-
2004/01 P. Houston, I. Perugia, A. Schneebeli and D. Schotzau.
-
Interior Penalty Method for the Indefinite
Time-Harmonic Maxwell Equations
In this paper, we introduce and analyze the interior
penalty discontinuous Galerkin method for the numerical
discretization of the {\em indefinite} time-harmonic
Maxwell equations in high-frequency regime. Based on
suitable duality arguments, we derive a-priori error
bounds in the energy norm and the $L^2$-norm. In
particular, the error in the energy norm is shown to
converge with the optimal order ${\mathcal
O}(h^{\min\{s,\ell\}})$ with respect to the mesh size~$h$,
the polynomial degree~$\ell$, and the regularity
exponent~$s$ of the analytical solution. Under additional
regularity assumptions, the $L^2$-error is shown to
converge with the optimal order ${\mathcal
O}(h^{\ell+1})$. The theoretical results are confirmed in
a series of numerical experiments.
Available as:
gzipped PostScript (.ps.gz)
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2004/02 P. Houston, D. Schotzau and T. Wihler.
-
Mixed hp-Discontinuous Galerkin Finite Element Methods for the Stokes Problem in Polygons
No abstract.
Available as:
gzipped PostScript (.ps.gz)
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2004/03 R. Hartmann and P. Houston.
-
Adaptive Discontinuous Galerkin Finite Element
Methods with Interior Penalty for the Compressible
Navier-Stokes Equations
No abstract.
Available as:
gzipped PostScript (.ps.gz)
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2004/04 P.Houston, I. Perugia, A. Schneebeli and D. Schotzau.
-
Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations
No abstract.
Available as:
gzipped PostScript (.ps.gz)
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2004/05 P. Houston, Ch. Schwab and E. Suli.
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On the Design of hp-Adaptive Finite Element Methods for Elliptic Partial Differential Equations.
We introduce an $hp$--adaptive finite element algorithm
based on a combination of reliable and efficient residual
error indicators and a new $hp$--extension control
technique which assesses the local regularity of the
underlying analytical solution on the basis of its local
Legendre series expansion. Numerical experiments confirm
the robustness and reliability of the proposed algorithm.
Available as:
gzipped PostScript (.ps.gz)
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2004/06 Yifeng
Chen and Zhiming
Liu.
-
Integrating Temporal Logics
In this paper, we study the predicative semantics of
different temporal logics and the relationships between
them. We use a notation called generic composition to
simplify the manipulation of predicates. The modalities of
possibility and necessity become generic composition and
its inverse of converse respectively. The relationships
between different temporal logics are also characterised
as such modalities. Formal reasoning is carried out at the
level of predicative semantics and supported by the
higher-level laws of generic composition and its inverse.
Various temporal domains are unified under a notion called
resource cumulation. Temporal logics based on these
temporal domains can be readily defined, and their axioms
identified. The formalism provides a framework in which
human experience about system development can be
formalised as refinement laws. The approach is
demonstrated in the transformation from Duration Calculus
to Temporal Logic of Actions. A number of common design
patterns are studied. The refinement laws identified are
then applied to the case study of water pump controlling.
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2004/07 Edward L. Green and Nicole Snashall.
-
The Hochschild Cohomology Ring Modulo Nilpotence of a Stacked Monomial
Algebra
This paper studies the Hochschild cohomology of finite-dimensional
monomial algebras. If A = KQ/I with I an admissible monomial ideal, then
we give sufficient conditions for the existence of an embedding of K[x_1,
..., x_r]/( x_ax_b for a \neq b) into the Hochschild cohomology ring
HH^*(A). We also introduce stacked algebras, a new class of monomial
algebras which includes Koszul and D-Koszul monomial algebras. If A is a
stacked algebra, we prove that HH^*(A)/N is isomorphic to K[x_1, ...,
x_r]/( x_ax_b for a \neq b), where N is the ideal in HH^*(A) generated by
the homogeneous nilpotent elements. In particular, this shows that the
Hochschild cohomology ring of A modulo nilpotence is finitely generated as
an algebra.
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2004/08 Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten and Gordana Todorov.
