Browsing by Author "Ellis, George F R"

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    Achieving baseline states in sparsely connected spiking-neural networks: stochastic and dynamic approaches in mathematical neuroscience
    (2015) Antrobus, Alexander Dennis; Murugan, Jeffrey; Ellis, George F R
    Networks of simple spiking neurons provide abstract models for studying the dynamics of biological neural tissue. At the expense of cellular-level complexity, they are a frame-work in which we can gain a clearer understanding of network-level dynamics. Substantial insight can be gained analytically, using methods from stochastic calculus and dynamical systems theory. This can be complemented by data generated from computational simulations of these models, most of which benefit easily from parallelisation. One cubic millimetre of mammalian cortical tissue can contain between fifty and one-hundred thousand neurons and display considerable homogeneity. Mammalian cortical tissue (or grey matter") also displays several distinct firing patterns which are widely and regularly observed in several species. One such state is the "input-free" state of low-rate, stochastic firing. A key objective over the past two decades of modelling spiking-neuron networks has been to replicate this background activity state using "biologically plausible" parameters. Several models have produced dynamically and statistically reasonable activity (to varying degrees) but almost all of these have relied on some driving component in the network, such as endogenous cells (i.e. cells which spontaneously fire) or wide-spread, randomised external input (put down to background noise from other brain regions). Perhaps it would be preferable to have a model where the system itself is capable of maintaining such a background state? This a functionally important question as it may help us understand how neural activity is generated internally and how memory works. There has also been some contention as to whether driven" models produce statistically realistic results. Recent numerical results show that there are connectivity regimes in which Self-Sustained, Asynchronous, Irregular (SSAI) firing activity can be achieved. In this thesis, I discuss the history and analysis of the key spiking-network models proposed in the progression toward addressing this problem. I also discuss the underlying constructions and mathematical theory from measure theory and the theory of Markov processes which are used in the analysis of these models. I then present a small adjustment to a well known model and provide some original work in analysing the resultant dynamics. I compare this analysis to data generated by simulations. I also discuss how this analysis can be improved and what the broader future is for this line of research.
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    Aspects of modern cosmology
    (1997) Bassett, Bruce Adrian Charles; Ellis, George F R; Fairall, Anthony Patrick
    The main work of this thesis can be summarised as: ■ An implementation of canonical quantisation to the covariant and gauge-invariant approach to cosmological perturbations. Standard results are reproduced. We discuss the advantages of this formalism over non-covariant and non gauge-invariant formalisms. ■ A characterisation of linear gravitational waves in a covariant way is achieved. The evolution equations for the electric and magnetic parts of the Weyl tensor are shown to be of different order. In particular, the electric part appears to have a third order evolution equation, while the magnetic part has a second order evolution equation. It is shown that the "silent" nature of the evolution equations for irrotational dust can be extended to the case of vortical dust. This may be relevant for the endpoints of gravitational collapse since the vorticity begins to grow as soon as density contrast becomes non-linear, as is the case in galaxies, showing that the irrotational silent universes are unstable. The main problem in accepting such vortical silent universes lies in proving integrability of the equations which has not been achieved so far, even in the irrotational case. ■ A review of issues in the Cosmic Microwave Background (CMB) is given, focussing particularly on points such as ergodicity, decaying modes, foreground contamination, recombination, spectral distortions and polarisation of the CMB. ■ A review of methods in gravitational lensing is presented, together with a hierarchy of distance measures in cosmology, forming an introduction to the following two chapters. ■ A common belief that photon conservation implies that the all-sky averaged area distance in inhomogeneous universes must be that of the background, matter-averaged Robertson-Walker area distance is dis proven. This means that there will in general be gravitational lensing effects even on large angular scales. ■ The realistic situation in which gravitational lensing leads to caustic formation is discussed. It is claimed that this invalidates many accepted beliefs concerning high-redshift observations in inhomogeneous universes. One application of importance is the CMB. Possible implications are discussed. ■ Random Gaussian fields are ubiquitous in modern statistical physics, and particularly important in CMB studies. Here we give accurate analytical functions approximating ∫e⁻ˣ²dx, the simplest of which is just the kink soliton.
