which are analytic in the open unit disk E. Also, A1 = A, the usual class of analytic functions defined in the open unit disk E = {z : |z| < 1}. Let f(z) and g(z) be analytic in E. We say that the function f is subordinate to the function g and write f(z) ≺ g(z), if and only if there exists Schwarz function w, analytic in E such that w(0) = 0, |w(z)| < 1 for z ∈ E, and f(z) = g(w(z)). In partic...

In this note we shall announce results concerning the structure of £^0^)> the space of 2s-valued functions integrable with respect to a measure m:H->L(E, F), where L(E,F) is the class of bounded operators from the Banach space E into the Banach space F. The bilinear integration theory introduced here is more restrictive than the one developed by Bartle [1], but it is general enough to allow a n...

Let E be a rearrangement-invariant Banach function space on [0, 1], and let Γ(E) denote the set of all p ≥ 1 such that any sequence {xn} in E converging weakly to 0 has a subsequence {yn} with supm m−1/p‖ ∑ 1≤k≤m yn‖ < ∞. The set Γi(E) is defined similarly, but only sequences {xn} of independent random variables are taken into account. It is proved (under the assumption Γ(E) = {1}) that if Γi(E...

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

Let P ⊂ R3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additionnal assumption of codecomposability. The proof relies on a result of independent interest concerni...

A nonerasing morphism σ is said to be weakly unambiguous with respect to a word s if σ is the only nonerasing morphism that can map s to σ(s), i. e., there does not exist any other nonerasing morphism τ satisfying τ(s) = σ(s). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-...

We formulate the problem of optimizing a convex function over the weakly e$cient set of a multicriteria a$ne fractional program as a special biconvex problem. We propose a decomposition algorithm for solving the latter problem. The proposed algorithm is a branch-and-bound procedure taking into account the a$ne fractionality of the criterion functions. c © 2001 Published by Elsevier Science B.V.