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In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2^nd order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.

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In this thesis, I present an experimental study of quantum control techniques for transferring population between vibrational states of atoms trapped in an optical lattice. Results from a range of techniques are compared, including techniques tested previously in the same system. In the study of the Adiabatic Rapid Passage (ARP) technique, control of population transfer is realized through varying the chirp rate and modulation amplitude of a frequency-chirped sinusoidal displacement of the lattice. Meanwhile, dependence of population transfer on the chirp direction is observed, which is explained by a model of ARP in a 3-level system. In the study of the coherent control technique, interference between a one-phonon transition at 2\omega and a two-phonon transition at omega is experimentally demonstrated. The omega and 2\omega transitions are realized by sinusoidally displacing the optical lattice at omega and sinusoidally modulating the lattice depth at 2\omega, respectively. The branching ratio of transitions to the first excited state and to higher excited states is controlled by varying the relative phase between these two pathways. The highest measured branching ratio of 17\pm2 is achieved among all the experiments using this coherent control scheme. In the study of the GRadient Ascent Pulse Engineering (GRAPE) technique, a "pulse" involving both displacement and depth-modulation of the lattice is used to transfer population. This pulse is theoretically engineered with the GRAPE algorithm to optimize the fidelity between the first excited state and the final state, when the lattice Hamiltonian without gravity for a specific lattice depth is considered. The experimental result shows that there is almost no excitation into higher excited states during population transfer from the ground to the first excited state, even when this process is affected by gravity and inhomogeneous broadening in reality. By comparing all the techniques, the GRAPE technique is found to be the best in terms of increasing population transfer into the first excited state while reducing excitation into higher excited states. On the other hand, the ARP technique creates the highest normalized population inversion, a ratio of the difference to the sum of the ground and the first excited state populations.

In this thesis, I present an experimental study of quantum control techniques for transferring population between vibrational states of atoms trapped in an optical lattice. Results from a range of techniques are compared, including techniques tested previously in the same system. In the study of the Adiabatic Rapid Passage (ARP) technique, control of population transfer is realized through varying the chirp rate and modulation amplitude of a frequency-chirped sinusoidal displacement of the lattice. Meanwhile, dependence of population transfer on the chirp direction is observed, which is explained by a model of ARP in a 3-level system. In the study of the coherent control technique, interference between a one-phonon transition at 2\omega and a two-phonon transition at omega is experimentally demonstrated. The omega and 2\omega transitions are realized by sinusoidally displacing the optical lattice at omega and sinusoidally modulating the lattice depth at 2\omega, respectively. The branching ratio of transitions to the first excited state and to higher excited states is controlled by varying the relative phase between these two pathways. The highest measured branching ratio of 17\pm2 is achieved among all the experiments using this coherent control scheme. In the study of the GRadient Ascent Pulse Engineering (GRAPE) technique, a "pulse" involving both displacement and depth-modulation of the lattice is used to transfer population. This pulse is theoretically engineered with the GRAPE algorithm to optimize the fidelity between the first excited state and the final state, when the lattice Hamiltonian without gravity for a specific lattice depth is considered. The experimental result shows that there is almost no excitation into higher excited states during population transfer from the ground to the first excited state, even when this process is affected by gravity and inhomogeneous broadening in reality. By comparing all the techniques, the GRAPE technique is found to be the best in terms of increasing population transfer into the first excited state while reducing excitation into higher excited states. On the other hand, the ARP technique creates the highest normalized population inversion, a ratio of the difference to the sum of the ground and the first excited state populations.

The interaction of a high power laser beam with the lattice of transparent optical materials induces internal strains in the latter. Attention is focused on optical materials of zinc blende structure and excitation at the 10.6-(mu)m wavelength of the laser. A lattice dynamical treatment is presented to obtain the laser induced internal strains in terms of the lattice Green's functions, the transverse effective charge and the Raman coefficient. The transverse effective charge and the Raman coefficient are then thoroughly discussed and discrepancies in previous works concerning these parameters are resolved. / The imperfect lattice Green's functions are utilized to calculate the expected magnitude of the actual strains induced in real crystals taking into account the effect of lattice point defects. The mechanical laser damage thresholds at the 10.6-(mu)m CO(,2) laser wavelength are then obtained specifically for SiC, GaAs, InSb, ZnS, ZnSe, and CdTe materials, in good agreement with the available measured values.

