A GENERALIZATION OF THE 0-NUMERICAL RANGE
Let H be a complex Hilbert space. Given a bounded linear operator A on H, we describe the set R n (A) = fV AW : V;W : C n ! H;V V = W W = In;V W = 0g. It is shown that the closed matricial convex hull of R n (A) is a closed ball of radius minfkA I k : 2 Cg centered at the origin.
GROWTH CONDITIONS AND THE NUMERICAL RANGE IN A BANACH ALGEBRA
0. Introduction. In this paper, we consider the numerical range in an arbitrary Banach algebra with identity, and study its relation to various growth conditions on the resolvent. In §1, we list several facts about this generalized numerical range. Some of these are more or less well known for concrete algebras, but do not seem to have been formulated in their proper generality. In particular, we recognize that part of this section is implicit or explicit in Lumer [7]. Consequently, we have included proofs here only when a result is new (or requires a new technique), or when the proof represents a substantial simplification. The main result of §2 is a Phragmen-Lindelδf theorem for quasi-nilpotent elements in a Banach algebra. This yields a sharper version of a similar theorem of Lumer and Phillips [8]. Finally, in §3 we apply the methods of §1 to study the numerical range in its usual setting, Hubert space. The notion of essential numerical range appears naturally here, and this set is shown to be chracterized in the way one would expect by analogy with the essential spectrum.
The infection process, sporulation and survival of Alternaria helianthi on sunflower
SUMMARY 
 
Electron microscopy was used to study the infection of sunflower leaves by Alternaria helianthi. Conidia germinated by producing one to many germ tubes which grew across the leaf surface before forming appressoria. The fungus directly penetrated its host through the cuticle and epidermis. Entry into the host through wounds and stomates was also observed. Extracellular sheaths were found to be associated with germ tubes and intercellular hyphae of A. helianthi. Conidiophores developed through collapsed stomates, from leaf veins, trichomes and also from mycelium growing across the host leaf surface. Microcylic conidia were produced directly from parent conidia under certain conditions. Studies using a volumetric spore trap showed that the airborne spore concentration followed a distinct periodicity with peaks occurring between 0900 and 1100 h each day. Laboratory studies showed that safflower, noogoora burr and bathurst burr could serve as alternative hosts for A. helianthi. The pathogen was readily isolated from sunflower crop debris from a diseased crop that had been harvested 1 yr earlier.