Dissertations / Theses on the topic 'Linear perturbation theory'

The thesis begins with a study of the origin of non-linear cosmological fluctuations. In particular, a class of models of multiple field inflation are considered, with specific reference to those cases in which the non-Gaussian correlation functions are large. The analysis shows that perturbations from an almost massless auxiliary field generically produce large values of the non-linear parameter fNL. Next, the effects of including non-Gaussian correlation functions in the statistics of cosmological structure are explored. For this purpose, a non-Gaussian probability distribution function (PDF) for the curvature perturbationR is required. Such a PDF is derived from first principles in the context of quantum field theory, with n-point correlation functions as the only input. Under reasonable power-spectrum conditions, an explicit expression for the PDF is presented, with corrections to the Gaussian distribution from the three-point correlation function hRRRi. The method developed for the derivation of the non-Gaussian PDF is then used to explore two important problems in the physics of primordial black holes (PBHs). First, the non-Gaussian probability is used to compute corrections to the number of PBHs generated from the primordial curvature fluctuations. Particular characteristics of such corrections are explored for a variety of inflationary models. The non-Gaussian corrections explored consist exclusively of non-vanishing three-point correlation functions. The second application concerns new cosmological observables. The formation of PBHs is known to depend on two main physical characteristics: the strength of the gravitational field produced by the initial curvature inhomogeneity and the pressure gradient at the edge of the curvature configuration. The latter has so far been ignored in the estimation of the probability of PBH formation. We account for this by using two parameters to describe the profile: The amplitude of the inhomogeneity and its second radial derivative, both evaluated at the centre of the configuration. The method developed to derive the non-Gaussian PDF is modified to find the joint probability of these two parameters. We discuss the implications of the derived probability for the fraction of mass in the universe in the form of PBHs.

In the work described in this thesis, the Feynman-Dyson perturbation theory, developed from quantum field theory, was employed in semiempirical calculations on trans - polyacetylene. A variety of soliton-like excited states of the molecule were studied by the PPP-UHF-RPA method. The results of this study provide useful information on the nature of these states, which are thought to account for the unique electrical conduction properties of trans - polyacetylene and similar conducting polymers. Feynman-Dyson perturbation theory was also used to extend Hartree-Fock theory by the inclusion of time-independent second-order self-energy insertions. The results of calculations on polyenes show that consideration of this approach is warranted, as the contribution of the second- order terms is significant. The computer program, written during the course of the research reported here, is discussed as well.
Ph. D.

We propose and construct a two-parameter expansion around a Friedmann-Lemaitre- Robertson-Walker geometry which uses both large-scale and small-scale perturbations analogous to cosmological perturbation theory and post-Newtonian gravity. We justify this observationally, derive a set of field equations valid on a fraction of the horizon size and perform a detailed investigation of the associated gauge problem. We find only the Newtonian gauge, out of the standard gauges used in cosmological perturbation theory, is applicable to post-Newtonian perturbations; we can identify a consistent set of perturbed quantities in the matter and gravity sectors and construct corresponding gauge-invariant quantities. The field equations, written in terms of these quantities, takes on a simpler form, and allows the effects of small-scale structure on the large-scale properties of the Universe to be clearly identified and discussed for different physical scenarios. With a definition of statistical homogeneity, we find that the cosmological constant and the average energy density, of radiation and dust, source the Friedmann equation, whereas only the inhomogeneous part of the Newtonian energy density sources the Newton-Poisson equation { even though both originate from the same equation. There exists field equations at new orders in our formalism, such as a frame-dragging field equation a hundred times larger than expected from using cosmological perturbation theory alone. Moreover, we find non-linear gravity, mode-mixing and a mixing-of-scales at orders one would not expect from intuition based on cosmological perturbation theory. By recasting the field equations as an effective fluid we observe that these non-linearities lead to, for example, a large-scale effective pressure and anisotropic stress. We expect our formalism to be useful for accurately modelling our Universe, and for investigating the effects of non-linear gravity in the era of ultra-large-scale surveys.