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    Data bias
    (2021-02-10) Felin, Teppo; Koenderink, Jan; Krueger, Joachim I; Noble, Denis; Ellis, George F R
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    The data-hypothesis relationship
    (2021-02-10) Felin, Teppo; Koenderink, Jan; Krueger, Joachim I; Noble, Denis; Ellis, George F R
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    Études on fuzzy geometry and cosmology
    (2007) Murugan, Jeffrey; Ellis, George F R
    We investigate various aspects of noncommutative geometry and fuzzy field theory and their relations to string theory. In particular, we study the BPS and non-BPS solutions of the CJPN nonlinear sigma model on the noncommutative plane in some detail and show among other things that a class of its solitonic excitations may be built from bound states of noncommutative scalar solitons. We then go on to construct a fuzzy extension of the semilocal SU(N)a x U(l)L Yang-Mills-Riggs model. We find that not only does this noncommutative model support a large class of BPS vortex solutions but, unlike in the commutative model, these are exact solutions of the BPS equations. We also study the large coupling limit of the semilocal model and demonstrate conclusively the metamorphosis of the semilocal vortex to an appropriate degree instanton of the fuzzy CJPN model. In the second part of this work, we study the perpendicular intersection of Dl- and D7-branes in type liB string theory and the fuzzy 6-sphere that resolves the singularity of the intersection. We demonstrate the equivalence of the D7 and dual D-string descriptions by computing the energy, charge and radial profiles of the solution in each description. We conclude the thesis with a foray into cosmology by constructing a realisation of a recently proposed singularity-free inflating universe. We discuss the basic characteristics of this model and show that none are at odds with current observations.
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    Fluid and gas models in FLRW and almost FLRW universes
    (1996) Gebbie, Timothy John; Ellis, George F R; Maartens, Roy; Dunsby, Peter K S
    Recently the universe has been modeled in the covariant sense, in terms of fluid models and perturbations thereof, leading to Gauge Invariant Covariant (GIC) perturbations of these fluid models. It is well known that kinetic theory provides a physically sound and consistent description of the matter and radiation in the universe, so a perturbative theory of gas models using kinetic theory would be most helpful. This has been done to a large degree in the Gauge Invariant (GI) Bardeen approach to perturbation theory by studies of gases based on the relativistic Boltzmann equation. These treatments, however, were not fully covariant. The GI Bardeen approach is dependent on a co-ordinate choice, while in the full GIC perturbation theory full covariance is maintained along with gauge invariance by describing the theory in a particular set of perturbation variables that differ from the Bardeen choice but can be related to the Bardeen variables. The covariant formulation of the relativistic Boltzmann equation in terms of variables that are of use in the GIC theory for gases has been well described. In this thesis, I provide both a good introduction to the full GIC perturbation theory of a photon gas and matter fluid system in the linear theory as well as a solid grounding with respect to the exact FLRW fluid model upon which most of the original ideas and concepts of modern cosmology are based. The introduction to the exact FLRW model is done in the sense of the dynamical systems approach to cosmology which provides the easiest access to understanding the evolution of single and multi-fluid FLRW models.
