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In this thesis, we look at how the Cross-Entropy (CE) method can be used to solve various optimisation and estimation problems in telecommunication systems and network planning, especially in the presence of noise. In Chapter 2, we mention what comprises optimisation problems. Various optimisation problems, such as constrained optimization and convex optimisation, are discussed. We also address noisy optimisation and how dealing effectively with noise plays a crucial role in locating a global optimal solution. A brief overview of three algorithms that have successfully been applied to noisy optimisation problems is also given. Chapter 3 explores a short overview of the methodology behind the CE method. We discuss how the CE method requires two simple iterative stages to locate an optimal "degenerate" sampling distribution, and hence an optimal solution to the optimisation problem. We also show how a simple modification of the algorithm can tackle noisy optimisation problems. Numerical experiments for solving both non-noisy and noisy multi-extremal continuous optimisation problems are conducted. Three test functions are used to investigate the performance of the CE algorithm on both non-noisy and noisy cases. The results suggest that the proposed algorithm can locate a global optimal solution accurately. Also, we show that the performance of the CE algorithm can be improved using the injection method. Chapter 4 and Chapter 5 discuss two types of the Network Planning Problem (NPP): single-type NPP and multi-type NPP. The aim is to determine which links in the system should be purchased in order to provide the highest possible service to the consumers, subject to a constraint on the total budget. We introduce various CE-based algorithms to tackle such non-linear combinatorial optimisation problem. Numerical experiments suggest that the proposed algorithms perform effectively and reliably in all test cases. Chapter 6 is concerned with estimating the blocking probabilities in circuit switched networks. We look at how Importance Sampling and Sequential Importance Sampling can be used to estimate the blocking probabilities. Here, the CE method is used to find optimal sampling parameters to be used in Importance Sampling and Sequential Importance Sampling. Numerical experiments suggest that Sequential Importance Sampling achieves a variance reduction over Importance Sampling in almost all cases at a cost of increased simulation time. Using CE further increases the efficiency of both.

In this thesis we provide a better understanding of the role of the inter-cluster filamentary environment in the evolution of galaxy properties, relating that to the physical mechanisms involved. We examine a poorer supercluster structure and describe how it compares with the richer regime, deciding whether the same processes are responsible for driving galaxy evolution in both cases. We develop a new approach to automated galaxy morphology classification, undertake an extensive observational survey of a nearby supercluster and investigate the evolution of galaxy properties within this structure, considering the filamentary environments that we confirm exist. Firstly, we present an application of Mathematical Morphology (MM) for the classification of astronomical objects, both for star/galaxy differentiation and galaxy morphology classification. We demonstrate that, for CCD images, 99.3 ± 3.8% of galaxies can be separated from stars using MM, with 19.4±7.9% of the stars being misclassified. We demonstrate that, for photographic plate images, the number of galaxies correctly separated from the stars can be increased using our MM diffraction spike tool, which allows 51.0±6.0% of the high-brightness galaxies that are inseparable in current techniques to be correctly classified, with only 1.4 ± 0.5% of the high-brightness stars contaminating the population. We demonstrate that elliptical (E) and late-type spiral (Sc-Sd) galaxies can be classified using MM at an accuracy of 91.4 ± 7.8%. It is a method involving less ‘free parameters’ than current techniques, especially automated machine learning algorithms. The limitations of MM due to seeing and distance are also discussed. We examine various star/galaxy differentiation and galaxy morhpology classification techniques commonly used today, and show that the above MM techniques compare very favourably. Secondly, we present a wide-field photometric and spectroscopic study of the supercluster SCL266 (z = 0.068). We have obtained CCD imaging in B-, R- and I-bands covering 5 clusters within the structure, and spectroscopy within the region of 3 of the clusters and the filamentary environment between them. We have over 950 redshifts obtained and 161 galaxies confirmed as members of the supercluster structure. The galaxy morphologies are derived using the MM techniques we have developed. We show that the population of passive, early-type galaxies extends into the dense regions of the filaments and is not just confined to the higher-density clusters, indicating that the denser filamentary environment hastens the galaxy evolution within it. The star-forming, late-type galaxies are distributed throughout the entirety of the supercluster structure. We show that both the mean star formation rate and the fraction of star-forming galaxies decrease strongly as a function of distance along the filaments towards the cluster centres. Given these observations we conclude that the evolution occurring within this poor supercluster begins within the filaments up to 4 Mpc away from the cluster centres, and is driven by long time-scale mechanisms involving encounters between neighbouring galaxies.