Alors que les approches standards de résolution de la structure électronique présentent un coût de calcul à la puissance 3 par rapport à la complexité du problème, des solutions permettant d’atteindre un régime asymptotique linéaire,O(N), sont maintenant bien connues pour le calcul de l'état fondamental. Ces solutions sont basées sur la "myopie" de la matrice densité et le développement d'un cadre théorique permettant de contourner le problème aux valeurs propres. La théorie des purifications de la matrice densité constitue une branche de ce cadre théorique. Comme pour les approches de type O(N) appliquées à l'état fondamental,la théorie des perturbations nécessaire aux calculs des fonctions de réponse électronique doit être révisée pour contourner l'utilisation des routines coûteuses.L'objectif est de développer une méthode robuste basée uniquement sur la recherche de la matrice densité perturbée, pour laquelle seulement des multiplications de matrices creuses sont nécessaires. Dans une première partie,nous dérivons une méthode de purification canonique qui respecte les conditions de N-representabilité de la matrice densité à une particule. Nous montrons que le polynôme de purification obtenu est auto-cohérent et converge systématiquement vers la bonne solution. Dans une seconde partie, en utilisant une approche de type Hartree-Fock, nous appliquons cette méthode aux calculs des tenseurs de réponses statiques non-linéaires pouvant être déterminés par spectroscopie optique. Au delà des calculs à croissance linéaire réalisés, nous démontrons que les conditions N-representabilité constituent un prérequis pour garantir la fiabilité des résultats
Whereas standard approaches for solving the electronic structures present acomputer effort scaling with the cube of the number of atoms, solutions to overcomethis cubic wall are now well established for the ground state properties, and allow toreach the asymptotic linear-scaling, O(N). These solutions are based on thenearsightedness of the density matrix and the development of a theoreticalframework allowing bypassing the standard eigenvalue problem to directly solve thedensity matrix. The density matrix purification theory constitutes a branch of such atheoretical framework. Similarly to earlier developments of O(N) methodology appliedto the ground state, the perturbation theory necessary for the calculation of responsefunctions must be revised to circumvent the use of expensive routines, such asmatrix diagonalization and sum-over-states. The key point is to develop a robustmethod based only on the search of the perturbed density matrix, for which, ideally,only sparse matrix multiplications are required. In the first part of this work, we derivea canonical purification, which respects the N-representability conditions of the oneparticledensity matrix for both unperturbed and perturbed electronic structurecalculations. We show that this purification polynomial is self-consistent andconverges systematically to the right solution. As a second part of this work, we applythe method to the computation of static non-linear response tensors as measured inoptical spectroscopy. Beyond the possibility of achieving linear-scaling calculations,we demonstrate that the N-representability conditions are a prerequisite to ensurereliability of the results

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft

Dans le premier chapitre, nous commençons par définir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques propriétés des classes de Schatten. La fin du chapitre est dédiée à l’étude des espaces de Bochner à valeurs dans l'espace des opérateurs factorisables par un espace de Hilbert. Le deuxième chapitre est consacré aux multiplicateurs de Schur linéaires. Nous caractérisons les multiplicateurs bornés sur B(Lp, Lq) lorsque p est inférieur à q puis appliquons ce résultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisième chapitre, nous caractérisons, au moyen de multiplicateurs de Schur linéaires, les multiplicateurs de Schur bilinéaires continus à valeurs dans l'espace des opérateurs à trace. Dans le quatrième chapitre, nous donnons divers résultats concernant les opérateurs intégraux multiples. En particulier, nous caractérisons les opérateurs intégraux triples à valeurs dans l'espace des opérateurs à trace puis nous donnons une condition nécessaire et suffisante pour qu'un opérateur intégral triple définisse une application complètement bornée sur le produit de Haagerup de l'espace des opérateurs compacts. Enfin, le cinquième chapitre est dédié à la résolution des problèmes de Peller. Nous commençons par étudier le lien entre opérateurs intégraux multiples et théorie de la perturbation pour le calcul fonctionnel des opérateurs autoadjoints pour finir par la construction de contre-exemples à ces problèmes
In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems

In this thesis we present research into linear perturbations in Lemaître-Tolman-Bondi (LTB) and Assisted Coupled Quintessence (ACQ) Cosmologies. First we give a brief overview of the standard model of cosmology. We then introduce Cosmological Perturbation Theory (CPT) at linear order for a at Friedmann-Robertson-Walker (FRW) cosmology. Next we study linear perturbations to a Lemaître-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities in LTB. We show, using the perturbed energy conservation equation, that there is a conserved quantitiy in LTB which is conserved on all scales. We then briefly extend our discussion to the Lemaître spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime. We also study the behaviour of linear perturbations in assisted coupled quintessence models in a FRW background. We provide the full set of governing equations for this class of models, and solve the system numerically. The code written for this purpose is then used to evolve growth functions for various models and parameter values, and we compare these both to the standard CDM model and to current and future observational bounds. We also examine the applicability of the "small scale approximation", often used to calculate growth functions in quintessence models, in light of upcoming experiments such as SKA and Euclid. We nd the results of the full equations deviates from the approximation by more than the experimental uncertainty for these future surveys. The construction of the numerical code, Pyessence, written in Python to solve the system of background and perturbed evolution equations for assisted coupled quintessence, is also discussed.

One seminal question that faces a vehicle's driver (either human or computer) is predicting the capability of the vehicle as it encounters upcoming terrain. A Location-Aware Adaptive Vehicle Dynamics (LAAVD) System is being developed to assist the driver in maintaining vehicle handling capabilities through various driving maneuvers. In contrast to current active safety systems, this system is predictive, not reactive. The LAAVD System employs a predictor-corrector method in which the driver's input commands (throttle, brake, steering) and upcoming driving environment (terrain, traffic, weather) are predicted. An Intervention Strategy uses a novel measure of handling capability, the Performance Margin (PM), to assess the need to intervene. The driver's throttle and brake control are modulated to affect desired changes to the PM in a manner that is minimally intrusive to the driver's control authority. This system depends heavily on an understanding of the interplay between the vehicle's longitudinal, lateral, and vertical forces, as well as their resulting moments. These vehicle dynamics impact the PM metric and ultimately the point at which the Intervention Strategy will modulate the throttle and brake controls. Real-time implementation requires the development of computationally efficient predictive models of the vehicle dynamics. In this work, a method for predicting future vehicle states, based on current states and upcoming terrain, is developed using perturbation theory. An analytical relationship between the change in the spindle forces and the resulting change in the PM is derived, and the inverse relationship, between change in PM and resulting changes in longitudinal forces, is modeled. This model is implemented in the predictor-corrector algorithm of the Intervention Strategy. Corrections to the predicted states are made at each time step using a detailed, full, non-linear vehicle model. This model is run in real-time and is intended to be replaced with a drive-by-wire vehicle. Finally, the impact of this work on the automotive industry is discussed and recommendations for future work are given.
Master of Science

