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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

On small scales, the universe is inhomogeneous. However, on scales many times greater than the average distances between galaxies, the distribution of galaxies, galaxy clusters, and dark matter begins to look more uniform throughout the universe. This study aims to analyze the large-scale structure of matter in the universe by looking at the time evolution and spatial development of the linear perturbations to the average density and average velocity of matter in the universe. These linear perturbations convey information about structure formation and distribution in the cosmos. In particular, this research investigates how retaining decaying terms and rotational velocities in the calculations ( which are often ignored to facilitate the mathematics) affects the higher order terms in the density and velocity perturbations for a matter-dominated universe. On such large scales, the matter in the universe is assumed to behave like a pressure-less fluid permeating the cosmos. The gravitational instability model and the Newtonian fluid flow equations are used as bases for analyzing the density and velocity perturbations. So far, results show that retaining decaying terms and rotational velocities when solving for the density and velocity perturbations produces possibly significant terms that are otherwise overlooked. In the next step of this project, the statistics describing the spatial distribution of the density and velocity perturbations, in particular the two-point correlation functions, will be calculated and investigated.

We study the masses of the low-lying charm and bottom mesons within the framework of heavy-hadron chiral perturbation theory. We work to third order in the chiral expansion, where meson loops contribute. In contrast to previous approaches, we use physical meson masses in evaluating these loops. This ensures that their imaginary parts are consistent with the observed widths of the D-mesons. The lowest odd- and even-parity, strange and non-strange mesons provide enough constraints to determine only certain linear combinations of the low-energy constants (LECs) in the effective Lagrangian. We comment on how lattice QCD could provide further information to disentangle these constants. Then we use the results from the charm sector to predict the spectrum of odd- and even-parity of the bottom mesons. The predicted masses from our theory are in good agreement with experimentally measured masses for the case of the odd-parity sector. For the even-parity sector, the $B$-meson states have not yet been observed; thus, our results provide useful information for experimentalists investigating such states. The near degeneracy of nonstrange and strange scalar $B$ mesons is confirmed in our predictions using $\mathrm{HHChPT}$. Finally,we show why previous approaches of using $\mathrm{HHChPT}$ in studying the mass degeneracy in the scalar states of charm and bottom meson sectors gave unsatisfactory results. Interactions between these heavy mesons are treated using effective theories similar to those used to study nuclear forces. We first look at a strongly-interacting channel which produces a bound or virtual state and a dimer state which couples weakly to a weakly-interacting channel to produce a narrow resonance. We also look at the short-range interactions in two channels. We consider two cases: two channels where one has a strong $s$-wave interaction which produces bound or virtual states, and a dimer state which couples weakly to weakly-coupled channels which in turn can produce narrow resonances. For each of these systems, we use well-defined power-counting schemes. The results can be used to investigate resonances in the charmonium and bottomonium systems. We demonstrate how the method can be applied to the $X(3872)$. The widths of the $X(3872)$ for decay processes to $\bar{D}^0 D^{*0}$ and $\bar{D}^0D^0\pi$ are calculated. We use these results to obtain the line shapes of the $X(3872)$ under different assumptions about the nature of this state.

The binary neutron star merger of 2017 and its corresponding electromagnetic signature resembling that of a kilonova has been one of the most groundbreaking astrophysical occurrences in the last decade. Indications of r-process nucleosynthesis in this event presents new opportunities for learning about the astrophysical origin of heavy elements. This has been a long-standing mystery in our understanding of the chemical evolution of the universe. This leads to the requirement of developing more accurate calculations of the corresponding atomic properties. In this work, I have studied the potential of utilizing a combined configuration-interaction and many-body perturbation theory approach.The goal is to study if a generalized and computationally efficient method is possible with this approach, so that it can be used to develop accurate and complete atomic structures applicable to any ion of any element in the periodic table. Focusing on the lanthanide group of elements, and in Ce in particular, the method in this work builds on including the bulk of strong correlation contributions in the configuration-interaction model, complemented with perturbation theory corrections in subsequent many-body perturbation theory calculations. For all Ce ions, this was however not possible without having to compromise the completeness condition. In conclusion, in the scope of this project, we find that it is challenging to generalize a procedure for near-neutral systems due to the amount of correlation needed to be treated with configuration-interaction. / <p>Project made as part of the course: "Project in physics and astronomy - 1FA195, 15 ECTS" at Uppsala University</p>

