastrophes that might be expected in an accreted stellar halo.Comment: Accepted for publication in ApJ Letters. Minor revisions from version