The radial growth of Bloch functions has been extensively studied. Using integral means estimates and the Hardy Littlewood theorem, Makarov proved the so called law of iterated logarithm for Bloch functions. This result has also been obtained using probabilistic arguments. In this paper we present another method of studying the radial growth of Bloch functions, having the integral means estimates as starting point and using certain results about normal functions.

Optimal control problem of semilinear evolutionary distributed parameter systems with impulse controls is considered. Necessary conditions of optimal controls are derived. The result generalises the usual Pontryagin's maximum principle.

We prove that there exist skew type Hadamard matrices of order 4n for n = 67, 113, 127, 157, 163, 181 and 241 which have not been constructed so far. In particular there exists a Hadamard matrix of order 4 · 163, which was unknown until now. We mention that very recently we have constructed skew type Hadamard matrices of orders 4n for n = 37 and 43.

This paper examines conditions for the monotone convergence of general Newton-like methods generated by point-to-point mappings. The speed of convergence of such mappings is also examined.

Topological degree theory can be applied to maps defined from Linear Complementarity Problems, as has been done by Howe and Stone, Ha, and Stewart. It is shown here that the definitions of Howe and Stone, and Stewart, are equivalent. Also a new family of matrices is defined whose degrees' magnitudes increase exponentially as 2n/√2πn, whereas Howe and Stone give examples whose degrees go as (22/5)n.

We give a simple geometric proof that the Jones polynomial at the value i of an oriented link is invariant under pass-equivalence.

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

Popov has recently introduced a class of subspaces of Lp(μ) (μ nonatomic) which generalise the finite codimensional ones, and proved that for p ≠ 2 any projection onto such a subspace has a norm strictly greater than one. In this paper we give the quantitative version of Popov's result computing the best possible lower bound for the norms of the considered projections.

In this paper, we deal with the following question: What kind of open covers are normal if they have cushioned open refinements? For this, we prove that an open cover consisting of members with compact closure is a desired one.

Let ϕ be a nonnegative, nondecreasing and nonconstant function defined on [0, ∞) such that Φ(t) = ϕ(et) is a convex function on (-∞, ∞). The Hardy-Orlicz space H (ϕ) is defined to be the class of all those functions f holomorphic in the open unit disc of the complex plane C satisfying The subclass H(ϕ)+ of H(ϕ) is defined to be the class of all those functions f ∈ H(ϕ) satisfying for almost all points eit of the unit circle. In 1990, Z. Jianzhong conjectured that H(ϕ)+ = H(ψ)+ if and only if H(ϕ) = H(ψ). In the present paper we prove that it is true not only on the unit disc of C but also on the unit ball of Cn.

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

We establish the Karlovitz lemma for a nonexpansive self mapping of a nonempty weak* compact convex set in a weak* orthogonal dual Banach lattice.

The loxodromic mapping conjecture of J. Harrison is affirmed for diffeomorphisms of the 2-sphere that embed in flows.