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    Global dynamics of the universe
    (2000) Boersma, Jelle Pieter; Ellis, George F R
    In this thesis we consider four different topics in the field of cosmology, namely, black hole topology, the averaging problem, the effect of surface terms on the dynamics of classical and quantum fields, and the generation of an open universe through inflation with random initial conditions. It should be mentioned that while the research for this thesis was being done, no large effort was made to pursue a single theme. One reason for the diversity of the topics in this thesis is that the results which came out of this research were not always the results which were expected to be found when the investigation was started. Another reason for looking at several topics is simply that once a problem has been solved, then it is natural to move on to another problem which has not yet been solved. For those readers who value that a thesis is centered around a single unifying theme, let me mention that each of the four topics in this thesis are indeed related. Namely, each topic which we discuss focuses on an aspect of the global dynamics of the universe, in a situation where this is non-trivially different from the local dynamics. The non-trivial relation between global and local dynamics is rarely addressed in cosmology. Partially this is because of the difficulties which arise when one considers a realistic universe with infinitely many coupled degrees of freedom. Hence, it is a common practice to rely on simplifications which reduce the number of degrees of freedom, or the couplings between them. Further, there are few direct observations which probe the large-scale dynamics of the universe, or none at all, depending on the length scale and the type of cosmological model which one considers. As a consequence, there is a considerable freedom in choosing a priori assumptions or simplifications in the field of cosmology, without being able to falsify the validity thereof. For instance, when we analyse the relation between field perturbations at spatial infinity and perturbations here and now, we assume that quantum field theory, as we know it, is valid everywhere between here and spatial infinity. Although one cannot avoid making certain fundamental assumptions, the type of simplifications which are adopted in a calculation plays a less fundamental role. It is the objective of this thesis to improve our understanding of the large scale dynamics of the universe by showing rigorously what one can and what one cannot derive from certain fundamental assumptions. Interestingly, our results are often quite different from the results which are based on the same assumptions, but which involve certain commonly made simplifications as well.
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    Global dynamics of the universe
    (2000) Boersma, Jelle Pieter; Ellis, George F R
    In this thesis we consider four different topics in the field of cosmology, namely, black hole topology, the averaging problem, the effect of surface terms on the dynamics of classical and quantum fields, and the generation of an open universe through inflation with random initial conditions. It should be mentioned that while the research for this thesis was being done, no large effort was made to pursue a single theme. One reason for the diversity of the topics in this thesis is that the results which came out of this research were not always the results which were expected to be found when the investigation was started. Another reason for looking at several topics is simply that once a problem has been solved, then it is natural to move on to another problem which has not yet been solved. For those readers who value that a thesis is centred around a single unifying theme, let me mention that each of the four topics in this thesis are indeed related. Namely, each topic which we discuss focuses on an aspect of the global dynamics of the universe, in a situation where this is non-trivially different from the local dynamics. The non-trivial relation between global and local dynamics is rarely addressed in cosmology. Partially this is because of the difficulties which arise when one considers a realistic universe with infinitely many coupled degrees of freedom. Hence, it is a common practice to rely on simplifications which reduce the number of degrees of freedom, or the couplings between them. Further, there are few direct observations which probe the large-scale dynamics of the universe, or none at all, depending on the length scale and the type of cosmological model which one considers. As a consequence, there is a considerable freedom in choosing a priori assumptions or simplifications in the field of cosmology, without being able to falsify the validity thereof. For instance, when we analyse the relation between field perturbations at spatial infinity and perturbations here and now, we assume that quantum field theory, as we know it, is valid everywhere between here and spatial infinity. Although one cannot avoid making certain fundamental assumptions, the type of simplifications which are adopted in a calculation plays a less fundamental role. It is the objective of this thesis to improve our understanding of the large-scale dynamics of the universe by showing rigorously what one can and what one cannot derive from certain fundamental assumptions. Interestingly, our results are often quite different from the results which are based on the same assumptions, but which involve certain commonly made simplifications as well. This thesis is structured as follows. In the first chapter it is shown how different sections of the Kruskal geometry can be identified in a way which preserves time-orient ability of the spacetime. The existence of topologically different but locally identical solutions of Einstein's equations is well known, and not surprising considering the