La tesi riguarda la formazione di strutture a larga scala nell'universo, cioè l'origine degli addensamenti di materia che hanno portato alla formazione dei cluster di galassie. La maggior parte del lavoro ha riguardato aspetti non lineari della Teoria delle Perturbazioni Cosmologiche, trattando in particolare il periodo di transizione tra epoca della radiazione e epoca della materia. In questo contesto si è considerato un modello non-standard di materia, analizzando il ruolo dell'indice barotropico nell'evoluzione del contrasto di densità . La nota approssimazione Meszaros è stata generalizzata ad una analisi non lineare che ha permesso di trovare la skewness della distribzione di materia, un importante indice di non-Gaussianità rilevabile dai dati osservativi. Nel contesto delle perturbazioni cosmologiche è stata formulata la teoria Post-Newtoniana (1PN) con lo scopo di ottenere un set di equazioni valido per ogni range di distanze, in particolare per le scale intermedie. I risultati finali coincidono sia con la teoria lineare relativistica per grandi scale sia con la teoria non lineare Newtoniana per piccole scale; quest'ultima connessione fornisce una chiara visione della relazione fra Relatività Generale e teoria Newtoniana.
The subject matter of this thesis is the formation of large-scale structure in the universe, describing the clustering of matter in galaxies and clusters of galax- ies. Most of the study has dealt with the non-linear evolution of cosmological fluctuations, focusing on the scalar sector of perturbation theory. The period of transition between the radiation era and the matter era has been largely examined, extending the already known linear results to a non-standard matter model and to a non-linear analysis. The obtained second order solutions for the matter fluctuations variables have been used to find the skewness of the density and velocity distributions, an important statistic estimator measuring the level of non-Gaussianity of a statistic ensamble. In the contest of cosmological perturbations a complete Post-Newtonian (1PN) treatment is presented with the aim of obtain a set of equations suitable in particular for the intermediate scales. The final result agrees with both the non linear Newtonian theory of small scales and the linear general relativistic theory of large scales. Analyzing the limit cases of our approach to 1PN cosmology, we have clarified the link between the Newtonian theory of gravity and General Relativity.

Many social, biological and technological systems can be viewed as complex networks with a large number of interacting components. However despite recent advancements in network theory, a satisfactory description of dynamic processes arising in such cooperative systems is a subject of ongoing research. In this dissertation the emergence of dynamical complexity in networks of interacting stochastic oscillators is investigated. In particular I demonstrate that networks of two and three state stochastic oscillators present a second-order phase transition with respect to the strength of coupling between individual units. I show that at the critical point fluctuations of the global order parameter are characterized by an inverse-power law distribution and I assess their renewal properties. Additionally, I study the effect that different types of perturbation have on dynamical properties of the model. I discuss the relevance of those observations for the transmission of information between complex systems.

Electromechanical systems dynamics analysis is approached through nonlinear differential equations and further creating a state space model for the system. There are three modules analyzed and validated, first module consists two magnet coupled with a mass spring damper system as a band-pass system, Low-pass equivalent system and Low-pass equivalent system through perturbation analysis. Initially Band Pass frameworks for the systems are formulated considering the relation between the mechanical forcing and current. Using Mathematical tools such as Hilbert transforms, Low-Pass equivalent of the systems are derived. The state equations of the systems are then used to design a working model in MATLAB and simulations investigated completely. The scope of the modules discussed for further development of tools various applications.

Thesis (M.S.)--East Tennessee State University, 2003.
Title from electronic submission form. ETSU ETD database URN: etd-0331103-140715. Includes bibliographical references. Also available via Internet at the UMI web site.