The primary aim of this thesis is to develop a minimal model Hamiltonian to describe the electronic properties of ceria (CeO$_{2}$) and its reduced phases. In order to do this, several energy scales of the problem were explored to determine their relative significance to the problem. These included the crystal electric field ($\Delta_{CEF}$), the spin-orbit coupling ($\lambda_{so}$), electron hopping ($t$), on-site Coulomb repulsion for the Ce $4f$ states ($U$), the reorganization energy ($\lambda_{0}$), the direct exchange ($J$) and the energy gap between the oxygen valence band level and the cerium $4f$ states ($\Gamma\epsilon$). Once the relative magnitudes of the various energy scales were determined, it was then possible to define a minimal set of degrees of freedom required to obtain a meaningful description of the system in the minimal model. The first task was to obtain baseline data for both CeO$_{2}$ and Ce$_{2}$O$_{3}$ as well as the metallic phases of Ce. Density Functional Theory (DFT) calculations were performed on these materials to obtain band structures as well as structural properties. The DFT results indicated that both the LDA and GGA functionals perform poorly for $\gamma$-Ce and Ce$_{2}$O$_{3}$. In the case of Ce$_{2}$O$_{3}$, both LDA and GGA give a metallic ground state contrary to the insulator observed in experiment. The first set of energy scales that were investigated included the crystal electric field and the spin-orbit coupling. Since these calculations, except for the presence of a crystal field, involve an isolated Ce ion, the energy scales associated with spin and charge fluctuations at a Ce site are ignored. In order to perform these calculations, it was necessary to determine the $f^{n}$ configuration at each Ce site in the various phases of the oxides. This was achieved by a simple empirical model called the bond valence model. The bond valence model provides a method for calculating site valences in crystals from bond length data alone. Apart from getting information about the $f^{n}$ configurations at the various Ce sites, the results of the bond valence analysis revealed that there was significant mixed valence in the different phases of the cerium oxies. In addition, it was possible to characterize the charge distribution in the local environment of the oxygen vacancies from the bond valence results. This analysis led to the important result that the charge prefers Ce sites farthest away from the oxygen vacancy. This is contrary to the widely accepted view that the extra charge resulting from oxygen vacancy formation localizes on Ce sites closest to the O vacancy. The LDA calculations which support the widely accepted view do not properly treat electron correlations. Thus this is an important result which emphasizes the role of strong electron correlations in the processes of oxidation and reduction in these materials. The results of the crystal field calculations were not quantitative because we could not find data on crystal field parameters which is required to evaluate the matrix elements. However, it is expected that the splitting of the $f$ manifold by the crystal fields will be smaller than that of the spin-orbit coupling energy which is of the order of $0.07\unit{eV}$. Results for the Hubbard $U$ parameter in Ce oxides were obtained from the literature but these vary widely and fall in the range $1.0 - 10.5\unit{eV}$. The reorganization energy reported from spectroscopic results of the intervalence transition band for reduced ceria is $1.4\unit{eV}$. Tight-binding calculations were performed for CeO$_{2}$ to obtain the energy band structure for this material. It was then possible to extract from these results the matrix elements for electron hopping. The sizes of the matrix elements obtained suggested that the direct $f$-$f$ hopping between Ce sites ($t_{ff}$) is of order $0.02\unit{eV}$ which is negligible compared to the indirect hopping via an O atom (i.e., $t_{eff}$) which is of order $0.2\unit{eV}$. The hopping matrix element between a Ce site and a neighbouring O atom ($t_{fp}$) is even higher $\sim 0.5\unit{eV}$. The role of multiple orbitals on a Ce site was explored to determine the minimal set of orbitals to include in the effective model Hamiltonian. The results indicated that the essential physics could be described well by replacing the $4f$ manifold with two orbitals on a Ce site. The magnitudes of the energy scales of the problem decrease in the order $U > \lambda_{0} > t_{fp}>t_{eff} >\lambda_{so} \gg t_{ff} \approx \Delta_{CEF} \approx J$ with a high uncertainty on the last two relationships. The results of the above analysis led us to the conclusion that the only relevant energy scales for the effective model Hamiltonian are the On-site Coulomb repulsion for the Ce $4f$ states ($U$) and electron hopping between a Ce site and a neighbouring O site ($t_{fp}$). A minimal model Hamiltonian was then constructed to take into account the spin and charge fluctuations on a Ce site due to electron hopping between the Ce site and neighbouring O atoms. In this description, the strongly correlated ground states of cerium oxides is then described by a variational wavefunction whose main feature is that double occupancy of a Ce $f$ orbital is prohibited.

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