This paper considers the negative gradient trajectories associated with the modified total squared curvature functional ∫ k2 + ν ds. The focus is on the limiting behavior as ν tends to zero from the positive side. It is shown that when ν = 0 spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. A...

w x Hodge and Pedoe 5 constructed bases of the homogeneous coordinate rings of the Schubert varieties in the Grassmannian, consisting of certain products of Plucker coordinates called standard monomials. Lakshmibai, ̈ w x Musili, and Seshadri 6 have generalized this theory extensively, giving standard monomial bases of the spaces of global sections of certain line bundles over unions of Schubert...

An infrared-imaging system has been used to study the influence of gravity on the kinetics of first positive phototropism. The development of phototropic curvature of etiolated seedlings of Arabidopsis thaliana was measured in the absence of visible radiation. Following a pulse of blue light, stationary seedlings curved to a maximum of approximately 160 about 80 minutes after stimulation. The s...