-
Tilting Theory and Cluster Combinatorics
We introduce a new category C, which we call the cluster
category, obtained as a quotient of the bounded derived
category D of the module category of a finite-dimensional
hereditary algebra H over a field. We show that, in the
simply-laced Dynkin case, C can be regarded as a natural
model for the combinatorics of the corresponding
Fomin-Zelevinsky cluster algebra. In this model, the
tilting modules correspond to the clusters of
Fomin-Zelevinsky. Using approximation theory, we
investigate the tilting theory of C, showing that it is
more regular than that of the module category itself, and
demonstrating an interesting link with the classification
of self-injective algebras of finite representation type.
This investigation also enables us to conjecture a
generalisation of APR-tilting.
Available as:
gzipped PostScript (.ps.gz)
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2004/09 Damian Brown, Leevan Ling, Edward Kansa and Jeremy Levesley.
-
On Approximate Cardinal Preconditioning Methods for
Solving PDEs with Radial Basis Functions
The approximate cardinal basis function (ACBF)
preconditioning technique has been used to solve partial
differential equations (PDEs) with radial basis functions
(RBFs). In \cite{LK02}, a preconditioning scheme that is
based upon constructing the least-squares approximate
cardinal basis function from linear combinations of the
RBF-PDE matrix elements has shown very attractive
numerical results. This preconditioning technique is
sufficiently general that it can be easily applied to many
differential operators.
In this paper, we review the ACBF preconditioning techniques
previously used for interpolation problems and investigate a class
of preconditioners based on the one proposed in \cite{LK02} when a
cardinality condition is enforced on different subsets. We
numerically compare the ACBF preconditioners on several numerical
examples of Poisson's, modified Helmholtz and Helmholtz equations,
as well as a diffusion equation and discuss their performance.
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2004/10 Aslak Bakke Buan, Robert
Marsh and Idun Reiten.
-
Cluster-Tilted Algebras
We introduce a new category C, which we call the cluster
category, obtained as a quotient of the bounded derived
category D of the module category of a finite-dimensional
hereditary algebra H over a field. We show that, in the
simply-laced Dynkin case, C can be regarded as a natural
model for the combinatorics of the corresponding
Fomin-Zelevinsky cluster algebra. In this model, the
tilting modules correspond to the clusters of
Fomin-Zelevinsky. Using approximation theory, we
investigate the tilting theory of C, showing that it is
more regular than that of the module category itself, and
demonstrating an interesting link with the classification
of self-injective algebras of finite representation type.
This investigation also enables us to conjecture a
generalisation of APR-tilting.
Available as:
gzipped PostScript (.ps.gz)
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2004/11 Rob
Brownlee, Will Light, D. Schotzau and T.P. Wihler.
-
Approximation Orders for Interpolation by Surface Splines
to Rough Functions
In this paper we consider the approximation of functions
by radial basic function interpolants. There is a plethora
of results about the asymptotic behaviour of the error
between appropriately smooth functions and their
interpolants, as the interpolation points fill out a
bounded domain in R^d. In all of these cases, the analysis
takes place in a natural function space dictated by the
choice of radial basis function - the native space. In
many cases, the native space contains functions possessing
a certain amount of smoothness. We address the question of
what can be said about these error estimates when the
function being interpolated fails to have the required
smoothness. These are the rough functions of the title. We
limit our discussion to surface splines, as an exemplar of
a wider class of radial basic functions, because we feel
our techniques are most easily seen and understood in this
setting.
Available as:
gzipped PostScript (.ps.gz)
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2004/12 Gabriel
Davis.
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Finiteness Conditions on the Ext-algebra of a Cycle
Algebra
Let $A$ be a finite-dimensional algebra given by quiver
and monomial relations. In [E.L. Green, D. Zacharia,
Manuscripta Math.\ 85 (1994) 11-23] we see that the
Ext-algebra of $A$ is finitely generated if and only if
all the Ext-algebras of certain cycle algebras overlying
$A$ are finitely generated. Here a cycle algebra
$\Lambda$ is a finite-dimensional algebra given by quiver
and monomial relations where the quiver is an oriented
cycle. The main result of this paper gives necessary and
sufficient conditions for the Ext-algebra of such a
$\Lambda$ to be finitely generated; this is achieved by
defining a computable invariant of $\Lambda$, the
smo-tube. We also give necessary and sufficient
conditions for the Ext-algebra of $\Lambda$ to be
Noetherian.
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2004/13 Rob Brownlee and Jeremy Levesley.
