A complete solution to the infinite Oberwolfach problem

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we characterize the infinite subgraphs $L$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(L)$

Further results on nearly Kirkman triple systems with subsystems

AbstractIn this paper we further discuss the embedding problem for nearly Kirkman triple systems and get the result that: (1) For u≡v≡0(mod6), v⩾78, and u⩾3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36,42,48,54,60,66 or 72, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0(mod6) and u⩾3v

On the minisymposium problem

The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${\sum_{i=1}^t a_i = v}$, find a seating arrangement over the $\frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {\em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $\left\lfloor\frac{m-1}{2}\right\rfloor$ of the days. When the cycles are as long as possible, i.e.\ $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v \equiv m \equiv 2 \pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $m\mid v$, except possibly when $k$ is odd and $v$ is even.Comment: 25 page

Combinatorial designs and their automorphism groups

This thesis concerns the automorphism groups of Steiner triple systems and of cycle systems. Although most Steiner triple systems have trivial automorphism groups [2], it is widely known that for every abstract group, there exists a Steiner triple system whose automorphism is isomorphic to that group [16]. The well-known Bose construction [4] for Steiner triple systems, which has a number of variants, has a particularly nice structure, which makes it possible to say much about the automorphism group, and in the case of the construction based on an Abelian group, to derive the full automorphism group. The thesis contains a full analysis of these matters. Some of these results have been published by the author in [14]. The thesis also proves new results concerning the automorphism group for Steiner triple systems constructed using the tripling construction. An m-cycle system is a decomposition of a complete graph into cycles of length m. A Steiner triple system is thus a 3-cycle system. The thesis proves the result that for all m > 3, and for each abstract finite group, there exists an m-cycle system whose automorphism group is isomorphic to that group. In addition, the thesis contains a collection of new results concerning the conjecture by Furedi that every Steiner triple system is decomposable into triangles. Although this conjecture is expected to remain open for some time, it is possible to prove it for a number of standard constructions. It is further shown that for sufficiently large v, the number of Steiner triple systems of order v that are decomposable into triangles is at least vv2(1/54-0(1))

Sustainable Development: The Nature of Change and the Influence of Cultural Traits

It is only in the last few years that researchers have started to investigate the impact of cultural characteristics on the approach nations take to addressing the demands of sustainable development. The basis of such work tends to resort to the use of frameworks that were developed by Hofstede and the GLOBE project. Rogge, Dessein, and and Verhoeve (2013) argue that research into subjects as complex as this should commence with work of a more exploratory nature. This thesis is multidisciplinary, and uses a variety of methods to provide that first exploration into the nature of this relationship. My review of literature reveals that there is general acceptance that paradigmatic change is necessary but that decision making, behaviour and politics all tend to “safer” incremental steps. Using systems theory to examine the nature of paradigm change I identify the potential scope of government influence. The remainder of my work concentrates on the development of case studies of Japan, Denmark, Sweden and the United Kingdom, focussing on the manner in which the governments concerned support small to medium sized enterprises (SMEs) within the country in developing more sustainable practices. I use these case studies to identify the impact of culture. I find that the cultural dimension as defined in classic frameworks is not the pertinent issue, but the nation is likely to be sensitive to environmental demands if the environmental damage impinges on areas of their life that they value. If there are characteristics within the nation that provide for care and support then action is more likely to be taken. The other characteristics necessary to see through financing such action are perserverence and a long-term view. My other finding is that the use of statistical analysis and frameworks of cultural characteristics is problematic in that they both simplify a subject that should be understood in all its complexity

California Extremely Large Telescope: Conceptual Design for a Thirty-Meter Telescope

Following great success in the creation of the Keck Observatory, scientists at the California Institute of Technology and the University of California have begun to explore the scientific and technical prospects for a much larger telescope. The Keck telescopes will remain the largest telescopes in the world for a number of years, with many decades of forefront research ahead after that. Though these telescopes have produced dramatic discoveries, it is already clear that even larger telescopes must be built if we are to address some of the most profound questions about our universe. The time required to build a larger telescope is approximately ten years, and the California community is presently well-positioned to begin its design and construction. The same scientists who conceived, led the design, and guided the construction of the Keck Observatory have been intensely engaged in a study of the prospects for an extremely large telescope. Building on our experience with the Keck Observatory, we have concluded that the large telescope is feasible and is within the bounds set by present-day technology. Our reference telescope has a diameter of 30 meters, the largest size we believe can be built with acceptable risk. The project is currently designated the California Extremely Large Telescope (CELT)

Paying 'Due Regard'? The Impact of the Public Sector Equality Duty on Service Provision for Single Mothers

The purpose of this thesis was to examine the impact of the Public Sector Equality Duty (‘the Duty’) on the lives of single mother, local authority service users, focusing on their experiences ‘on the ground’. The discourse surrounding lone mothers has long been highly politicised, entrenched in a paradigm of dependency. This in-depth, qualitative study, undertaken between February 2013 and May 2015, used an alternative perspective of gender equality through considering the positive rights of this group. Using Bristol and Bristol City Council (BCC) as a case study, a socio-legal approach was utilised through desktop, analytical work to explore the theoretical underpinning of the Duty as ‘reflexive’ law, assess local policy and decision-making processes, and ascertain the services available to single mothers. Semi-structured interviews were conducted with 14 single mothers and 11 stakeholders to support a contextualised understanding of the way these services were used. Based on the analysis and findings of this research, BCC had established significant structures to deliver the Duty, effectively engaging local actors to improve organisational mechanisms and support equality. The Duty was identified as having a positive impact on single mother service users, potentially supporting their participation in public life. However, organisational mechanisms stimulated by the Duty were identified as vulnerable to ongoing austerity measures. This study contributes to knowledge in three respects. Firstly, it addresses a gap in evidence identified in the 2013 Governmental review of the Equality Duty, by showing how the Duty underpins transparent decision-making processes and, through localised, reflexive mechanisms informs service delivery that better meets the needs of service users. Furthermore, the systematic and detailed sociological study of the Duty’s mechanisms explores its operation ‘on the ground’ from a novel perspective. Finally, through positioning single mothers as knowledgeable social actors, it offers an alternative paradigm to existing work that portrays them as passive recipients