differential structure of these equations. also discuss the occurrence of Hawking radiation in topologically different black-hole geometries. Furthermore, we study the relation between black-hole solutions and circular cosmic strings. Assuming the existence of circular 1 cosmic string with deficit angle ranging between 0 and 211", we are able to construct a class of non-trivial vacuum solutions with properties similar to black-hole solutions but with a more complicated topology. In the second chapter of this thesis, we focus on the averaging problem in cosmology. The averaging problem occurs when one attempts to model a realistic inhomogeneous universe by a more symmetric model. Although averaging is often implied when studying realistic cosmological models, a rigorous treatment of averaging in cosmology appears to be surprisingly difficult. One difficulty which occurs when one tries to specify an averaging procedure is related to the large number of unphysical degrees of freedom which are present in the problem, namely, the coordinate freedom and the gauge freedom. The coordinate freedom manifests itself when one tries to evaluate the average of tensorial quantities, since the components of a tensor depend on the local choice of a frame. One may attempt to avoid this problem by specifying a local frame and evaluating some kind of average for each component separately. However, since there is no choice of frame which is preferred for physical reasons, this gives rise to a considerable amount of ambiguity. When one follows a perturbative approach, there is an additional freedom of choosing a gauge, which makes it ambiguous what one means by a perturbation of a physical quantity, even when this quantity does not depend on the local choice of frame. By specifying a choice of gauge, it becomes well defined what one means by a perturbation, but once again no choice of gauge seems to be preferred for physical reasons. In addition to these problems, there is an inherent ambiguity which is related to the freedom in choosing an averaging operation. Since there is generally more than one choice of averaging operation which is mathematically consistent, one needs to impose additional constraints which restrict the freedom of choosing an averaging operation. However, one would like to do so on the basis of a minimal set of assumptions. It is shown that each of these problems can be resolved in the case where perturbations theory can be applied. We use our results to calculate the lowest order non-trivial correction to the expansion of the observable universe, which is due to the fact that averaging does not commute with evaluating the (nonlinear) Einstein equations. In the third chapter of this thesis, we investigate the relation between surface terms which are evaluated at spatial infinity, and the local dynamics of a scalar field. Starting from the path-integral approach to quantum field theory, it is shown that the contribution of surface terms to the variation of the action functional cannot in general be neglected. The classical field equations can be derived by requiring that the variation of the action vanishes for all field perturbations, and it is shown that a surface term generally contributes a non-trivial source term to the classical field equations. This source term appears to vanish in spatially flat geometries, but it diverges in a spatially open geometries with super curvature perturbations. Rather surprisingly, it appears that the degrees of freedom of the scalar field which generate surface terms must have zero norm in the space of square integrable field 2 perturbations. Without restricting these zero-norm degrees of freedom, it follows that the local dynamics of the field are sensitive to details of the spacetime at spatial infinity. The main difficulty which we are confronted with consists of quantifying the zero-norm degrees of freedom. We briefly discuss a strategy for resolving this problem. In the fourth chapter we discuss different types of inflation. As is well known, the standard idea of inflation provides a simple explanation for the homogeneity of the observed universe. However, it appears to be much less straightforward to reconcile a period of inflation with the observed negative spatial curvature in the universe. Bubble inflation combines these two aspects, but it requires a rather restricted type of potential. After introducing the established ideas of standard inflation and bubble inflation, we focus on the dynamics of bubble spacetimes. It is shown that the often used thin-wall approach is not consistent with the assumption that the stress-energy is generated by a scalar field, although this assumption plays a crucial role in the theory of bubble-dynamics. In order to resolve this problem, we derive a simplified set of equations which describe the exact dynamics of a general spherically symmetric bubble spacetime. We then focus on the question of whether the restrictions on the shape of the potential, which are essential in the bubble inflation scenario, are necessary in order to explain the generation of negative spatial curvature during inflation. By studying the most generic situation where constant-scalar field hypersurfaces make a transition from being spacelike to being time like , it is shown that negative spatial curvature is generated under conditions which are more generic than the conditions which are generally assumed. The results which are presented in this thesis have been obtained through independent research, which was conducted by the author on an individual basis. The contents of the first three chapters have been published, [1] - [3], excluding the third section of the first chapter, which was added recently. The contents of the last chapter are currently being prepared for submission. None of the results which are obtained in this thesis have, to the best of my knowledge, been published elsewhere, or the original work has been cited.