One of the most interesting challenges of present cosmology concerns the understanding of structure formation and evolution. This physical process encodes fundamental information about the nature of our Universe. The analysis of the growth of density and velocity perturbations gives us an excellent opportunity to address basic questions such as the amount of dark matter or the reason for the present accelerated expansion of the Universe (known as the Dark Energy problem). On large scales the growth of perturbations can be described by linear perturbation theory. The results on these scales have been largely confirmed by numerical simulations and employed in real data analysis. Future galaxy surveys will be able to give constraints on the perturbations growth with unprecedent precision. Moreover relevant physical information is encoded in scales smaller than O(100 Mpc), where where the structures are more clustered and linear perturbation theory breaks down. Higher orders in the perturbative expansion are not manifestly suppressed by a small expansion parameter, so that their use depends on the scale, redshift, and on features of initial conditions such as the shape of the primordial power spectrum and of the other statistical correlators (bispectrum, trispectrum, etc.). The main implemented tools to face the problem of nonlinear structure formation are the N-body simulations, even though they show inconveniences. Indeed, simulations give less physical insight with respect to analytical or semi-analytical approach. Furthermore simulations are limited by the long computational time involved. Given the motivations above, different semi-analytical approaches to the problem have appeared in the last years. These are based on the possibility of recasting the cosmological perturbation series in an appropriate way. Most of these new theories reorganize the series expansion by redefining the building blocks of the perturbative expansion. In particular the nonlinear propagator gains a fundamental role and it has been analytically computed by means of different approximations. It represents the cross-correlation between the final density, or velocity, perturbations and the initial ones. The main purpose of this thesis is to go beyond the known nonlinear propagator approximations leveraging on the exact evolution equations for the propagator which we derive. We analytically develop a well motivated approximation scheme by including a larger class of perturbative corrections which are neglected in other analytic re-summations. More specifically, this method allows us to take into account also the corrections given by a generic non linear matter power spectrum. Furthermore we compute the non linear propagator considering two different approximations for the full nonlinear power spectra. This problem has been addressed both with analytic and numerical techniques. As a general result we find that the precedent approaches lead to a wrong prediction on the sign of the propagator corrections. This clarifies a controversial problem and points out that further comparisons with N-body simulations are necessary. We find that the new corrections are significant at the relevant scales and therefore cannot be neglected in a re-summation scheme aiming at an accuracy compatible with future generation galaxy surveys. Furthermore we propose a method to implement the power spectrum computation taking into account this results.
Una delle sfide più importanti della cosmologia riguarda la comprensione della formazione e dell'evoluzione delle strutture. Questi processi fisici ci forniscono informazioni fondamentali sulla natura del nostro Universo. L'analisi della crescita delle perturbazioni di densita' e di velocita' ci offre un'importante opportunita' per quantificare con piu' accuratezza la percentuale di materia oscura o comprendere le cause che guidano l'espansione accelerata dell'Universo (il cosiddetto problema dell'Energia Oscura). A grandi scale la crescita delle fluttuazioni puo' essere descritta con la teoria delle perturbazioni lineari. I risultati a queste scale sono stati largamente confermati dalla simulazioni numeriche e utilizzati nell'analisi dei dati reali. Le future osservazioni di galassie ci permetteranno di porre dei vincoli sulla crescita delle perturbazioni con una precisione mai raggiunta prima. Inoltre, informazioni fisiche rilevanti sono presenti a scale piu' piccole di O(100 Mpc), dove le strutture sono molto piu' raggruppate e la teoria delle perturbazioni lineare non e' piu' valida. Le tecniche piu' utilizzate per affrontare il problema della formazione delle strutture sono le simulazioni ad N-corpi, anche se mostrano alcuni inconvenienti. Infatti, le simulazioni rendono piu' difficile la comprensione della fisica rispetto all'approccio analitico e semi-analitico. Inoltre le simulazioni sono limitate dal lungo tempo di calcolo che caratterizza questi studi. Date le precedenti motivazioni, differenti approcci semi-analitici al problema sono apparsi negli ultimi anni. Essi sono basati sulla possibilita' di riformulare la serie di perturbazioni cosmologiche in maniera appropriata. La maggior parte di queste nuove teorie riorganizza l'espansione in serie ridefinendo le grandezze fondamentali dello sviluppo perturbativo. In particolare il propagatore non lineare assume un ruolo fondamentale ed e' stato calcolato analiticamente con approcci differenti. Esso rappresenta il cross-correlatore tra le perturbazioni finali e iniziali di densita', o di velocita'. L'obiettivo principale di questa tesi e' quello di andare oltre le approssimazioni note relative al calcolo del propagatore non lineare facendo uso delle equazioni esatte di evoluzione per il propagatore. Abbiamo sviluppato e motivato un metodo di approssimazione analitico che ci ha permesso di includere una piu' ampia classe di correzioni perturbative che sono state trascurate in altre risommazioni analitiche. In particolare, questo metodo ci permette di considerare le correzioni date da un generico spettro di potenza non lineare. Inoltre abbiamo calcolato il propagatore non lineare considerando due differenti approssimazioni per lo spettro di potenza esatto. Questo problema e' stato trattato sia con tecniche analitiche che numeriche. Come risultato generale, abbiamo trovato che gli approcci predenti portano ad un risultato errato per quanto riguarda il segno delle correzioni del propagatore. Questo chiarisce un problema controverso e suggerisce che sono necessari ulteriori confronti con le simulazioni a N-corpi. Abbiamo trovato che le nuove correzioni sono significative alle scale rilevanti e quindi non possono essere trascurate in uno schema di risommazione che mira a raggiungere un'accuratezza compatibile con le future osservazioni astrofisiche. Inoltre proponiamo un metodo per calcolare lo spettro di potenza tenendo conto di questi risultati.

Effective design of liquid fuel injection systems is a function of good understanding of liquid breakup mechanisms. A transient liquid breakup model is developed on the classical interfacial breakup theory by modifying the classical linear perturbation process to include time-dependent base and perturbed flow parameters. The non-isothermal condition on liquid jet instability and breakup is theoretically modelled; with the particular consideration of a spatially variation of surface tension along the liquid-gas interface. The model combines the classical interface hydrodynamic instability and breakup theory and heat-transfer through semi-infinite medium. Analytical liquid breakup model, which combines transient and non-isothermal effects on liquid jet breakup, is suggested. The suggested model could be simplified to the transient breakup model and the non-isothermal breakup model equivalents. A novel mechanistic model, which is based on a simple momentum balance between the injected jet and the aerodynamic drag force, is suggested for breakup length. A new model, which combines energy criterion and dual-timescale for turbulent shear in droplet dispersion, is suggested for droplet breakup criteria on the basis of critical Webber number. All developed models showed good predictions of available experimental data, and established empirical correlation, within the operational conditions of contemporary ICEs, specifically diesel engines. Continued research in these areas could benefit the development of the next generation of liquid fuel injectors and combustors – by accounting for transient effects and non-isothermal conditions in liquid jet breakup, and turbulent shear in droplet breakup.

