http://hdl.handle.net/11343/159 2019-10-15T04:10:52Z http://hdl.handle.net/11343/228926 Singular vectors for the WN algebras and the BRST cohomology for relaxed highest-weight Lk(sl(2)) modules Siu, Steve Wai Chun This thesis presents the computation of singular vectors of the W_n algebras and the BRST cohomology of modules of the simple vertex operator algebra L_k(sl2) associated to the affine Lie algebra of sl2 in the relaxed category We will first recall some general theory on vertex operator algebras. We will then introduce the module categories that are relevant for conformal field theory. They are the category O of highest-weight modules and the relaxed category which contains O as well as the relaxed highest-weight modules with spectral flow and non-split extensions. We will then introduce the W_n algebras and the simple vertex operator algebra L_k(sl2). Properties of the Heisenberg algebra, the bosonic and the fermionic ghosts will be discussed as they are required in the free field realisations of W_n and L_k(sl2) as well as the construction of the BRST complex. We will then compute explicitly the singular vectors of W_n algebras in their Fock representations. In particular, singular vectors can be realised as the image of screening operators of the W_n algebras. One can then realise screening operators in terms of Jack functions when acting on a highest-weight state, thereby obtaining explicit formulae of the singular vectors in terms of symmetric functions. We will then discuss the BRST construction and the BRST cohomology for modules in category O. Lastly we compute the BRST cohomology for L_k(sl2) modules in the relaxed category. In particular, we compute the BRST cohomology for the highest-weight modules with positive spectral flow for all degrees and the BRST cohomology for the highest-weight modules with negative spectral flow for one degree. © 2019 Steve Wai Chun Siu 2019-01-01T00:00:00Z http://hdl.handle.net/11343/228925 Missing data analysis, Combinatorial Model Selection and Structure Learning Kwok, Chun Fung This thesis examines three problems in statistics: the missing data problem in the context of extracting trends from time series data, the combinatorial model selection problem in regression analysis, and the structure learning problem in graphical modelling / system identification. The goal of the first problem is to study how uncertainty in the missing data affects trend extraction. This work derives an analytical bound to characterise the error of the estimated trend in terms of the error of the imputation. It works for any imputation method and various trend-extraction methods, including a large subclass of linear filters and the Seasonal-Trend decomposition based on Loess (STL). The second problem is to tackle the combinatorial complexity which arises from the best-subset selection in regression analysis. Given p variables, a model can be formed by taking a subset of the variables, and the total number of models p is $2^p$. This work shows that if a hierarchical structure can be established on the model space, then the proposed algorithm, Gibbs Stochastic Search (GSS), can recover the true model with probability one in the limit and high probability with finite samples. The core idea is that when a hierarchical structure exists, every evaluation of a wrong model would give information about the correct model. By aggregating these information, one may recover the correct model without exhausting the model space. As an extension, parallelisation of the algorithm is also considered. The third problem is about inferring from data the systemic relationship between a set of variables. This work proposes a flexible class of multivariate distributions in a form of a directed acyclic graphical model, which uses a graph and models each node conditioning on the rest using a Generalised Linear Model (GLM), and it shows that while the number of possible graphs is $\Omega(2^{p \choose 2})$, a hierarchical structure exists and the GSS algorithm applies. Hence, a systemic relationship may be recovered from the data. Other applications like imputing missing data and simulating data with complex covariance structure are also investigated. © 2019 Chun Fung Kwok 2019-01-01T00:00:00Z http://hdl.handle.net/11343/228924 Mixed Spatial and Movement Representations in the Primate Posterior Parietal Cortex. Hadjidimitrakis, K; Bakola, S; Wong, YT; Hagan, MA The posterior parietal cortex (PPC) of humans and non-human primates plays a key role in the sensory and motor transformations required to guide motor actions to objects of interest in the environment. Despite decades of research, the anatomical and functional organization of this region is still a matter of contention. It is generally accepted that specialized parietal subregions and their functional counterparts in the frontal