-
A scale of improved error estimates for radial approximation in Euclidean space and on spheres
We adapt Schaback's error doubling trick~\cite{schaback} to give
error estimates for radial interpolation of functions with
smoothness lying (in some sense) between that of the usual native
space and the subspace with double the smoothness. We do this for
both bounded subsets of $\R^d$ and spheres. As a step on the way
to our ultimate goal we also show convergence of derivatives of
the interpolation error
-
2004/14 J. Levesley and X. Sun.
-
Approximation in Native Spaces
Within the conventional framework of a native
space structure, a smooth kernel generates a small native space,
and ``radial basis functions" stemming from the smooth kernel are
intended to approximate only functions from this small native
space. Therefore their approximation power is quite limited.
Recently, Narcowich, Schaback and Ward [NSW], and Narcowich and
Ward [NW], respectively, have studied two approaches that have led
to the empowerment of smooth radial basis functions in a larger
native space. In the approach of [NW], the radial basis function
interpolates the target function at some scattered (prescribed)
points. In both approaches, approximation power of the smooth
radial basis functions is achieved by utilizing spherical
polynomials of a (possibly) large degree to form an intermediate approximation
between the radial
basis approximation and the target function. In this paper, we take a
new approach.
We embed the smooth radial basis functions in a larger native
space generated by a less smooth kernel, and use them to
approximate functions from the larger native space. Among other
results, we characterize the best approximant with respect to the
metric of the larger native space to be the radial basis function
that interpolates the target function on a set of finite scattered
points after the action of a certain multiplier operator. We also
Michael Hoffmann
Richard M.Thomas
In the study of automatic groups, the geometrical
characterization of automaticity (in terms of the "fellow
traveller property") plays a fundamental role. When we move
to the study of automatic semigroups, we no longer have this
simple formulation. The purpose of this paper is to give a
general geometric characterization of automaticity in
semigroups.
establish the error bounds between the best approximant and the
target function.
-
2004/15 Rob Brownlee.
-
Error estimates for interpolation of rough data
using the scattered shifts of a radial basis function
D. Schotzau
T.P. Wihler
The error between appropriately smooth functions and their radial
basis function interpolants, as the interpolation points fill out
a bounded domain in $\R^d$, is a well studied artifact. In all of
these cases, the analysis takes place in a natural function space
dictated by the choice of radial basis function---the native
space. The native space contains functions possessing a certain
amount of smoothness. We address the question of what happens if
the function being approximated is conspicuously rough.
-
2004/16 Jurgen Muller.
-
Invariant Theory of Finite Groups
This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action. We will introduce the basic notions of invariant theory, discuss the structural properties of invariant rings, comment on computational aspects, and finally present two applications from function theory and number theory.
Keywords are: polynomial rings, graded commutative algebras, Hilbert series, Noether normalization, Cohen-Macaulay property, primary and secondary invariants symmetric groups, reflection groups.
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2004/17 P. Houston, D. Schotzau and T.P. Wihler.
-
hp-Adaptive Discontinuous Galerkin Finite Element Methods for the Stokes
ProblemError estimates for interpolation of rough data
using the scattered shifts of a radial basis function
In this paper, we derive an $hp$--version a posteriori error estimator for
mixed discontinuous Galerkin finite element methods for the Stokes problem.
The estimator is obtained by extending the $h$--version a posteriori
analysis of [10] to the context of the $hp$--version of the finite element
method. We then present a series of numerical experiments where we test the
performance of the proposed error estimator within an automatic
$hp$--adaptive refinement procedure.
Available as:
gzipped PostScript (.ps.gz)
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2004/18 P. Houston, I. Perugia and D. Schotzau.
-
A Review of Discontinuous Galerkin Methods for Maxwell's Equation D. Schotzau
T.P. Wihlers in
Frequency-Domain
In this paper, we review recent work on discontinuous
Galerkin (DG) methods for the discretization of the
time-harmonic Maxwell equations in frequency-domain,
based on the interior penalty discretization of the curl-curl
operator. Direct and mixed methods will be presented for
low-frequency and high-frequency cases.
The performance of the mixed DG method for the indefinite Maxwell
problem in the high-frequency case is tested in a
new set of numerical experiments carried out on a model problem
with a singular solution.