Ce travail porte sur l'étude de la propriété d'(A,B)-invariance de polyèdres convexes et son application à la commande sous contraintes et au problème l1. D'abord, nous proposons une caractérisation explicite de l'(A,B)-invariance de polyèdres convexes pour des systèmes en temps discret. Cette caractérisation se traduit par des conditions nécessaires et suffisantes sous la forme de relations matricielles linéaires, et présente deux avantages majeurs vis-à-vis de celles rencontrées dans la littérature : elle s'applique à tous les polyèdres convexes et elle ne nécessite pas le calcul de sommets. Ces avantages se font sentir notamment dans le calcul du domaine (a,b)-invariant supremal inclus dans un polyèdre donne, pour lequel nous proposons une méthode numérique. Le problème de calculer une loi de commande rendant positivement invariant en boucle fermée un polyèdre (a,b)-invariant est également traite. Les relations d'(A,B)-invariance sont alors généralisées à des systèmes soumis à des contraintes linéaires sur la commande et à des systèmes soumis à des perturbations additives bornées. Puis, les résultats obtenus en temps discret sont étendus aux systèmes en temps continu. Ensuite, le problème d'atténuation de perturbations additives persistantes, connu dans la littérature comme problème l1, est étudié. Les domaines (A,B)-invariants intérieurement stabilisables sont d'abord caractérises. Puis, nous proposons une approche décomposée pour le calcul du domaine intérieurement stabilisable supremal inclus dans le polyèdre défini par les contraintes de performance l1. Un niveau de performance donne est atteignable si et seulement si ce domaine supremal n'est pas vide. Cette approche géométrique permet notamment de déterminer directement la solution du problème l1 pour une classe importante de systèmes. Enfin, nous étendons l'étude de l'(A,B)-invariance de polyèdres à des systèmes dont le modèle est soumis à des incertitudes du type structure.

The possibility of a nonzero cosmological constant has been invoked several times in the past, both for theoretical and observational motivations. It has been often discarded by particle physicists, due to the huge difficulties in justifying a value of vacuum energy tiny enough to allow the universe to survive more than 10-41 s after the Big Bang. At present, the cosmological constant problem is still, probably, the most ununderstood issue of the physics. However, in recent times, it has again come into vogue, and again as a consequence of a number of observational evidences. Despite their apparent simplicity, the results of observations of more than 40 distant type lA supernovae seem to converge on the astonishing evidence that the present expansion of the universe is accelerating; this fact, together with many other experimental evidences of a lowdensity universe, and with the most recent CMB data indicating that the universe is very near to the flatness, has opened again the difficult question of what is the unobserved energy component that would balance a low-density universe with a flat one. At the same time, a new branch of cosmology has been opened, involving scalar fields as candidates for such "missing" energy. Motivated by the difficulties of cosmological constant models and by the most recent observational case of an accelerating universe, many alternative scenarios have been proposed: amongst them, the Quintessence scenarios, which are the subject of this thesis. In these models, the "missing energy" should reside in a dynamical scalar field rather than in a pure vacuum state; the dynamics of the field plays an important role, since the field energy density can adjust in a way that it comes to dominate at late times. The most important and distinctive feature of a scalar field vs. a cosmological constant, is that a field will develop fluctuations, that interact gravitationally with those of matter. To obtain the correct predictions of their impact on the Cosmic Microwave Background radiation and on the evolution of perturbations, the formalism of linear perturbation theory must be widely used. In this thesis we will focus on some basic issues connected with the attempt to build predictions on the cosmological impact of such scalar fields, following the results discussed in refs. [173, 174, 10, 15]. The first chapter is introductory and aims at giving an overview of the current observational evidences from which the case of a positive non-zero vacuum energy arises, motivating the consideration of the cosmological constant problem. In Chapter 2, the gauge-invariant formalism of linear cosmological perturbation theory is described, with particular attention to quantities such as the gravitational potentials, entropy and curvature perturbations, which are used in the following of the thesis. In Chapter 3 we recall some basic ideas on the mechanisms that generated the observed Cosmic Microwave Background of radiation, whose small but detectable anisotropies contain a large amount of information on the history of the Universe. In particular, CMB anisotropies turned out to be a very rich ground of investigation for discriminating between Quintessence and cosmological constant scenarios. These chapters set the framework for the results that will be presented in the following chapters, containing the original work of the thesis. In Chapter 4, focusing on scalar-type perturbations, we settled the analytical initial conditions that must be imposed on the components of cosmic fluid involving a minimally-coupled scalar field, in order to produce purely adiabatic or purely isocurvature initial conditions on super-horizon scales. Thus, an interesting comparison with the "standard" pure CDM flat model is performed. The distinctive imprints of Quintessence on large scale structure and on CMB anisotropies, both of polarization and temperature, are extensively analyzed. Chapter 5 extends the concept of Quintessence to a larger class of scalar fields, having an explicit coupling with the Ricci scalar in the Lagrangian. These more general models, here named "Extended Quintessence", are shown to enrich the phenomenology with respect to the simple minimally-coupled Quintessence. In fact, the predictions for the CMB anisotropies show new distinctive features, directly related to the presence of a non-zero coupling of the field with the gravitational sector of the Lagrangian and, ultimately, with the time-variation of the gravitational constant. A problem of "fine tuning" is however inherent both to cosmological constant and quintessence models: in order to have today an amount of vacuum energy comparable with that of matter, the vacuum energy density should have been initially vanishingly small. A way out to such fine tuning problem is possible in Quintessence scenarios, where one can select a subclass of models which admit "tracking solutions". This means that the present value of scalar field energy density, once fixed, can be determined starting from a very wide set of initial conditions, even though the tracking solutions are not perfect attractors and do not solve the problem of why the field energy density should have this value just today. Chapter 6 considers tracking behaviors in Extended Quintessence scenarios and presents a description of the rich phenomenology that arises from the corresponding dynamics; in particular, we show that the coupling with the Ricci scalar can act initially as an effective potential pushing the field in the tracking trajectory ( "R-boost"). The dependence of the phenomenology on the sign of the coupling constant is also described. Finally, Chapter 7 presents the conclusions and faces the future observational perspectives, on the light of the most recent data from MAXIMA and BOOMERANG-98 balloon experiments and of the future satellite missions MAP and Planck.

