1,895 research outputs found
Boundary smoothness of analytic functions
We consider the behaviour of holomorphic functions on a bounded open subset
of the plane, satisfying a Lipschitz condition with exponent , with
, in the vicinity of an exceptional boundary point where all such
functions exhibit some kind of smoothness. Specifically, we consider the
relation between the abstract idea of a bounded point derivation on the algebra
of such functions and the classical complex derivative evaluated as a limit of
difference quotients. We obtain a result which applies, for example, when the
open set admits an interior cone at the special boundary point.Comment: 14 pages. This revision corrects a misprint on p.12: In equation (3),
should have been . Also a misprint on page 14 in the
formula for . The validity of the argument is not affected and the
result stand
Pervasive Algebras and Maximal Subalgebras
A uniform algebra on its Shilov boundary is {\em maximal} if is
not and there is no uniform algebra properly contained between and
. It is {\em essentially pervasive} if is dense in whenever
is a proper closed subset of the essential set of . If is maximal,
then it is essentially pervasive and proper. We explore the gap between these
two concepts. We show the following: (1) If is pervasive and proper, and
has a nonconstant unimodular element, then contains an infinite descending
chain of pervasive subalgebras on . (2) It is possible to imbed a copy of
the lattice of all subsets of into the family of pervasive subalgebras of
some . (3) In the other direction, if is strongly logmodular, proper
and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is
removed. We discuss further examples, involving Dirichlet algebras,
algebras, Douglas algebras, and subalgebras of . We
develop some new results that relate pervasiveness, maximality and relative
maximality to support sets of representing measures
Conjugacy of real diffeomorphisms. A survey
Given a group G, the conjugacy problem in G is the problem of giving an
effective procedure for determining whether or not two given elements f, g of G
are conjugate, i.e. whether there exists h belonging to G with fh = hg. This
paper is about the conjugacy problem in the group Diffeo(I) of all
diffeomorphisms of an interval I in R. There is much classical work on the
subject, solving the conjugacy problem for special classes of maps.
Unfortunately, it is also true that many results and arguments known to the
experts are difficult to find in the literature, or simply absent. We try to
repair these lacunae, by giving a systematic review, and we also include new
results about the conjugacy classification in the general case.Comment: 53 page
Factoring Formal Maps into Reversible or Involutive Factors
An element of a group is called reversible if it is conjugate in the
group to its inverse. An element is an involution if it is equal to its
inverse. This paper is about factoring elements as products of reversibles in
the group of formal maps of , i.e.
formally-invertible -tuples of formal power series in variables, with
complex coefficients. The case was already understood.
Each product of reversibles has linear part of determinant .
The main results are that for each map with det is the
product of reversibles, and may also be factored as the product of
involutions, where is the smallest integer .Comment: 20 page
Geometry in the Transition from Primary to Post-Primary
This article is intended as a kind of precursor to the document Geometry for
Post-primary School Mathematics, part of the Mathematics Syllabus for Junior
Certicate issued by the Irish National Council for Curriculum and Assessment in
the context of Project Maths.
Our purpose is to place that document in the context of an overview of plane
geometry, touching on several important pedagogical and historical aspects, in
the hope that this will prove useful for teachers.Comment: 19 page
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