Available as:
gzipped PostScript (.ps.gz)
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2004/19 P. Houston, I. Perugia, A. Schneebeli and D. Schotzau.
-
Mixed Discontinuous Galerkin Approximation of the Maxwell Operator: The
Indefinite Case.
We present and analyze an interior penalty D. Schotzau
T.P. Wihler
method for the numerical discretization of the indefinite
time-harmonic Maxwell equations in mixed form.
The method is based on the mixed
discretization of the curl-curl operator developed
in [11] and can
be understood as a non-stabilized variant
of the approach proposed in [19].
We show the well-posedness of this approach and
derive optimal a-priori error estimates in the energy-norm
as well as the $L^2$-norm. The theoretical results are
confirmed in a series of numerical experiments.
Available as:
gzipped PostScript (.ps.gz)
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2004/20 P. Houston, I. Perugia and D. Schotzau.
-
Recent Developments in Discontinuous Galerkin Methods for the Time-Harmonic
Maxwell's Equations.
In this article, we review recent work on discontinuous
Galerkin (DG, for short) methods for the discretization of the
time-harmonic Maxwell's equations,
based on the interior penalty discretization of the curl-curl
operator. Direct and mixed methods will be presented for both the
low- and high-frequency cases.
The performance of the proposed DG methods will be highlighted
in a series of numerical examples with known analytical solutions.
Available as:
gzipped PostScript (.ps.gz)
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2004/21 G.N. Milstein and M.V. Tretyakov.
-
Numerical algorithms for forward-backward stochastic differential equations connected with semilinear parabolic equations
Efficient numerical algorithms are proposed for a class of forward-backward stochastic differential equations (FBSDEs) connected with semilinear parabolic partial differential equations. As in [J. Douglas, Jr., J. Ma, P. Protter, Ann. Appl. Prob., 6 (1996), 940--968.], the algorithms are based on the known four-step scheme for solving FBSDEs. The corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach. The derivatives of the solution u of the semilinear equation are found by finite differences. The forward equation is simulated by mean-square methods of order 1/2 and 1. Corresponding convergence theorems are proved. Along with the algorithms for FBSDEs on a fixed finite time interval, we also construct algorithms for FBSDEs with random terminal time. The results obtained are supported by numerical experiments.
AMS 2000 subject classification. Primary 60H35; secondary 65C30, 60H10, 62P05.
Keywords. Forward-backward stochastic differential equations, numerical integration, mean-square convergence, semilinear partial differential equations of parabolic type.
-
2004/22 Duncan W. Parkes, V. Yu. Shavrukov and Richard M. Thomas.
-
Monoid Presentations of Groups by Finite
Special String-Rewriting Systems
We show that the class of groups which
have monoid presentations by means of finite special
$\lambda$-confluent string-rewriting systems strictly
contains the class of plain groups (the groups which
are free products of a finitely generated free group
and finitely many finite groups), and that any group
which has an infinite cyclic central subgroup can be
presented by such a string-rewriting system if and only
if it is the direct product of an infinite cyclic group
and a finite cyclic group.
-
2004/23 Derek F. Holt, Sarah Rees, Claas E. Rover and Richard M. Thomas.
-
Groups with context-free co-word problem
We study the class of co-context-free groups. We define a
co-context-free group to be one whose co-word problem (the complement of its word
problem) is context-free. This class is larger than the subclass of context-free
groups, being closed under the taking of finite direct products, restricted
standard wreath products with context-free top groups, and passing to finitely generated subgroups
and finite index overgroups. But we do not know of other examples of
co-context-free groups. We prove that the only examples amongst polycyclic groups or the Baumslag-Solitar groups
are virtually abelian. We do this by proving that languages with certain
purely arithmetical properties cannot be context-free; this result may
be of independent interest.
-
2004/24 P. Houston, J. Robson and E. Suli.
-
Discontinuous Galerkin finite element approximation of quasilinear elliptic
boundary value problems I: The scalar case.