La gyrocinétique est un modèle clef pour la microturbulence en physique des plasmas. Elle présente encore plusieurs difficultés, qui pourraient invalider ses équations. Ce rapport de thèse clarifie trois d'entre elles. Tout d'abord, une de des coordonnées causait des soucis, d'un point de vue tant physique que mathématique ; une coordonnée adéquate est introduite, qui dissipe les difficultés et explique les structures intrinsèques sous-jacentes. Ensuite, des relations de récurrence explicites sont obtenues pour tous les ordres du développement perturbatif. Enfin, en utilisant la structure hamiltonienne de la dynamique, le couplage plasma-champ électromagnétique est implémenté d'un façon plus adaptée, avec d'importantes conséquences sur les équations gyrocinétiques.Plusieurs autres résultats sont obtenus, e.g. sur l'origine de l'invariant adiabatique centre-guide, sur une transformation centre-guide minimale très efficace, ou sur un modèle hamiltonien intermédiaire entre Vlasov-Maxwell et la gyrocinétique, dont les caractéristiques de Vlasov contiennent à la fois la dynamique lente centre-guide et la dynamique rapide du gyro-angle. Diverses méthodes de réduction sont utilisées, développées ou introduites, e.g. une transformée de Lie du mouvement, un relèvement transférant les réductions de la dynamique des particules à la dynamique des champs, ou une troncature reliée à la fois à la théorie des contraintes de Dirac et à une projection sur une sous-algèbre. Outre la gyrocinétique, cela clarifie d'autres réductions hamiltoniennes en plasmas, e.g. pour une dynamique incompressible ou électrostatique, pour la MHD, ou pour des fermetures fluides avec tenseur de pression
Gyrokinetics is a key model for plasma micro-turbulence. It still suffers from several issues, which could imply to reconsider the equations. This thesis dissertation clarifies three of them. First, one of the coordinates caused questions, both from a physical and from a mathematical point of view; a suitable constrained coordinate is introduced, which removes the issues from the theory and explains the intrinsic structures underlying the questions. Second, explicit induction relations are obtained to go arbitrary order in the perturbative expansion. Third, using the Hamiltonian structure of the dynamics, the coupling between the plasma and the electromagnetic field is implemented in a more appropriate way, with strong consequences on the gyrokinetic equations. Several other results are obtained, for instance about the origin of the guiding-center adiabatic invariant, about a very efficient minimal guiding-center transformation, or about an intermediate Hamiltonian model between Vlasov-Maxwell and gyrokinetics, where the characteristics include both the slow guiding-center dynamics and the fast gyro-angle dynamics. In addition, various reduction methods are used, introduced or developed, e.g. a Lie-transform of the equations of motion, a litfing method to transfer particle reductions to the corresponding Hamiltonian field dynamics, or a truncation method related both to Dirac's theory of constraints and to projections onto Lie-subalgebras. Besides gyrokinetics, this is useful to clarify other Hamiltonian reductions in plasma physics, e.g. for incompressible or electrostatic dynamics, for magnetohydrodynamics, or for fluid closures including moments of order two

