Torus knot polynomials and susy Wilson loops

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the $(m,n)\leftrightarrow (n,m)$ symmetry and the leading polynomial at large $N$ are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in $\mathcal{N}$=4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of $q$-harmonic oscillators.Comment: 17 pages, v2: More concise (published) version; typos correcte

Optimising Parameters in Recurrence Quantification Analysis of Smart Energy Systems

Recurrence Quantification Analysis (RQA) can help to detect significant events and phase transitions of a dynamical system, but choosing a suitable set of parameters is crucial for the success. From recurrence plots different RQA variables can be obtained and analysed. Currently, most of the methods for RQA radius optimisation are focusing on a single RQA variable. In this work we are proposing two new methods for radius optimisation that look for an optimum in the higher dimensional space of the RQA variables, therefore synchronously optimising across several variables. We illustrate our approach using two case studies: a well known Lorenz dynamical system, and a time-series obtained from monitoring energy consumption of a small enterprise. Our case studies show that both methods result in plausible values and can be used to analyse energy data

Spectral dimension in graph models of causal quantum gravity

The phenomenon of scale dependent spectral dimension has attracted special interest in the quantum gravity community over the last eight years. It was first observed in computer simulations of the causal dynamical triangulation (CDT) approach to quantum gravity and refers to the reduction of the spectral dimension from 4 at classical scales to 2 at short distances. Thereafter several authors confirmed a similar result from different approaches to quantum gravity. Despite the contribution from different approaches, no analytical model was proposed to explain the numerical results as the continuum limit of CDT. In this thesis we introduce graph ensembles as toy models of CDT and show that both the continuum limit and a scale dependent spectral dimension can be defined rigorously. First we focus on a simple graph ensemble, the random comb. It does not have any dynamics from the gravity point of view, but serves as an instructive toy model to introduce the characteristic scale of the graph, study the continuum limit and define the scale dependent spectral dimension. Having defined the continuum limit, we study the reduction of the spectral dimension on more realistic toy models, the multigraph ensembles, which serve as a radial approximation of CDT. We focus on the (recurrent) multigraph approximation of the two-dimensional CDT whose ensemble measure is analytically controlled. The latter comes from the critical Galton-Watson process conditioned on non-extinction. Next we turn our attention to transient multigraph ensembles, corresponding to higher-dimensional CDT. Firstly we study their fractal properties and secondly calculate the scale dependent spectral dimension and compare it to computer simulations. We comment further on the relation between Horava-Lifshitz gravity, asymptotic safety, multifractional spacetimes and CDT-like models.Comment: DPhil (Phd) thesis (2013), University of Oxford, 145 pages, 18 figures, new results/analysis on 2+1 model, comments are welcom

Aspects of dynamical dimensional reduction in multigraph ensembles of CDT

We study the continuum limit of a "radially reduced" approximation of Causal Dynamical Triangulations (CDT), so-called multigraph ensembles, and explain why they serve as realistic toy models to study the dimensional reduction observed in numerical simulations of four-dimensional CDT. We present properties of this approximation in two, three and four dimensions comparing them with the numerical simulations and pointing out some common features with 2+1 dimensional Horava-Lifshitz gravity.Comment: 4 pages, 1 figure, Presented at "Gravity, Quantum, and Black Holes" session of IC-MSQUARE 2012, Budapest, to appear in the proceedings, IOP Conference Serie

Investigating Robustness of Energy Management Maps for SMEs

Using the data from three small businesses, we are investigating robustness of the recently proposed Recurrence Quantitive Analysis (RQA) based method for energy management of small and medium enterprises. The method consists of two phases, the training phase where the map or maps of ‘usual’ behaviour is obtained, and the operational phase where the new data is tested against the existing map(s). We measure how the output changes when there is a small change in input, with respect to the sampling rate, missing data and noise. Our results over three qualitatively different datasets show that the method is relatively robust and can be used for different SMEs

Dynamical dimensional reduction in toy models of 4D causal quantum gravity

In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimension can be determined from a model with fewer degrees of freedom obtained from the CDTs by "radial reduction". In the resulting "toy" model we can take the continuum limit analytically and obtain a scale dependent spectral dimension varying from four to two with scale and having functional behaviour exactly of the form which was conjectured on the basis of the numerical results.Comment: 12 pages, 2 figures, v3: improved discussion, results unchanged, as publishe

Multigraph models for causal quantum gravity and scale dependent spectral dimension

We study random walks on ensembles of a specific class of random multigraphs which provide an "effective graph ensemble" for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the spectral dimension and Hausdorff dimension are equal and show that this occurs when rho, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent spectral dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton-Watson process which has a scale dependent spectral dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four dimensional CDT which has a spectral dimension of four at large scales and two at small scales.Comment: 30 pages, 3 figures, references added, minor changes in the abstract to match the published versio

Spectral dimension flow on continuum random multigraph

We review a recently introduced effective graph approximation of causal dynamical triangulations (CDT), the multigraph ensemble. We argue that it is well suited for analytical computations and that it captures the physical degrees of freedom which are important for the reduction of the spectral dimension as observed in numerical simulations of CDT. In addition multigraph models allow us to study the relationship between the spectral dimension and the Hausdorff dimension, thus establishing a link to other approaches to quantum gravityComment: 6 pages, 1 figure, to appear in the Proceedings of Sixth International School on Field Theory and Gravitation 2012 (Petropolis, Brazil

Supersymmetric gauge theories, Coulomb gases and Chern-Simons matrix models

We develop Coulomb gas pictures of strong and weak coupling regimes of supersymmetric Yang-Mills theory in five and four dimensions. By relating them to the matrix models that arise in Chern-Simons theory, we compute their free energies in the large N limit and establish relationships between the respective gauge theories. We use these correspondences to rederive the N^3 behaviour of the perturbative free energy of supersymmetric gauge theory on certain toric Sasaki-Einstein five-manifolds, and the one-loop thermal free energy of N=4 supersymmetric Yang-Mills theory on a spatial three-sphere.Comment: 17 pages; v2: reference adde

A genetic algorithm approach for modelling low voltage network demands

Distribution network operators (DNOs) are increasingly concerned about the impact of low carbon technologies on the low voltage (LV) networks. More advanced metering infrastructures provide numerous opportunities for more accurate load flow analysis of the LV networks. However, such data may not be readily available for DNOs and in any case is likely to be expensive. Modelling tools are required which can provide realistic, yet accurate, load profiles as input for a network modelling tool, without needing access to large amounts of monitored customer data. In this paper we outline some simple methods for accurately modelling a large number of unmonitored residential customers at the LV level. We do this by a process we call buddying, which models unmonitored customers by assigning them load profiles from a limited sample of monitored customers who have smart meters. Hence the presented method requires access to only a relatively small amount of domestic customers' data. The method is efficiently optimised using a genetic algorithm to minimise a weighted cost function between matching the substation data and the individual mean daily demands. Hence we can show the effectiveness of substation monitoring in LV network modelling. Using real LV network modelling, we show that our methods perform significantly better than a comparative Monte Carlo approach, and provide a description of the peak demand behaviour