We develop a one--parameter family of $hp$--version discontinuous
Galerkin finite element methods, parameterised by $\theta \in
[-1,1]$, for the numerical solution of quasilinear elliptic
equations in divergence--form on a bounded open set $\Omega \subset
\mathbb{R}^d$, $d \geq 2$. In particular, we consider the analysis
of the family for the equation
$-\nabla\cdot\left\{\mu(x,|\nabla u|)\nabla u\right\} = f(x)$ subject
to mixed Dirichlet--Neumann
boundary conditions on $\partial \Omega$. It is assumed that $\mu$
is a real--valued function, $\mu \in C(\bar{\Omega}\times[0,\infty))$,
and there exist positive
constants $m_\mu$ and $M_\mu$ such that $m_{\mu} (t-s) \leq
\mu(x,t)t - \mu(x,s)s \leq M_{\mu} (t-s)$ for $t \geq s \geq 0$
and all $x \in \bar{\Omega}$. Using Brouwer's Fixed Point Theorem,
for any value of $\theta \in [-1,1]$ the corresponding method is
shown to have a unique solution $u_{\rm DG}$ in the finite element
space. If $u \in C^1(\Omega) \cap H^k(\Omega)$, $k \geq 2$,
then, with discontinuous piecewise polynomials of degree $p\geq
1$, the error between $u$ and $u_{\rm DG}$, measured in the broken
$H^1(\Omega)$--norm, is $\mathcal{O}(h^{s-1}/p^{k-3/2})$, where
$1 \leq s \leq \min\{p+1,k\}$.
Available as:
gzipped PostScript (.ps.gz)
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2004/25 Edward L. Green, Nicole Snashall and Oeyvind Solberg.
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The Hochschild cohomology ring modulo nilpotence of a monomial algebra
For a finite dimensional monomial algebra A over a field K we show that
the Hochschild cohomology ring of A modulo the ideal generated by
homogeneous nilpotent elements is a commutative finitely generated
K-algebra of Krull dimension at most one. This finite generation was
conjectured to be true for any finite dimensional algebra over a field by
N. Snashall and O. Solberg in "Support varieties and Hochschild cohomology
rings", Proc. London Math. Soc. 88 (2004) 705-732.
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2004/26 R. A. Brownlee.
-
Error Estimates for Interpolation of Rough and Smooth Functions using Radial Basis Functions
In this thesis we are concerned with the approximation of
functions by radial basis function interpolants. There is a
plethora of results about the asymptotic behaviour of the error
between appropriately smooth functions and their interpolants, as
the interpolation points fill out a bounded domain in Euclidean
space. In all of these cases, the analysis takes place in a
natural function space dictated by the choice of radial basis
function{\Mdash}the native space.
This work establishes $L_p$-error estimates, for $1\leq p \leq
\infty$, when the function being interpolated fails to have the
required smoothness to lie in the corresponding native space;
therefore, providing error estimates for a class of rougher
functions than previously known. Such estimates have application
in the numerical analysis of solving partial differential
equations using radial basis function collocation methods. At
first our discussion focusses on the popular polyharmonic splines.
A more general class of radial basis functions is admitted into
exposition later on, this class being characterised by the
algebraic decay of the Fourier transform of the radial basis
function. The new estimates presented here offer some improvement
on recent contributions from other authors by having wider
applicability and a more satisfactory form. The method of proof
employed is not restricted to interpolation alone. Rather, the
technique provides error estimates for the approximation of rough
functions for a variety of related approximation schemes as well.
For the previously mentioned class of radial basis functions, this
work also gives error estimates when the function being
interpolated has some additional smoothness. We find that the
usual $L_p$-error estimate, for $1\leq p \leq \infty$, where the
approximand belongs to the corresponding native space, can be
doubled. Furthermore, error estimates are established for
functions with smoothness intermediate to that of the native space
and the subspace of the native space where double the error is
observed.
-
2004/27 P. Kosiuczenko.
-
Proof Transformation via Interpretation Functions
Abstract: In this paper we prove several useful facts concerning term mappings, proof transformation and preservation of different logical systems by term mappings. This research is motivated by a practical approach to an automatic transformation of class invariants and pre- and post-conditions. We show a method of extending mappings from a set of terms to, what we call interpretation functions, which are defined on larger sets of terms obtained by substitution. We provide a sufficient condition for compositional functions to preserve proofs in equational logic, as well as proofs in other logical systems like propositional logic with modus ponens or proofs using resolution rule. Consequently, constraint transformation via interpretation functions preserves entailment relations in several logical systems.
-
2004/28 I. Ulidowski (editor) .
-
Preliminary Proceedings
6th AMAST Workshop on Real-Time Systems
ARTS 2004
Stirling, U.K., July 12, 2004
No abstract.