La composition interne de notre planète est un vaste sujet d’étude auquel participent de nombreuses disciplines scientifiques. Les conditions extrêmes de pression et de température qui règnent à l’intérieur du noyau (constitué principalement de fer et de nickel) et du manteau terrestre (à base de pérovskites) rendent très difficile la détermination de leur compositions exactes. Ce projet de thèse contribue aux études récentes dont l’enjeu est de déterminer plus précisément le chimisme des minéraux présents. Il a pour objet le développement d’un outil de calcul des vitesses de propagation des ondes sismiques a l’aide d’une méthode fondée sur les simulations ab initio. Ces vitesses sont déduites du tenseur élastique complet, incluant la relaxation atomique et les modifications induites du champ cristallin. Nous utilisons l’approche de la théorie de perturbation de la fonctionnelle de la densité (DFPT) qui permet de s'affranchir des incertitudes numériques qu’impliquent les méthodes classiques basées sur des différences finies. Nous combinons cette approche avec le formalisme « Projector Augmented-Wave » (PAW) qui permet, avec un coût de calcul faible, de prendre en compte tous les électrons du système. Nous avons appliqué la méthode sur des matériaux du noyau et du manteau terrestre. Nous avons déterminé les effets de différents éléments légers (Si, S, C, O et H) sur les vitesses de propagation des ondes sismiques dans le fer pur ainsi que celui de l’aluminium dans la pérovskite MgSiO3
The internal composition of our planet is a large topic of study and involves many scientific disciplines. The extreme conditions of pressure and temperature prevailing inside the core (consisting primarily of iron and nickel) and the mantle (consisting mainly of perovskites) make the determination of the exact compositions very difficult. This thesis contributes to recent studies whose aim is to determine more accurately the chemistry of these minerals. Its purpose is the development of a tool for the calculation of seismic wave velocities within methods based on ab-initio simulations. These velocities are calculated from the full elastic tensor, including the atomic relaxation and induced changes in the crystal field. We use the approach of the density functional perturbation theory (DFPT) to eliminate numerical uncertainties induced by conventional methods based on finite differences. We combine this approach with the « Projector Augmented-Wave » (PAW) formalism that takes into account all the electrons of the system with a low computational cost. We apply the method on core and mantle materials and we determine the effects of various lights elements (Si, S, C, O and H) on the seismic wave velocities of pure iron, as well as the effect of aluminum in the perovskite MgSiO3

In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning.

Computation of parameters often includes as a part solution of linear system of equations Ax = b. The corresponding pseudoinverse solution depends on the properties of the matrix A and vector b. The singular value decomposition of A can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution x. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of A into account

O comportamento infra-vermelho dos propagadores de glúons e de ghosts é de fundamental importância para o entendimento do limite de baixas energias da cromodinâmica quântica (QCD), especialmente no que diz respeito ao problema do confinamento de quarks e de glúons. O objetivo desta tese é implementar um novo método para o estudo do propagador de glúons no calibre covariante linear para a QCD na rede. Em particular, analisamos em detalhe a nova implementação proposta e estudamos os algoritmos para fixação numérica deste calibre. Note que a fixação numérica da condição de calibre de Feynman apresenta vários problemas não encontrados nos casos de Landau e de Coulomb, o que impossibilitou por longo tempo o seu estudo adequado. De fato, a definição considerada inicialmente, por Giusti et. al., é de difícil implementação numérica e introduz condições espúrias na fixação de calibre. Como consequência, os únicos estudos efetuados anteriormente referem-se aos propagadores de glúons e de quarks em redes relativamente pequenas, não permitindo uma análise cuidadosa do limite infra-vemelho da QCD neste calibre. A obtenção de novas soluções para a implementação do calibre de Feynman na rede é portanto de grande importância para viabilizar estudos numéricos mais sistemáticos dos propagadores e dos vértices neste calibre e, em geral, no calibre covariante linear.
The infrared behavior of gluon and ghost propagators is of fundamental importance for the understanding of the low-energy limit of quantum chromodynamics (QCD), especially with respect to the problem of the confinement of quarks and gluons. The goal of this thesis is to implement a new method to study the gluon propagator in the linear covariant gauge in lattice QCD. In particular, we analyze in detail the newly proposed implementation and study the algorithms for numerically fixing this gauge. Note that the numerical fixing of the Feynman gauge condition poses several problems that are not present in the Landau and Coulomb cases, which prevented it from being properly studied for a long time. In fact, the definition considered initially, by Giusti et. al., is of difficult numerical implementation and introduces spurious conditions into the gauge fixing. As a consequence, the only studies carried out previously involved gluon and quark propagators on relatively small lattices, hindering a careful analysis of the infrared limit of QCD in this gauge. Obtaining new solutions for the implementation of the Feynman gauge on the lattice is therefore of great importance to enable more systematic numerical studies of propagators and vertices in this gauge and, in general, in the linear covariant gauge.

Neste trabalho estudamos os modelos de Gross-Neveu e Sigma Não-Linear empregando a expansão 1/N no espaço não-comutativo. Consideramos três formalismos distintos para implementar a não-comutatividade. Dois deles tratam a não-comutatividade através da deformação do produto entre funções, porém, um dos formalismos é invariante de Lorentz e o outro não. O terceiro método trata a não-comutatividade através dos estados coerentes. Nesses diferentes contextos, calculamos o propagador do campo auxiliar de ambos os modelos, além da correção radiativa para o propagador do campo básico no modelo Sigma Não-Linear.
ln this work we studied the Gross-Neveu and nonlinear sigma models employing the 1/N expansion in the non-commutative espace. We describe three different manner of introduce non-commutativity. Two of them treat the non-commutativity through the deformation of the product of functions, however one of the method is Lorentz invariant and the other not. The third manner treat the non-commutativity through coherent states. We calculated the propagator of the auxiliary field in both models in these different settings and, besides that, we compute the leading radiative correction to propagator of the basic field of the nonlinear sigma model.

