Representing measures on the Royden boundary for solutions of Δu=Pu on a Riemannian manifold

Consider the Royden compactification R* of a Riemannian n-manifold R, Γ = R*\R its Royden boundary, Δ its harmonic boundary and the elliptic differential equation Δu = Pu, P ≥ 0 on R. A regular Borel measure m^P can be constructed on Γ with support equal to the closure of Δ^P = {q ϵ Δ : q has a neighborhood U in R* with _U^ʃ_ᴖR^{P ˂ ∞} }. Every enegy-finite solution to u (i.e. E(u) = D(u) + ^ʃ_Ru²P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ^ʃ_Γu(q)K(z,q)dm^P(q) where K(z,q) is a continuous function on ^Rx Γ . A _P^~_E-function is a nonnegative solution which is the infimum of a downward directed family of energy-finite solutions. A nonzero _P^~_E-function is called _P^~_E-minimal if it is a constant multiple of every nonzero _P^~_E-function dominated by it. THEOREM. There exists a _P^~_E-minimal function if and only if there exists a point in q ϵ Γ such that m^P(q) > 0. THEOREM. For q ϵ Δ^P , m^P(q) > 0 if and only if m⁰(q) > 0 .

Talbot-SNOM method for non-contact evaluation of high-density gratings

A Talbot scanning near-field optical microscopy (SNOM) method for non-contact evaluating of high-density gratings was described. This method combines the Talbot self-imaging effect of the gratings and the conventional SNOM technique without damage. The significant advantages of this method are its simple structure, reliable and fast measurement for the surface quality of the tested gratings. Experimental results of three different kinds of gratings were demonstrated to indicate that this method is effective for evaluation surface quality of high-density gratings. (c) 2004 Elsevier B.V. All rights reserved.

Effective herding of flatfish by cables with minimal seafloor contact

Otter trawls are very effective at capturing flatfish, but they can affect the seaf loor ecosystems where they are used. Alaska f latf ish trawlers have very long cables (called sweeps) between doors and net to herd fish into the path of the trawl. These sweeps, which ride on and can disturb the seaf loor, account for most of the area affected by these trawls and hence a large proportion of the potential for damage to seaf loor organisms. We examined modifications to otter trawls, such that disk clusters were installed at 9-m intervals to raise trawl sweeps small distances above the seafloor, greatly reducing the area of direct seafloor contact. A critical consideration was whether flatfish would still be herded effectively by these sweeps. We compared conventional and modified sweeps using a twin trawl system and analyzed the volume and composition of the resulting catches. We tested sweeps raised 5, 7.5, and 10 cm and observed no significant losses of flatfish catch until sweeps were raised 10 cm, and those losses were relatively small (5–10%). No size composition changes were detected in the flatfish catches. Alaska pollock (Theragra chalcogramma) were captured at higher rates with two versions of the modified sweeps. Sonar observations of the sweeps in operation and the seaf loor after passage confirmed that the area of direct seafloor contact was greatly reduced by the modified sweep