cortex participate in distinct segregated networks related to eye, arm and hand movements. However, experimental evidence obtained primarily from single neuron recording studies in non-human primates has demonstrated a rich mixing of signals processed by parietal neurons, calling into question ideas for a strict functional specialization. Here, we present a brief account of this line of research together with the basic trends in the anatomical connectivity patterns of the parietal subregions. We review, the evidence related to the functional communication between subregions of the PPC and describe progress towards using parietal neuron activity in neuroprosthetic applications. Recent literature suggests a role for the PPC not as a constellation of specialized functional subdomains, but as a dynamic network of sensorimotor loci that combine multiple signals and work in concert to guide motor behavior. 2019-01-01T00:00:00Z http://hdl.handle.net/11343/228923 Retinal Neurovascular Coupling in Streptozotocin Diabetic Rats Wang, Joe Yuchi Diabetic retinopathy (DR) is a significant cause of vision impairment worldwide, and it is projected to incur a rising global disease burden. Although the pathophysiology of diabetic retinopathy had been historically characterised as a vascular disease, there is a growing body of evidence to suggest that DR is also a process of retinal neurodegeneration. There is also evidence that functional changes occur to the retinal vasculature’s capacity to respond to physiological stimuli, before anatomical changes manifest. Additionally, there is evidence to suggest that neuronal dysfunction precedes anatomical evidence of neurodegeneration in both clinical and experimental studies of diabetes. These findings were examined in Chapter 2, and collectively suggest that functional deficits of both the vascular and neuronal retinal components may be key components of DR pathogenesis. Several research questions were proposed in this thesis in order to further understand vascular and neuronal interactions in the retina, at the earliest stages of diabetes. In order to model an early stage of diabetic disease, 4 weeks of hyperglycaemia was introduced to a cohort of dark Agouti laboratory rats using streptozotocin (STZ). Flickering light was used to stimulate neuronal driven vasodilation in the retina, and the autoregulatory capacity of retinal vasculature was challenged through inhalation of oxygen and carbon dioxide. Electroretinography was conducted to assess retinal function, and the scotopic threshold response (STR) was recorded during gas inhalation as a measure of inner retinal functional response to an autoregulatory challenge. A series of pilot studies were undertaken to optimise the parameters of flicker stimulation, gas delivery, retinal imaging and electroretinography. These materials and methods were described in Chapter 3. Flicker stimulation reliably produced vasodilation of inner retinal arteries and veins, although this response was not significantly affected by 4 weeks of STZ hyperglycaemia. Oxygen and carbon dioxide breathing introduced vasodilation and vasoconstriction of the inner retinal arteries and veins, respectively. The speed of venous vasoconstriction was reduced in STZ animals during oxygen breathing, and carbon dioxide breathing revealed a reduced arterial vasodilatory capacity in STZ animals. These findings were described and discussed in Chapter 4, and they indicate that, despite normal retinal neurovascular coupling, subtle autoregulatory deficits in the retinal vasculature are present at an early stage of diabetes. The oscillatory potentials and STR were found to be reduced after 4 weeks of hyperglycaemia, suggesting a manifest deficit of inner retinal function. Additionally, carbon dioxide breathing introduced an increase in the peak positive STR (pSTR) amplitude in both normal and STZ animals, whereas oxygen breathing resulted in a decrease of pSTR amplitude that was more significant in the STZ cohort. These findings were described and discussed in Chapter 5, and they seem to suggest that that relative inner retinal ischemia may be present early in the course of diabetes. Furthermore, this effect could be exacerbated by vasoconstrictive stimuli. The overall experimental findings suggest that neurovascular coupling is unaffected at an early stage of diabetes, despite findings of inner retinal dysfunction and subtle deficit of vascular autoregulation in the retina. This suggests, in addition to neuronal and vascular activity, that other physiological processes, likely mediated by glial cells, may be implicated in the modulation neurovascular interactions in the retina. These experimental findings and implications were discussed in Chapter 6, as well as study limitations and future directions of research. © 2019 Joe Yuchi Wang 2019-01-01T00:00:00Z