-
2004/29 Y. Chen.
-
A Language of Flexible Objects
In this paper, we introduce a new language called \flexibo.
\flexibo\ is an executable object-oriented specification language
designed for open-source software development with different levels
of {\em trust} in a {\em decentralized} programming environment.
\flexibo\ provides a number of new programming mechanisms for both
systematic static analysis and dynamic control of {\em correctness},
{\em ownership} and {\em resources}. All language ingredients are
values. Many programming constructs have their corresponding {\em
mirror classes}, each of which can be extended to programmer-defined
sub-mirrorclasses. Various language operators such as method
invocation are overridable. Unlike other OO languages that follow
specific binding rules, \flexibo\ allows programmers to choose the
binding mechanism of a variable by explicitly denoting its binding
direction. Variables are untyped in \flexibo. Types are introduced
as constraints and checked in runtime. However, if \flexibo\ is used
for language translation, the checkings are actually done in the
compilation phase of the whole process. Programmers are allowed to
define their own types. Each type system then corresponds to a
particular \flexibo\ program. That means \flexibo\ programs are able
to compile themselves: a source \flexibo\ program can be evaluated
for type checking and translated to another programming language, if
the type checking succeeds. Unlike other languages providing one
particular set of language mechanisms, \flexibo\ is designed to
represent various language mechanisms systematically and flexibly.
-
2004/30 J. Hunton and Bjorn Schuster.
-
Subalgebras of Group Cohomology Defined by Infinite Loop Spaces
We study natural subalgebras $Ch_E(BG;R)$ of group cohomology
$H^*(BG;R)$ defined in terms of the infinite loop spaces in spectra $E$
and give representation theoretic descriptions of those based on $QS^0$
and the Johnson-Wilson theories
$E(n)$. We describe the subalgebras arising from the Brown-Peterson
spectra $BP$ and as a result give a simple reproof of Yagita's theorem
that the image of $BP^*(BG)$ in $H^*(BG;F_p)$ is $F$-isomorphic to the
whole cohomology ring; the same result is shown to hold with $BP$
replaced by any complex oriented theory $E$ with a non-trivial map of
ring
spectra $E\rightarrow HF_p$. We also
extend our constructions to define subalgebras of $H^*(X;R)$ for any
space $X$; when $X$ is a finite CW complex we show that the subalgebras
$Ch_{E(n)}(X;R)$ give a natural unstable chromatic filtration of
$H^*(X;R)$.
-
2004/31 G.N. Milstein, M.V. Tretyakov and M.Hoffmann.
-
Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients
We propose a new conception which allows us to apply any
numerical method of weak approximation to stochastic
differential equations (SDEs) with nonglobally Lipschitz
coefficients. Following this conception, we do not take into
account the approximate trajectories which leave a
sufficiently large sphere. We prove that accuracy of any
method of weak order $p$ is estimated by $\varepsilon
+O(h^{p})$, where $\varepsilon $ can be made arbitrarily
small with increasing the radius of the sphere. The results
obtained are supported by numerical experiments.
AMS 2000 subject classification. Primary 60H35; secondary
65C30, 60H10.
Keywords. SDEs with nonglobally Lipschitz coefficients,
numerical integration of SDEs in the weak sense.
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2004/32 P. Houston, D. Schotzau and T.P. Wihler.
-
Energy Norm A Posteriori Error Estimation of hp-Adaptive Discontinous Galerkin Methods for Elliptic Problems
In this paper, we develop the a posteriori error estimation of
hp-version interior penalty discontinuous Galerkin
discretizations of elliptic boundary-value
problems. Computable upper and lower bounds on the error
measured in terms of a natural (mesh-dependent) energy
norm are derived. The bounds are explicit in the local
mesh sizes and approximation orders. A series of
numerical experiments illustrate the performance of the
proposed estimators within an automatic hp-adaptive
refinement procedure.
Available as:
gzipped PostScript (.ps.gz)
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2004/33 Tim Hardcastle.
