On the sense preserving mappings in the Helm topology in the plane

∗Research supported by the grant No. GAUK 186/96 of Charles University.We introduce the helm topology in the plane. We show that (assuming the helm local injectivity and the Euclidean continuity) each mapping which is oriented at all points of a helm domain U is oriented at U

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials

Extensions of Normed Algebras

We review and analyse techniques from the literature for extending a normed algebra, A to a normed algebra, B, so that B has interesting or desirable properties which A may lack. For example, B might include roots of monic polynomials over A. These techniques have been important historically for constructing examples in the theory of Banach algebras. We construct new examples in this way. Elsewhere we contribute to the related programme of determining which properties of an algebra are shared by certain extensions of it. Similarly, we consider the relations between the topological spaces, M(A) and M(B), of closed, maximal ideals of A and B respectively. For example, it is shown that if B is one of the types of 'algebraic extensions' of A constructed in the thesis and M(B) has trivial first Cech-cohomology group then so has M(A). The invertible group of a normed algebra is studied in Chapter 4; it is shown that if a Banach algebra, A, has dense invertible group then so has every integral extension of A. The context for this work is also explained: some new results characterising trivial uniform algebras by means of approximation by invertible elements are given. We show how these results partially answer a famous, open problem of Gelfand. Results in Chapter 4 lead to the conjecture that a uniform algebra is trivial if the group of exponentials of its elements is dense in the algebra. We investigate this conjecture in Chapter 5. In the search for a counterexample, we construct and establish some properties of `logarithmic extensions' of a regular uniform algebra.Comment: Ph.D. Thesis, University of Nottingham, 2003. 76 pages; plain Te

125th Anniversary Edition

The Atlanta University Center Robert W. Woodruff Library acknowledges the generous support of the Council on Library and Information Resources (CLIR) in supporting the processing and digitization of a number of historic collections as part of the project: Our Story: Digitizing Publications and Photographs of the Historically Black Atlanta University Center Institutions