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    The KLT relations in unimodular gravity
    (2016) Burger, Daniel; Weltman, Amanda; Murugan, Jeffrey; Ellis, George F R
    Here we initiate a systematic study of some of the symmetry properties of unimodular gravity, building on much of the known structure of general relativity, and utilizing the powerful technology developed in that context, such as the spinor helicity formal-ism. In particular, we show, up to five-points and tree-level, that the KLT relations of perturbative gravity hold for trace free or unimodular gravity. This work is in conjunction with a paper written with A. Welman, J. Murugan and G.F.R. Ellis (ARXIV: 1511.08517)
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    Manifold joins and jump conditions in general relativity
    (1982) Baston, Robert James; Ellis, George F R
    This thesis has as its aim the analysis of a possible manifold structure on V, a join of two individual manifolds V⁺ and V⁻, and analysing the physics across the join, as implied by Einstein's theory of General Relativity. There are several reasons why one might want to study such a situation. Firstly, the joining of manifolds is useful in the study of shock waves, be they of gravitational or other origin - we will be able to characterise the propagation of energy in the join. Secondly piecing together manifolds is a potentially fruitful way of obtaining exact solutions of Einstein's equations which do not exhibit any symmetries in the large, and are yet sufficiently homogeneous (in some sense) to enable one to model the apparent Universe - the prototype of this is the Swiss-Cheese model, used to study light transmission in an inhomogeneous Universe. Thirdly, discontinuities in the fundamental quantities in Relativity are of prime importance in the study of singularities and in particular, it is of prime importance to single out the contributions of the differential geometry and metric structure of the Universe to the existence and nature of such singularities. Closely linked to these problems is the problem of linking the small scale structure of the Universe (which is manifestly complicated and potentially full of singularities) and the large scale structure, which seems so well modelled by assumptions of homogeneity and isotropy. In this regard, the techniques of Regge (1961), originally proposed to provide approximate solutions to the Einstein equations, assume a new theoretical importance, for the delta-type singularities in the curvature he used, in a smoothing process, to represent the (assumed) continuous curvature of space, could in themselves play a distinguished role representing the small scale structure of the Universe. Furthermore, the matching together of blocks of space-time with sharp edges and corners may enable to develop a manifold like structure in which the tangent spaces of some points had a surfeit or deficiency of vectors, so that the differential geometry of the resulting space-time forced discontinuities and singularities in the metric structure of the Universe. Although this may be aphysical, it may be a reasonable way of seeking further understanding of the Universe.
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    Observations of galaxies in a cosmological context.
    (1982) Sievers, A W; Sievers, A W; Ellis, George F R
    The basic theory of observations of galaxies in a cosmological context is reviewed and extended to include e.g. the pointspread effect of the atmosphere. From this the relation between the sources (objects) and the images of these sources is derived (the observational map). A program is developed to calculate this map and some results are given.
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    Phase planes in the universe : chaotic cyclic universes and kicking Chameleons
    (2016) Platts, Emma; Weltman, Amanda; Ellis, George F R
    This thesis consists of two main sections: chaotic cyclic cosmology and Chameleon gravity in the early universe. Both sections invoke a phase plane analysis as their commonality. The first explores a cyclic model, proposed by Ellis et al, that is in keeping with current observations. No exotic nor new physics is needed for the bounce nor the turnaround. The model is chaotic in nature and requires only that the universe is closed and that dark energy (at some time) decays. The second section contests the claim by Burrage et al. that Chameleon gravity is inconsistent in the early universe, unless constraints on its coupling mechanism are significantly increased. It is shown that the addition of a Dirac-Borne-Infeld (DBI) correction - a consistent, high energy modification - to the Chameleon dynamically renders it weakly coupled to matter. This is done without any fine-tuning and ensures the consistency of the Chameleon at all scales without infringing upon its crucial feature as a dark energy candidate: its elusive but prominent coupling to matter.
