On the asphericity of a family of positive relative group presentations

Excluding four exceptional cases, the asphericity of the relative presentation P= ⟨G; x|xmgxh⟩ for m ≥ 2 is determined. If H = ⟨g; h⟩ ≤ G, then the exceptional cases occur when H is isomorphic to C5 or C6

Asphericity of positive free product length 4 relative group presentations

© 2018 Walter de Gruyter GmbH, Berlin/Boston. Excluding some exceptional cases, we determine the asphericity of the relative presentation P = ,where a, b ∈ G \ {1} and 1 ≤ m ≤ n. If H = ≤ G, the exceptional cases occurwhen a = b2 or when H is isomorphic to C6.

Asphericity of length six relative group presentations

Combinatorial group theory is a part of group theory that deals with groups given by presentations in terms of generators and defining relations. Many techniques both algebraic and geometric are used in dealing with problems in this area. In this thesis, we adopt the geometric approach. More specifically, we use so-called pictures over relative presentations to determine the asphericity of such presentations. We remark that if a relative presentation is aspherical then group theoretic information can be deduced. In Chapter 1, the concept of relative presentations is introduced and we state the main theorems and some known results. In Chapter 2, the concept of pictures is introduced and methods used for checking asphericity are explained. Excluding four unresolved cases, the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|x^{m}gxh\rangle$ for $m\geq2$ is determined in Chapter 3. If $H=\langle g, h\rangle$ $\leq G$, then the unresolved cases occur when $H$ is isomorphic to $C_{5}$ or $C_{6}$. The main work is done in Chapter 4, in which we investigate the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|xaxbxcxdxexf\rangle$, where the coefficients $a, b, c, d, e, f\in G$ and $x \notin G$ and prove the theorems stated in Chapter 1

Asphericity of length six relative group presentations

Combinatorial group theory is a part of group theory that deals with groups given by presentations in terms of generators and defining relations. Many techniques both algebraic and geometric are used in dealing with problems in this area. In this thesis, we adopt the geometric approach. More specifically, we use so-called pictures over relative presentations to determine the asphericity of such presentations. We remark that if a relative presentation is aspherical then group theoretic information can be deduced. In Chapter 1, the concept of relative presentations is introduced and we state the main theorems and some known results. In Chapter 2, the concept of pictures is introduced and methods used for checking asphericity are explained. Excluding four unresolved cases, the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|x^{m}gxh\rangle$ for $m\geq2$ is determined in Chapter 3. If $H=\langle g, h\rangle$ $\leq G$, then the unresolved cases occur when $H$ is isomorphic to $C_{5}$ or $C_{6}$. The main work is done in Chapter 4, in which we investigate the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|xaxbxcxdxexf\rangle$, where the coefficients $a, b, c, d, e, f\in G$ and $x \notin G$ and prove the theorems stated in Chapter 1

Investigating relationships between cost and CO₂ emissions in reinforced concrete structures using a BIM-based design optimisation approach

An integrated design approach for the cost and embodied carbon optimisation of reinforced concrete structures is presented in this paper to inform early design decisions. A BIM-based optimisation approach that utilises Finite Element Modelling (FEM) and a multi-objective genetic algorithm with constructability constraints is established for that purpose. A multilevel engineering analysis model is developed to perform structural layout optimisation, slab and columns sizing optimisation, and slab and columns reinforcement optimisation. The overall approach is validated using real buildings and the relationships between cost and carbon optimum solutions are explored. The study exhibits how cost effective and carbon efficient solutions could be obtained without compromising the feasibility of the optimised designs. Results demonstrate that the structural layout and the slab thickness are amongst the most important design optimisation parameters. Finally, the overall analysis suggests that the building form can influence the relationships between cost and carbon for the different structural components

Intensity analysis to unify measurements of size and stationarity of land changes by interval, category, and transition

This article presents a quantitative method to analyze maps of land categories from several points in time for a single site by considering cross-tabulation matrices, where one matrix summarizes the change in each time interval. There are three levels of analysis, starting from general to more detailed levels, where each level exposes different types of information given the previous level of analysis. First, the interval level examines how the size and speed of change vary across time intervals. Second, the category level examines how the size and intensity of gross losses and gross gains in each category vary across categories for each time interval. Third, the transition level examines how the size and intensity of a category\u27s transitions vary across the other categories that are available for that transition. At each level, the method tests for stationarity of patterns across time intervals. The unique contribution of this article is that it combines all three levels of analysis into one unified framework that we call intensity analysis, where the more detailed levels are conditional on the less detailed levels. The illustrative case study is for seven categories at the Plum Island Ecosystems site in northeastern Massachusetts, USA, where the largest transition is from Forest to Built during 1985, 1991, and 1999. We compare our approach to other established methods such as the Markov approach in order to show how our proposed intensity analysis gives more information concerning five possible reasons to explain why the transitions vary across time and space. © 2012 Elsevier B.V

Map errors that could account for deviations from a uniform intensity of land change

Intensity Analysis is a mathematical framework that compares a uniform intensity to observed intensities of temporal changes among categories. Our article summarizes Intensity Analysis and presents a new method to compute the minimum hypothetical error in the data that could account for each observed deviation from a uniform intensity. A larger hypothetical error gives stronger evidence against a hypothesis that a change is uniform. The method produces results for five groups of measurements, which are organized into three levels of analysis: interval, category, and transition. The method applies generally to analysis of changes among categories during time intervals, because the input is a standard contingency table for each time interval. We illustrate the method with a case study concerning change during three time intervals among four land categories in northeastern Massachusetts, USA. Modelers can perform the analysis using our computer program, which is free. © 2013 Taylor & Francis