Rapid behavioral transitions produce chaotic mixing by a planktonic microswimmer

Despite their vast morphological diversity, many invertebrates have similar larval forms characterized by ciliary bands, innervated arrays of beating cilia that facilitate swimming and feeding. Hydrodynamics suggests that these bands should tightly constrain the behavioral strategies available to the larvae; however, their apparent ubiquity suggests that these bands also confer substantial adaptive advantages. Here, we use hydrodynamic techniques to investigate "blinking," an unusual behavioral phenomenon observed in many invertebrate larvae in which ciliary bands across the body rapidly change beating direction and produce transient rearrangement of the local flow field. Using a general theoretical model combined with quantitative experiments on starfish larvae, we find that the natural rhythm of larval blinking is hydrodynamically optimal for inducing strong mixing of the local fluid environment due to transient streamline crossing, thereby maximizing the larvae's overall feeding rate. Our results are consistent with previous hypotheses that filter feeding organisms may use chaotic mixing dynamics to overcome circulation constraints in viscous environments, and it suggests physical underpinnings for complex neurally-driven behaviors in early-divergent animals.Comment: 20 pages, 4 figure

Geometric Proofs of Horn and Saturation Conjectures

We provide a geometric proof of the Schubert calculus interpretation of the Horn conjecture, and show how the saturation conjecture follows from it. The geometric proof gives a strengthening of Horn and saturation conjectures. We also establish transversality theorems for Schubert calculus in non-zero characteristic. Some parts of the version posted in Nov 2002 (concerning explicit invariants constructed from Schubert calculus) have been removed from this version. They have appeared separately in IMRN 2004, no. 69, pages 3709--3721 " Invariant theory of GL(n) and Intersection theory of Grassmannians"Comment: 36 pages, accepted for publication in the Journal of Algebraic Geometr

Geometric Proof of a Conjecture of Fulton

We give a geometric proof of a conjecture of W. Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).Comment: 10 pages, no figure