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This thesis consists of two parts, both being concerned with operads related to the ribbon braid groups. In the first part, we define a notion of semidirect product for operads and use it to study the framed $n$-discs operad (the semidirect product $f\mathcal{D}_n=\mathcal{D}_n\rtimes SO(n)$ of the little $n$-discs operad with the special orthogonal group). This enables us to deduce properties of $f\mathcal{D}_n$ from the corresponding properties for $\mathcal{D}_n$. We prove an equivariant recognition principle saying that algebras over the framed $n$-discs operad are $n$-fold loop spaces on $SO(n)$-spaces. We also study the operations induced on homology, showing that an $H(f\mathcal{D}_n)$-algebra is a higher dimensional Batalin-Vilkovisky algebra with some additional operators when $n$ is even. Contrastingly, for $n$ odd, we show that the Gerstenhaber structure coming from the little $n$-discs does not give rise to a Batalin-Vilkovisky structure. We give a general construction of operads from families of groups. We then show that the operad obtained from the ribbon braid groups is equivalent to the framed 2-discs operad. It follows that the classifying spaces of ribbon braided monoidal categories are double loop spaces on $S^1$-spaces. The second part of this thesis is concerned with infinite loop space structures on the stable mapping class group. Two such structures were discovered by Tillmann. We show that they are equivalent, constructing a map between the spectra of deloops. We first construct an `almost map', i.e a map between simplicial spaces for which one of the simplicial identities is satisfied only up to homotopy. We show that there are higher homotopies and deduce the existence of a rectification. We then show that the rectification gives an equivalence of spectra.

We prove two theorems concerning the action of the Steenrod algebra in cohomology and homology. (i) Let A denote a finitely generated graded F_p polynomial algebra over the Steenrod algebra whose generators have dimensions not divisible by p. The possible sets of dimensions of the generators for such A are known. It was conjectured that if we replaced the polynomial algebra A by a polynomial algebra truncated at some height greater than p over the Steenrod algebras, the sets of all possible dimensions would coincide with the former list. We show that the conjecture is false. For example F₁₁[x₆,x₁₀]¹² truncated at height 12 supports an action of the Steenrod algebra but F₁₁[x₆,x₁₀] does not. (ii) Let V be an elementary abelian 2-group of rank 3. The problem of determining a minimal set of generators for H*(BV,F₂) over the Steenrod algebra was an unresolved problem for many years. (A solution was announced by Kameko in June 1990, but is not yet published.) A dual problem is to determine the subring M of the Pontrjagin ring H*(BV,F₂). We determine this ring completely and in particular give a verification that the minimum number of generators needed in each dimension in cohomology is as announced by Kameko, but by using completely different techniques. Let v ε V - (0) and denote by a_5(v) ε H*(BV,F₂) the image of the non-zero class in H_2s-1(RP^∞,F₂) imeq F₂ under the homomorphism induced by the inclusion of F_{2 → V onto (0,v). We show that M is isomorphic to the ring generated by (a}_s(v),s ≥ 1, v ε V - (0)) except in dimensions of the form 2^r+3 + 2^r+1 + 2^r - 3, r ≥ 0, where we need to adjoin our additional generator.

In this thesis I present a mathematical tool for understanding the spin networks that arise from the study of the loop states of quantum gravity. The spin networks that arise in quantum gravity possess more information than the original spin networks of Penrose: they are embedded within a manifold and thus possess topological information. There are limited tools available for the study of this information. To remedy this I introduce a slightly modi ed mathematical object - Braided Ribbon Networks - and demonstrate that they can be related to spin networks in a consistent manner which preserves the di eomorphism invariant character of the loop states of quantum gravity. Given a consistent de nition of Braided Ribbon Networks I then relate them back to previous trinion based versions of Braided Ribbon Networks. Next, I introduce a consistent evolution for these networks based upon the duality of these networks to simplicial complexes. From here I demonstrate that there exists an invariant of this evolution and smooth deformations of the networks, which captures some of the topological information of the networks. The principle result of this program is presented next: that the invariants of the Braided Ribbon Networks can be transferred over to the original spin network states of loop quantum gravity. From here we represent other advances in the study of braided ribbon networks, accompanied by comments of their context given the consistent framework developed earlier including: the meaning of isolatable substructures, the particular structure of the capped three braids in trivalent braided ribbon networks and their application towards emergent particle physics, and the implications of the existence of microlocal topological structures in spin networks. Lastly we describe the current state of research in braided ribbon networks, the implications of this study on quantum gravity as a whole and future directions of research in the area.

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Topology optimisation techniques help designers to nd the best layout of structural members. When followed by shape and sizing optimisation, these techniques result in far greater savings than shape and sizing optimisation alone. During the last three decades extensive research has been carried out in the topology optimisation area. Consequently topology optimisation techniques have been considerably improved and successfully applied to a range of physical problems. These techniques are now regarded as invaluable tools in mechanical, aerostructural and structural design. In spite of great potential in geomechanical problems, however, the application of topology optimisation techniques in this eld has not been studied thoroughly. This thesis explores the state-of-the-art topology and shape optimisation methods in excavation design. The main problems of concern in this thesis are to nd the optimum shape of an underground opening and to optimise the reinforcement distribution around it. To tackle these problems, new formulations for some topology optimisation techniques are proposed in this thesis to match the requirements in excavation problems. Although linear elastic material models have limited applications in excavation design, these models are used in the rst part of this thesis to introduce the proposed optimisation technique and to verify it. Simultaneous shape and reinforcement optimisation is considered as well. Using the proposed optimisation techniques, it is shown that the computational effort needed for this mixed optimisation problem is almost the same as the effort required to solve each of shape or reinforcement optimisation problems alone. In the next part of this thesis, reinforcement optimisation of tunnels in massive rocks is addressed where the behaviour of the rock mass is in uenced by few major discontinuities. Although discontinuities exist in the majority of rock masses, due to its complexities, optimising the excavations in these types of rocks has not been considered by any other researcher before. A method for reinforcement optimisation of tunnels in such rock masses is proposed in this thesis and its capability is demonstrated by means of numerical examples. Lastly, shape optimisation of excavations in elasto-plastic soil is addressed. In this problem the excavation sequence is also taken into account. A stressbased parameter is dened to evaluate the efficiency of the soil elements assuming Mohr-Coulomb material model. Some examples are solved to illustrate and verify the application of the proposed technique. Being one of the rst theses on the topic, this work concentrates on the theoretical background and the possibility of applying topology optimisation techniques in excavation designs. It has been demonstrated that a properly tailored topology optimisation technique can be applied to nd both the optimum shape and the optimum reinforcement design of openings. Optimising the excavations in various types of grounds including elastic homogeneous rock masses, massive rocks, and elasto-plastic soil and rock media have been considered. Different objective functions, namely, mean compliance, oor heave, and tunnel convergence have been selected and successfully minimised using the proposed techniques. The results obtained in this thesis illustrate that the proposed topology optimisation techniques are very useful for improving excavation designs.

vi, 99 leaves : ill. ; 26 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1974

Thesis (Ph.D.)--University of Texas at Dallas, 2008. / Includes vita. Includes bibliographical references (leaves 157-161)

Thesis--Columbia University. / Issued also in microfilm form. Includes bibliographical references.

Thesis (M.A. Sc.)--University of Toronto, 2006. / Source: Masters Abstracts International, Volume: 45-03, page: 1548.