Abstract:
Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a H\"older continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two $C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic Cantor sets are $C^1$-conjugate, then the conjugacy (with a different extension) is in fact already $C^{k+\gamma}, C^\infty$ or $C^\omega$. Within one $C^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Conjugacy classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the $C^1$ norm.

Abstract:
This paper reviews the role of expert judgement to support reliability assessments within the systems engineering design process. Generic design processes are described to give the context and a discussion is given about the nature of the reliability assessments required in the different systems engineering phases. It is argued that, as far as meeting reliability requirements is concerned, the whole design process is more akin to a statistical control process than to a straightforward statistical problem of assessing an unknown distribution. This leads to features of the expert judgement problem in the design context which are substantially different from those seen, for example, in risk assessment. In particular, the role of experts in problem structuring and in developing failure mitigation options is much more prominent, and there is a need to take into account the reliability potential for future mitigation measures downstream in the system life cycle. An overview is given of the stakeholders typically involved in large scale systems engineering design projects, and this is used to argue the need for methods that expose potential judgemental biases in order to generate analyses that can be said to provide rational consensus about uncertainties. Finally, a number of key points are developed with the aim of moving toward a framework that provides a holistic method for tracking reliability assessment through the design process.

Abstract:
It is well-known that perturbative calculations in field theory can lead to far simpler answers than the Feynman diagram approach might suggest. In some cases scattering amplitudes can be constructed for processes with any desired number of external legs yielding compact expressions which are inaccessible from the point of view of conventional perturbation theory. In this thesis we discuss some attempts to address the nature of this underlying simplicity and then use the results to calculate some previously unknown amplitudes of interest. Witten's twistor string theory is introduced and the CSW rules at tree-level and one-loop are described. We use these techniques to calculate the one-loop gluonic MHV amplitudes in N=1 super-Yang-Mills as a verification of their validity and then proceed to evaluate the general MHV amplitudes in pure Yang-Mills with a scalar running in the loop. This latter amplitude is a new result in QCD. In addition to this, we review some recent on-shell recursion relations for tree-level amplitudes in gauge theory and apply them to gravity. As a result we present a new compact form for the n-graviton MHV amplitudes in general relativity. The techniques and results discussed are relevant to the understanding of the structure of field theory and gravity and the non-supersymmetric Yang-Mills amplitudes in-particular are pertinent to background processes at the LHC. The gravitational recursion relations provide new techniques for perturbative gravity and have some bearing on the ultraviolet properties of Einstein gravity.

Abstract:
These notes are based on lectures given by Michael Green during Part III of the Mathematics Tripos (the Certificate for Advanced Study in Mathematics) in the Spring of 2003. The course provided an introduction to string theory, focussing on the Bosonic string, but treating the superstring as well. A background in quantum field theory and general relativity is assumed. Some background in particle physics, group theory and conformal field theory is useful, though not essential. A number of appendices on more advanced topics are also provided, including an introduction to orientifolds in various brane configurations which helps to populate a relatively sparse part of the literature.

Abstract:
Let f be a rational mapping of a space X . The complexity of (f,X) as a dynamical system is measured by the dynamical degrees $\delta_p(f)$, $1\le p\le {\rm dim}(X)$. We give the definition of the dynamical degrees show how they are computed in certain cases. For instance, we show that if the dynamical degree of an automorphism of a K\"ahler manifold is greater than one, then it must be irrational.

Abstract:
We discuss automorphisms and pseudo-automorphisms on blowups of complex projective space with an eye to finding ones with interesting dynamical behavior.

Abstract:
This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.

Abstract:
The role of prophetic witness of the churches in the public discourse of modern civil societies is analysed on the basis of three public memorandums of the German Protestant churches on economic questions and their impact on the public. Among the ten systematic conclusions which are drawn from this case study is the importance of the specific context for the role of prophetic statements. The article tries to show how prophetic witness is a necessary element of a public theology, which is not based on fundamental criticism, but develops both critical and constructive perspectives for politics and society. If such public theology is liberation theology for a democratic society it is the task of the church to get involved in the public debate in a ‘bilingual’ way, that is, on the basis of its biblical-theological sources but at the same time with the ability to engage in the secular language of pluralistic societies. How to cite this article: Bedford-Strohm, H., 2010, ’Prophetic witness and public discourse in European societies – a German perspective’, HTS Teologiese Studies/Theological Studies 66(1), Art. #784, 6 pages. DOI: 10.4102/hts.v66i1.784

Abstract:
The article dealt with the moral and political problem of international food justice in which the deep contradiction between the present situation of malnourishment and starvation in large parts of the global population on the one hand and the biblical notion of the preferential option for the poor on the other hand was described. This ecumenically widely accepted notion was clarified in several aspects. How deeply this is rooted in the history of Christian social thought was shown by Martin Luther’s writings on the economy which have remained relatively unknown in the churches and in the scholarly world. The article then presented three models of Christian economic ethic: the technical economic model, the utopian economic model and the public theological economic model. On the basis of the public theological model seven challenges for international food justice were presented. The basis for these challenges is an understanding of globalisation which guarantees just participation for everyone and deals with nature in an ecologically sustainable way. The interests of small farmers are the basis for judging the activities of big agro-corporations. Public theology is the background for an active involvement of the churches as agents of a global civil society to promote international food justice.