碩士
國立中央大學
數學系
101
We now go into the perturbation theory for the eigenvalue prob- lem in a nite-dimensional vector space X over C. A typical prob- lem of this theory is to investigate how the eigenvalues and eigen- vectors (or eigenspaces) of a linear operator T change when T is subjected to a small perturbation. We give introduction in the rst chapter , methametical basic theory and the knowledge in the sec- ond chapter and proofs in the third chapter. This article is mainly about Kato's book "Perturbation Theory of Linear Operators" and how to understand the theory and the content of the book through examples.

The main goal of this thesis is to present cosmological perturbation theory (based on the standard Friedmann cosmological model) in volume-preserving coordinates, which then provides a suitable basis for studies in cosmological averaging. We review perturbation theory to second order, allowing for averaging to second order in future research. To solve the averaging problem we need a method of covariantly and gauge invariantly averaging tensorial objects on a background manifold. This is a very difficult problem. However, the definition of an average takes on a particularly simple form when written in a system of volume-preserving coordinates. Therefore, we develop a three dimensional and a four dimensional volume-preserving coordinate gauge in this thesis that can be used for averaging in cosmological perturbation theory.

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework.
Mathematical Sciences
M. Sc. (Applied Mathematics)

We present a self-consistent calculation of the mono vacancy formation energy for seven simple metals Li, Na, K, Rb, Cs (all bcc), Al (fcc) and Mg (hcp) using both model and ab initio pseudopotential used in earlier unified studies. The local model pseudopotential calculations show small variations with respect to different exchange-correlation functions and the results are in fair agreement with other similar calculations and the available experimental data. The comparisons show that reliable model (pseudo) potential for simple metals can indeed be obtained for explaining a host of properties. Also, considering the importance of third order term in ab initio calculations, the results of our second order calculation appear fairly reasonable and are comparable with other first principle calculations. The perturbation series being an oscillating one, we hope to improve the calculational results using suitable series convergence acceleration method in the next part of our study.

We consider linear perturbations of the background type IIB SUGRA solutions and find the equations of motion for the moduli. In particular, we allow for spacetime fluctuations of the positions of D3-branes in the compact dimensions. We postulate an ansatz for the 5-form flux due to the motion of the D3-branes, and a corresponding first-order part of the metric. The movement of the D3-branes is then shown to affect the warp factor at linear order. Using the equations of motion for the D3-branes, the universal volume modulus, and the universal axion, we construct a second-order, effective action. Finally, based on the form of the effective action, we examine a Kahler potential for the moduli space.

Research Doctorate - Doctor of Philosophy (PhD)
Specific ion effects (SIEs) encompass any phenomenon induced by ions that is dependent on the identity of the ions, and not just their charge or concentration. These occur in salts, electrolyte solutions, ionic liquids, acids and bases and have been known for over 130 years, from which the Hofmeister series originated. They are important in biology, nutrition, electrochemistry and various interfacial or geophysico-phenomena. It is perhaps harder to find a “real-world” system in which specific ion effects don’t occur, than systems where they do. Nonetheless, despite such ubiquity and effect on our daily lives, our understanding of these salty solutions is limited. This thesis addresses the knowledge gap surrounding the lack of parameters (for both ion and solvent) for quantifying SIEs in aqueous and nonaqueous environments. This thesis begins with a deeper introduction to the topic of SIEs and highlights the current state-of-play. The theories underlying quantum mechanics and computational chemistry are discussed to highlight how they may be applied to elucidate some of the fundamental origins of SIEs. These methods were subsequently used to investigate possible energetic origins of counterion and solvent induced reversals to the Hofmeister series, and highlights that the Lewis acidity and basicity (collectively Lewis strength) indices of the cations and anions respectively, can quantify SIEs. Following this revelation, these empirical parameters were recast in terms of intermolecular forces. Electrostatics appeared to govern the Lewis strength indices, so these were replaced with an electrostatic parameter, ϸ (“sho”), that originates from Coulomb’s Law. For anions, ϸ is shown to quantify SIE trends observed in enthalpies of hydration, polymer lower critical solution temperatures, enzyme and viral activities, SN2 reaction rates and Gibbs free energies of transfer from water to nonaqueous solvents highlighting the versatility of ϸ as a new SIE parameter. Cation interactions are more prone to deviations from ϸ correlations. In the absence of any cosolute (i.e., pure ion-solvent interactions) however, cation solvent interactions follow a strong trend with Coulomb’s Law for ~15 different solvents. This supports a conclusion that competing electrostatic interactions between the solvent and a cosolute for the cation may mask each other allowing non-electrostatic contributions to play a dominant role. Furthermore, with similarity to the ion parameterisation, the ϸ values at the negative and positive solvent dipolar atoms correlate with the solvent’s Lewis basicity and acidity respectively. Additionally, these analyses can be related to macroscopic solvent parameters such as the relative permittivity. The data deficiency issue facing the SIE field was more generally addressed in this thesis by the generation of IonSolvR, a repository containing over 3000 distinct QM/MD trajectories of up to 52 ions in 28 bulk solvents on nanosecond-scales. Finally, the key findings of this thesis are summarised and an outlook on the field of SIEs and the broader implications arising from this thesis is presented.