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    Signature change in spherical vacuum spacetimes
    (1993) Sumeruk, H A; Ellis, George F R
    This thesis follows the approach of papers (1, 2) by exploring signature changes in other metrics. The metrics we chose to investigate are the Schwarzschild metric and the Tolman metric. The Schwarzschild metric was originally chosen in order to investigate whether the neighborhood of the singularity inside a black hole can be replaced with a Euclidean region, and also to see whether this Euclidean region can lead to new universes by providing "wormholes" through to other Lorentzian universes. By this we mean that, if one follows "time-like" geodesic paths from a Lorentzian region into a Euclidean region, they bounce (instead of hitting a singularity) and can then pass through a second signature change into another Lorentzian region. Consideration of how geodesics pass through a signature change naturally leads to the Tolman metric, whose vacuum cases cover the Schwarzschild/Kruskal-Szekeres manifold with all possible sets of radial geodesic coordinates. We take the opportunity to explore several cases of signature change in other Tolman models.
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    A study of integrability conditions for irrotational dust spacetimes
    (1998) Lesame, William Mphepeng; Ellis, George F R
    This thesis examines consistency conditions for fluid solutions of the field equations of general relativity. The exact non-linear dynamic equations for a generic irrotational dust spacetime are consistent. To analyse conditions characterizing pure gravity waves, linearization instability in general relativity and consistency of the so-called "silent universes", further exact conditions are imposed locally on irrotational dust. These are classified into Class II conditions, which change evolution equations into constraint equations, and Class I and III conditions, which do not doso-rather they add a new constraint, leaving the propagation equations unchanged in form. Class I conditions are imposed on terms in the constraint equations, while Class II and III conditions are imposed on terms in the evolution equations. In the Class I case it is shown that for irrotational dust space times the divergence-free magnetic Weyl tensor and the divergence-free electric Weyl tensor (necessary conditions for gravity waves interacting with matter), both imply integrability conditions in the exact non-linear case. The integrability conditions for the divergence-free magnetic Weyl tensor are identically satisfied in the linearized perturbation case, but are non-trivial in the exact non-linear case. This leads to a linearization instability in these models. The integrability conditions for the divergence-free electric Weyltensor are non-trivial in both the linear and non-linear cases. The Class II case focuses on irrotational silent cosmological dust models characterized by vanishing magnetic Weyl tensor and vanishing electric Weyl tensor. In both these models there exist a series of integrability conditions that need to be satisfied. Integrability conditions for the zero magnetic Weyl tensor condition hold identically for linearized case, but are non-trivial in the exact non-linear case. Thus there is also a linearization instability. The zero electric Weyl tensor condition leads to a chain of non-trivial integrability conditions in both the linear and non-linear cases. Because of the complexity of the integrability conditions, it is highly unlikely that there is a large class of models in both the silent zero magnetic Weyl tensor case and the silent zero electric Weyl tensor case.
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    The wave function of the universe
    (1994) Solomons, Deon Mark; Ellis, George F R
    In Quantum Cosmology, universe states are treated as wave function solutions to a zero-energy Schroedinger equation that is hyperbolic in its second derivatives of spatial geometries and matter-fields. In order to select one wave function (that would in principle correspond to our Universe) out of infinitely many, requires an appropriate boundary condition. The Hartle-Hawking No Boundary and the Vilenkin Tunneling proposals are examples of such boundary conditions. We review their applications and shortcomings in the context of the Inflationary Scenario. Another boundary condition is that of S.W. Hawing and D.N. Page (1990) in the context of wormholes. Wormholes are generally considered to play a major role in setting the cosmological constant to zero and to provide a mechanism for black hole evaporation. It is significant that we are able to show that even the class of bulk matter wormhole instantons found by Carlini and Mijic (1990) are predicted in the quantum theory. However, unresolved issues and newfound problems seem to threaten the wormhole theory. Furthermore, since there are no a priori notions of time (and space) present in the quantum theory, it is important to show exactly how the notion of time is recovered over distances much larger than the Planck scale. A good notion of time is also essential for any quantum theory to predict the correct classical behaviour for the Universe today. The issue of time inevitably re-emerges throughout our work.