-
Normal and characteristic strucute in quasigroups and loops - PhD Thesis
In this thesis I shall be exploring the normal and characteristic
structure of quasigroups and loops. In recent years there
has been a revival of interest in the theory of loops and
in particular in the relationship between the properties
of a loop and the properties of its multiplication group;
and several powerful new theorems have emerged which allow
the structural properties of a loop to be related to its
multiplication group. I shall combine these ideas with
tools developed at the beginning of loop theory to produce
some interesting new theorems, principally relating the
order of a finite multiplication group to the structure of
its loop.
-
2004/34 P. Houston, D. Schotzau and T.P. Wihler.
-
An hp-Adaptive Mixed Discontinuous Galerkin FEM
for Nearly Incompressible Linear Elasticity
We develop the a posteriori error estimation of mixed hp-version
discontinuous Galerkin finite element methods for nearly
incompressible elasticity problems. Computable upper and
lower bounds on the error measured in terms of a natural
(mesh-dependent) energy norm are derived. The bounds are
explicit in the local mesh sizes and approximation orders,
and are independent of the locking parameter. A series of
numerical experiments are presented which demonstrate the
performance of the proposed error estimator within an
automatic hp-adaptive refinement procedure.
-
2004/35 G.N. Milstein and M.V. Tretyakov.
-
Solving forward-backward stochastic differential equations and related quasilinear parabolic equations
Efficient numerical algorithms for a class of
forward-backward stochastic differential equations (FBSDEs)
and related quasilinear parabolic partial differential
equations (quasilinear PDEs) are proposed. The corresponding
quasilinear parabolic equation is solved by new layer
methods which are constructed by means of a probabilistic
approach. Efficiency of the methods is achieved due to
replacing derivatives of the solution to the quasilinear
equation by finite differences. The proposed algorithms for
solving FBSDEs are based on the four-step scheme of Ma,
Protter, Yong. Convergence theorems are proved. Results of
some numerical experiments are presented.
AMS 2000 subject classification. Primary 60H35; secondary 65C30,
60H10, 35K55.
Keywords. Forward-backward stochastic differential equations,
numerical integration, mean-square convergence, quasilinear partial
differential equations of parabolic type.
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2004/36 N.Snashall, K Erdmann, E L Green and R Taillefer.
-
Representation Theory of the Drinfel'd Doubles of a Family of Hopf Algebras
We investigate the Drinfeld doubles of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfeld doubles, from which we deduce properties of their representations, including their Auslander-Reiten quivers. We then determine decompositions of their tensor products of most of the representations described, and in particular give a complete description of the tensor product of two simple modules. This study also leads to explicit examples of Hopf bimodules over the original Hopf algebras.
-
2004/37 Michael Hoffmann and Richard M.Thomas.
-
A geometric characterization of automatic semigroups
In the study of automatic groups, the geometrical
characterization of automaticity (in terms of the "fellow
traveller property") plays a fundamental role. When we move
to the study of automatic semigroups, we no longer have this
simple formulation. The purpose of this paper is to give a
general geometric characterization of automaticity in
semigroups.
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2004/38 Graham P.Oliver and Richard M.Thomas.
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Finitely generated groups with automatic presentations
A structure is said to be computable if its domain can be represented by a set which is accepted by a Turing machine and if its relations can then be checked using Turing machines. Restricting the Turing machines in this defintion to finite automata gives us a class of structures with a particularly simple computational structure; these structures are said to have automatic presentations. Given their nice algorithmic properties, these have been of interest in a wide variety of areas.
An area of particular interest has been the classification of automatic structures. One of the prime examples has been the class of groups. We give a complete characterization in the case of finitely generated groups and show that such a group has an automatic presentation if and only if it is virtually abelian.
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2004/39 Stephen R. Lakin and Richard M.Thomas.
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Context-sensitive decision problems in groups
There already exists classifications of those groups which have regular, context-free or recursively enumerable word problem. The only remaining step in the Chomsky heirarchy is to consider those groups with a context-sensitive word problem. In this paper we consider this problem and prove some results about these groups. We also establish some results about other context-sensitive decision problems in groups.
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2004/40 Aslak Bakke Buan, Robert
Marsh and Idun Reiten.
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Cluster mutation via quiver representations
Matrix mutation appears in the definition of cluster
algebras of Fomin and Zelevinsky. We give a representation
theoretic interpretation of matrix mutation, using tilting
theory in cluster categories of hereditary algebras. Using
this, we obtain a representation theoretical interpretation
of cluster mutation in case of acyclic cluster algebras of
finite type.
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