Abstracts
On the Embedding of Zero-Dimensional Double Loops in Locally Euclidean
Double Loops
It is shown that a closed (hence locally compact)
zero-dimensional sub-double-loop
of a locally Euclidean double loop is always tamely embedded and that
its complement is simply connected.
Intuitive User Interfaces (IUI): A CASE Starting Point for Design and
Programming
A Rexx-Controlled Developing Environment for Implementing Intuitive
User Interfaces (IUI)
Automorphism groups of differentiable double loops
In this paper, we study local and global topological loops as well as
topological double loops having
a differentiable structure such that the loop operations
are differentiable. The main result states that
the group of differentiable automorphisms of a differentiable double loop
is compact with respect to the compact-open topology.
Automorphism groups of locally compact connected double loops are
locally compact
It is proved that the automorphism group of a locally
compact connected double loop is a locally compact transformation group
with respect to the compact-open topology.
On the dimensions of automorphism groups of four-dimensional double
loops
It is proved that a locally compact automorphism
group of a four-dimensional locally compact connected double loop is
at most four-dimensional.
Tables for an effective enumeration of real representations of quasi-simple
Lie groups
The purpose of this paper is to provide the data which are necessary to
calculate the (real) dimension, the centralizer and the kernel of a
finite-dimensional
irreducible real representation of a quasi-simple Lie group.
The authors have implemented a software package based on the LiE
programming language which allows the user to access these data easily.
An implementation in ANSI C is also available from the authors.
Fast Algorithms for Computing and Displaying Dose-Distributions in Tomogram-Oriented
Brachy-Radiotherapy
On the dimensions of automorphism groups of eight-dimensional double
loops
Let D be an eight-dimensional, locally compact, connected
double loop. It is proved that the dimension of its automorphism group
with respect to the compact-open topology is at most 16.
On the dimensions of automorphism groups of eight-dimensional ternary
fields II
Let T be an eight-dimensional, connected,
locally compact ternary field and let G denote
a connected closed Lie subgroup of its automorphism group
which is taken with the compact-open topology.
It is proved that if the ternary field of fixed elements of G
is connected, then G is either isomorphic to one of the
compact Lie groups G(2) or SU(3,C), or the
(covering) dimension of G is at most 7.
Differentiability of continuous homomorphisms between smooth loops
It is a well-known fact that a continuous homomorphism
between Lie groups is analytic. We prove a similar result for
continuous homomorphisms of differentiable left or right loops.
Another section of the paper deals with images and kernels of such homomorphisms.
Again, the results obtained are quite analogous to the Lie group case.
The paper ends with applications. For example, it turns out
that the group of continuous automorphisms of a smooth generalized polygon
is a Lie transformation group with respect to the compact-open topology.
On the dimensions of automorphism groups of eight-dimensional ternary
fields I
Let T be an eight-dimensional, connected,
locally compact ternary field and let G denote
a connected closed subgroup of its automorphism group
which is taken with the compact-open topology.
It is proved that G is either isomorphic to the
compact exceptional Lie group G(2),
or the (covering) dimension of G is at most 11.
This bound can be decreased
to 10, if the ternary field of fixed elements of
G is connected.
On Homomorphisms Between Generalized Polygons
The aim of this paper is to investigate homomorphisms of abstract,
topological and smooth generalized polygons. The first two sections deal with
generalized polygons and their homomorphisms. We have tried to make
this paper self-contained, so we give a short exposition of the
coordinatization and the algebraic operations (+,-,.,/)
of a generalized polygon. The most important theorems at this stage are
the characterization of injective homomorphisms, and the proof
of Pasini's Theorem that the fibers of a non-injective homomorphism
are infinite.
Topological polygons are introduced in section 3. Here, the main result
is that a homomorphism is either injective or locally constant; in
particular, a connected polygon admits only injective homomorphisms. For
topological projective planes, this is has been proved by Breitsprecher.
This result may be compared to Pasini's Theorem that
finite polygons admit only injective homomorphisms.
In the last section, we introduce smooth polygons. The main result of this
paper states that a continuous homomorphism between smooth polygons is always
a smooth imbedding. In particular, every continuous automorphism of
a smooth polygon is smooth, and thus the topological automorphism group of
the polygon, endowed with the compact-open topology, is a smooth Lie
transformation group.
As an application, we prove a strong inhomogeneity theorem for a
class of isoparametric hypersurfaces.
Smooth stable planes
This paper deals with smooth stable planes which generalize the notion of
differentiable (affine or projective) planes.
It is intended to be the first one of a series of papers
on smooth incidence geometry based on the Habilitationsschrift of the author.
It contains the basic definitions and results which are needed
to build up a foundation for a systematic study of smooth planes.
We define smooth stable planes, and we prove that
point rows and line pencils are closed submanifolds of the point set and line
set, respectively.
Moreover, the flag space is a closed submanifold of the product manifold
PxL, and the smooth structure
on the set P of points and on the set L of lines is uniquely
determined by the smooth structure of one single line pencil.
