Periodic Points and Topological Restriction Homology

We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.

Semiglobal Nonmemoryless Attitude Controls on the Special Orthogonal Group

This paper presents tracking strategies for the attitude dynamics of a rigid body that are global on the configuration space SO(3) and semiglobal over the phase space SO(3)×ℝ3. It is well known that global attractivity is prohibited for continuous attitude control systems on the special orthogonal group. Such topological restriction has been dealt with either by constructing smooth attitude control systems that exclude a set of zero measure in the region of attraction or by introducing discontinuities in the control input. This paper proposes nonmemoryless attitude control systems that are continuous in time, where the region of attraction guaranteeing exponential convergence completely covers the special orthogonal group. This provides a new framework to address the topological restriction in attitude controls. The efficacy of the proposed methods is illustrated by numerical simulations and an experiment.

Stability of capillary hypersurfaces in a manifold with density

We introduce capillary hypersurfaces and its stability in a manifold with density. We prove that stable [Formula: see text]-minimal hypersurfaces with free boundary in a geodesic ball in space form with suitable radial density must be totally geodesic. We also prove two criteria for instability of the capillary hypersurfaces in a Euclidean ball with suitable density. At last, we obtain a topological restriction on strongly stable capillary surfaces in a 3-manifold with density under certain conditions. These results generalize those in a manifold with constant density.

Note on the Support of Sobolev Functions

We prove a topological restriction on the support of Sobolev functions.

Topologically restricted appearance in the developing chick retinotectal system of Bravo, a neural surface protein: experimental modulation by environmental cues.

A novel neural surface protein, Bravo, shows a pattern of topological restriction in the embryonic chick retinotectal system. Bravo is present on the developing optic fibers in the retina; however, retinal axons in the tectum do not display Bravo. The appearance of Bravo in vitro is modulated by environmental cues. Axons growing out from retinal explants on retinal basal lamina, their natural substrate, express Bravo, whereas such axons growing on collagen do not. Retinal explants provide a valuable system to characterize the mechanism of Bravo restriction, as well as the cellular signals controlling it. Bravo was identified with monoclonal antibodies from a collection generated against exposed molecules isolated by using a selective cell surface biotinylation procedure. The NH2-terminal sequence of Bravo shows similarity with L1, a neural surface molecule which is a member of the immunoglobulin superfamily. This possible relationship to L1, together with its restricted appearance, suggests an involvement of Bravo in axonal growth and guidance.