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Abstract. Let B be the unit ball of C n with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into C n of "positive real part", is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S 0 (B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr's work, we consider subsets of S 0 (B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S 0 (B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

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In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space C n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric represen-, then a mapping f ∈ S(B n ) is Aasymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f (z, t) such that Df (0, t) = e At , {e −At f (·, t)} t≥0 is a normal family on B n and f = f (·, 0). In particular, a spirallike mapping with respect to A ∈ L(C n , C n ) with ∞ 0 e (A−2m(A)In )t dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A * = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.

The effects of calcium on the mixing of synthetic diacylphosphatidylcholines (PC's) and diacylphosphatidylethanolamines (PE's) with the corresponding phosphatidic acids (PA's) have been examined by high-sensitivity differential scanning calorimetry and by measurements of the fluorescence of labeled PA or PC species in PA-PC bilayers. Calorimetrically derived phase diagrams for dimyristoyl- and dielaidoyl-substituted PA-PC and PA-PE mixtures indicate that these species are readily miscible in the absence of calcium but phase-separate very extensively in the presence of high levels of calcium (30 mM). The limiting solubilities of PA (Ca2+) in liquid-crystalline PC or PE bilayers are less than or equal to 10 and approximately 5 mol %, respectively, while approximately 20 mol % of PC or PE can be introduced into the "cochleate" phase of PA (Ca2+) before a distinct PC-rich (or PE-rich) phase appears. The kinetics of calcium-induced lateral phase separations were examined for dioleoyl- and dielaidoyl-substituted PA-PC unilamellar vesicles labeled with fluorescent (C12-NBD-acyl) PA or PC, whose fluorescence becomes partially quenched upon phase separation. Our results indicate that, for the PA-PC system, lateral phase separation is very rapid (approximately less than 1 s) after calcium addition and develops partially (possibly in only one face of the bilayer) when calcium is present only on one side of the bilayer. Moreover, phase separations can develop at a rate faster than that of vesicle diffusion when calcium is added to dilute suspensions of vesicles, suggesting that interbilayer contacts are not essential to promote phase separations.