In the second section it is shown that for any point
p of P the tangent space T_pP
carries the structure of a locally
compact affine translation plane A_p.
Dually, we prove
that for any line l the tangent space T_lL
together with the set of all tangent spaces at p of lines through p
gives rise to some shear plane.
It turned out that the translation planes A_p are one of the most
important tools in the investigation of smooth incidence geometries.
The linearization theorems can be viewed
as the main results of this paper.
In the closing section we investigate some homogeneity properties of smooth
projective planes.
Collineations of smooth stable planes
We show that every continuous collineation between two smooth stable
planes is in fact a smooth collineation.
This implies that the group G of all continuous
collineations of a smooth stable plane is a Lie
transformation group on both the set P of points and the
set L of lines.
In particular, this shows that the point and line sets of a (topological)
stable plane S admit at most
one smooth structure such that S becomes a smooth stable plane.
The investigation of central and axial collineations in the case of
(topological) stable planes due to R. L\"owen
is continued for smooth stable planes. Many results of Loewen
which are only proved for low dimensional planes (dim S at
most 4)
are transferred to smooth stable planes of arbitrary finite dimension.
As an application of these transfers we show that the connected
componentes of the stabilizers
G_{[c,c]} and G_{[A,A]}
are closed, simply connected, solvable subgroups
of the automorphism group of S. Moreover, we show
that G_{[c,c]} is even abelian.
In the closing section we investigate the behaviour of reflections.
Collineation groups of smooth stable planes
Let S be a smooth stable plane of dimension n
and let G be a
closed subgroup of the collineation group of S which fixes
some point p. We derive some results on the group-theoretical structure
of G, e.g.\ that G is
a linear Lie group. As a by-product this shows that
no (affine or projective) Moulton plane can be turned into a smooth plane.
If G fixes some flag, then any semisimple subgroup of G
is a compact groupand G is contained in the flag stabilizer of the
classical Moufang plane of dimension n.
Let G fix three concurrent lines through the point p
If S is one of the
classical projective planes over the reals, the complex numbers,
the quaternions, or the Cayley numbers, then the dimension of
S is d(klass)=3, 6, 15, 38, respectively.
We show that for a smooth stable (projective) plane S
of dimension 2l
either S is an almost projective translation plane
(classical projective plane)
or that dim G is at most d(klass) - l
holds.
Solvable collination groups of smooth projective planes
Let G be a closed solvable subgroup of the
collineation group of one of the four classical Moufang planes
P_2(F) over one of the domains R, C, H, O,
where H denotes the quaternions and O the Cayley numbers.
Then dim G is at most 4,9,17,30 if F=R, C, H, O.
For arbitrary compact connected projective planes P
of finite dimension n M. Lueneburg derived upper bounds for the dimension
of G depending on n and the configuration of the
fixed elements of G. In this paper we consider
dim G in the case of smooth projective planes.
Compared to the results of Lueneburg, the upper bounds of dim G
can be lowered in almost every case; except for one single case
we even derive sharp bounds.
Smooth 16-dimensional projective planes
Smooth projective planes are projective planes defined on smooth manifolds
(i.e. the set of points and the set of lines are smooth manifolds)
such that the geometric operations of join and intersection are smooth.
A systematic study of such planes and of their collineation groups
can be found in the articles above
We prove in this paper that a 16-dimensional smooth projective plane
which admits a collineation group of dimension d greater than
38 is isomorphic to the octonion projective plane.
For topological compact projective planes this is true if d is
greater then 40. Note that there are non-classical
topological planes with a collineation group of dimension 40.
Smooth stable planes and the moduli spaces of locally compact translation planes
At every point p of a smooth stable plane S = (P,L,F)
the tangent spaces of the lines through d form a compact spread
on the tangent space T_p thus defining a locally compact topological
affine translation plane A_p.
We define a topological locally trivial bundle JP over P that
describes the classes of isomorphic spreads in the tangent spaces T_p.
We show that the topology of the total space of JP is not T_1 by
constructing a sequence of non-classical spreads in F^2
that converges to the classical spread in F^2,
where F^2 \in {\CC,\HH,\OO}.
Moreover, we prove that the isomorphism
type of A_p varies continuously with the point p.
Finally, we give examples of smooth affine planes
which have both classical and non-classical tangent translation planes.
Smooth Hughes planes
We prove that the only compact projective Hughes planes
which are smooth projective planes are the classical planes
over the quaternions and the Caley numbers.
As a by-product this shows
that an 8-dimensional smooth projective plane
which admits a collineation group of dimension d \geq 17
is isomorphic to the quaternion projective plane \cP_2\HH.
For topological compact projective planes this is true
if d \geq 19,and this bound is sharp.
Smooth flat projective planes
Regular smooth stable planes
Smooth flexible projective shift planes
We prove that the only four-dimensional flexible shift plane
which can be turned into a smooth projective plane is the complex
projective plane.
Smooth sphere bundles and geodesic stable planes
Riemannian projective planes