There are many problems in mathematical finance which require the evaluation of a multivariate integral. Since these problems typically involve the discretisation of a continuous random variable, the dimension of the integrand can be in the thousands, tens of thousands or even more. For such problems the Monte Carlo method has been a powerful and popular technique. This is largely related to the fact that the performance of the method is independent of the number of dimensions. Traditional quasi-Monte Carlo techniques are typically not independent of the dimension and as such have not been suitable for high-dimensional problems. However, recent work has developed new types of quasi-Monte Carlo point sets which can be used in practically limitless dimension. Among these types of point sets are Sobol' sequences, Faure sequences, Niederreiter-Xing sequences, digital nets and lattice rules. In this thesis, we will concentrate on results concerning lattice rules. The typical setting for analysis of these new quasi-Monte Carlo point sets is the worst-case error in a weighted function space. There has been much work on constructing point sets with small worst-case errors in the weighted Korobov and Sobolev spaces. However, many of the integrands which arise in the area of mathematical finance do not lie in either of these spaces. One common problem is that the integrands are unbounded on the boundaries of the unit cube. In this thesis we construct function spaces which admit such integrands and present algorithms to construct lattice rules where the worst-case error in this new function space is small. Lattice rules differ from other quasi-Monte Carlo techniques in that the points can not be used sequentially. That is, the entire lattice is needed to keep the worst-case error small. It has been shown that there exist generating vectors for lattice rules which are good for many different numbers of points. This is a desirable property for a practitioner, as it allows them to keep increasing the number of points until some error criterion is met. In this thesis, we will develop fast algorithms to construct such generating vectors. Finally, we apply a similar technique to show how a particular type of generating vector known as the Korobov form can be made extensible in dimension.

Bibliography: p. 169-172. / viii, 172 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Uses the tool of lattice regularisation to calculate the 2-point Euclidean Green's function in pure SU (3) Yang-Mills (quenched quantum chromodynamics), in Landau gauge. Adding improved gauge fixing and tree level correction to the improvement program developed by Symanzik [et al.], shows that the gluon propagator can be reliably calculated on lattices coarser than has hitherto been used. / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 2000?

Browsing marked-up documents by traversing hyperlinks has become probably the most important means by which documents are accessed, both via the World Wide Web (WWW) and organisational Intranets. However, there is a pressing demand for document management and retrieval systems to deal appropriately with the massive number of documents available. There are two classes of solution: general search engines, whether for the WWW or an Intranet, which make little use of specific domain knowledge or hand-crafted specialised systems which are costly to build and maintain. The aim of this thesis was to develop a document management and retrieval system suitable for small communities as well as individuals in specialised domains on the Web. The aim was to allow users to easily create and maintain their own organisation of documents while ensuring continual improvement in the retrieval performance of the system as it evolves. The system developed is based on the free annotation of documents by users and is browsed using the concept lattice of Formal Concept Analysis (FCA). A number of annotation support tools were developed to aid the annotation process so that a suitable system evolved. Experiments were conducted in using the system to assist in finding staff and student home pages at the School of Computer Science and Engineering, University of New South Wales. Results indicated that the annotation tools provided a good level of assistance so that documents were easily organised and a lattice-based browsing structure that evolves in an ad hoc fashion provided good efficiency in retrieval performance. An interesting result suggested that although an established external taxonomy can be useful in proposing annotation terms, users appear to be very selective in their use of terms proposed. Results also supported the hypothesis that the concept lattice of FCA helped take users beyond a narrow search to find other useful documents. In general, lattice-based browsing was considered as a more helpful method than Boolean queries or hierarchical browsing for searching a specialised domain. We conclude that the concept lattice of Formal Concept Analysis, supported by annotation techniques is a useful way of supporting the flexible open management of documents required by individuals, small communities and in specialised domains. It seems likely that this approach can be readily integrated with other developments such as further improvements in search engines and the use of semantically marked-up documents, and provide a unique advantage in supporting autonomous management of documents by individuals and groups - in a way that is closely aligned with the autonomy of the WWW.

Bibliography: p. 169-172. / viii, 172 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Uses the tool of lattice regularisation to calculate the 2-point Euclidean Green's function in pure SU (3) Yang-Mills (quenched quantum chromodynamics), in Landau gauge. Adding improved gauge fixing and tree level correction to the improvement program developed by Symanzik [et al.], shows that the gluon propagator can be reliably calculated on lattices coarser than has hitherto been used. / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 2000?