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Crossing symmetry relates scattering and annihilation processes to each other. Its derivation is straightforward in perturbative approaches to quantum field theory: it merely reflects the exchange of in- and outgoing states in Feynman diagram computations. In soliton models, the situation is much more complicated because the scattering and the annihilation processes concern distinct topological sectors that are not related by any continuous transformation. In this thesis a simple soliton model will be employed to address this problem numerically. First, in the unit topological sector we extract asymptotically the phase shift of the scattering process of a wave packet off the kink-solution. To this end we solve the time-dependent equation of motion of the non-integrable '4 field model in (1+1) spacetime dimensions for two distinct initial conditions: the wave packet in a trivial vacuum background and in the background of the kink-solution. Second, in the topologically trivial sector we present numerical solutions of the kink– antikink interaction in the same model. We find that the final state of this interaction varies dramatically with the impact velocity. As result, we analyze our numerical solutions for the kink–antikink collisions system in two regimes. For the initial velocity of the system less than some critical velocity, vc 0:26, the kink and the antikink either annihilate or inelastically scatter. On the other hand, the kink and the antikink always inelastically scatter when the initial velocity of the system is higher than this critical velocity. However, the scattering processes of the kink–antikink with initial velocity below and above the critical velocity are different. Below the critical velocity the kink and the antikink collide and always undergo n-bounces (n 2) before they depart to infinity. When the initial velocity of the system is higher than vc, the kink and the antikink depart to infinity after only one bounce. We present a qualitative description for these bounce effects between the kink and the antikink motivated by earlier studies as well as our numerical simulations. We utilize collective coordinates to study the dynamics of the kink–antikink system in two degrees of freedom. In this regime, we modify the ansätze of the kink–antikink system from earlier studies to account for relativistic effects. We perform a comparison between this approximation and the full system. We end our discussion of this sector by discussing the scattering data for the inelastic scattering and the annihilation processes of the kink–antikink. Third, we compare the extracted scattering data for the scattering process of a wave packet off the kink-solution and the annihilation process of the kink–antikink to each other. Finally, these studies of different sectors allow us to make a conjecture about the validity of crossing symmetry within the non-integrable '4 field model. / AFRIKAANSE OPSOMMING: Kruising-simmetrie beskryf ’n verband tussen verstrooiings- en vernietigingsprosesse. Die afleiding daarvan binne die raamwerk van steuringsteorie is eenvoudig: dit behels bloot die omruil van ingaande en uitgaande toestande in die Feynman-diagram. In soliton-modelle is die situasie egter meer ingewikkeld aangesien die verstrooiings- en vernietigingsprosesse in verskillende topologiese sektore plaasvind wat nie deur kontinue transformasies aan mekaar gekoppel is nie. In hierdie tesis word daar van ’n eenvoudige soliton-model gebruik gemaak om hierdie probleem numeries te ondersoek. Eerstens word die faseverskuiwing van die verstrooiingsproses van ’n golfpakkie vanaf ’n kinkoplossing asimptoties in die topologiese eenheidssektor bepaal. Vir hierdie doel word die tydafhanklike bewegingsvergelykings van die klassieke, nie-integreerbare 4-veldeteorie in (1+1) dimensionele ruimte-tyd opgelos. Twee beginkondisies word ondersoek: ’n golfpakkie in die triviale vakuum agtergrond asook in die kinkoplossing agtergrond. Tweedens ondersoek ons ook numeriese oplossings vir die kink-antikink wisselwerking binne die triviale topologiese sektor van dieselfde model. Hier vind ons dat die finale toestand van hierdie wisselwerkingsproses op ’n uiters sensitiewe wyse van die impaksnelheid afhang. Ons ondersoek gevolglik die numeriese oplossings vir die kink-antikink botsings in twee gebiede. Vir beginsnelhede onder die kritieke snelheid vc 0:26 sal die kink en antikink mekaar óf vernietig óf nie-elasties verstrooi. In teenstelling hiermee sal die kink-antikink altyd nie-elastiese verstrooiing ondergaan as die beginsnelheid die kritieke snelheid oorskry. Die aard van die verstrooiingsprosesse vir beginsnelhede bo en onder die kritieke snelheid is egter verskillend. Onder die kritieke snelheid sal die kink en antikink ’n n-bots proses (n 2) ondergaan voor hulle finaal van mekaar weg beweeg. Bo die kritieke snelheid sal die kink-antikink egter net ’n enkele botsing ondergaan en dan uitmekaar beweeg. Ons lewer ’n kwalitatiewe beskrywing vir die bons-effek tussen die kink en antikink wat deur vorige studies asook ons numeriese resultate gemotiveer word. Ons maak gebruik van ’n kollektiewe koördinaatstelsel om die dinamika van die kink-antikink in terme van twee vryheidsgrade te bestudeer. In hierdie gebied pas ons ook die ansatz vir die kink-antikink stelsel aan om relatiwistiese effekte in ag te neem. Ons vergelyk dan hierdie benadering met die oplossing van die volle sisteem. Die bespreking van hierdie sektor word afgesluit met ’n analise van die verstrooiingsdata vir die verstrooiing- en vernietingsprosesse van die kink-antikink. Derdens vergelyk ons die verstrooiingsdata vir die verstrooiing van ’n golfpakkie vanaf ’n kinkoplossing met die van die vernietigingsproses van die kink-antikink. Ons studie van die verskillende sektore laat ons dan toe om ’n vermoede te formuleer oor die geldigheid van kruissing-simmetrie binne die nie-integreerbare 4-model.

There are many problems that lead to analysis of dynamical systems with phase variables of two types, slow and fast ones. Such systems are called slow-fast systems. The dynamics of such systems is usually described by means of different versions of perturbation theory. Many questions about accuracy of this description are still open. The difficulties are related to presence of resonances. The goal of the proposed thesis is to establish some estimates of the accuracy of the perturbation theory for slow-fast systems in the presence of resonances. We consider slow-fast Hamiltonian systems and study an accuracy of one of the methods of perturbation theory: the averaging method. In this thesis, we start with the case of slow-fast Hamiltonian systems with two degrees of freedom. One degree of freedom corresponds to fast variables, and the other degree of freedom corresponds to slow variables. Action variable of fast sub-system is an adiabatic invariant of the problem. Let this adiabatic invariant have limiting values along trajectories as time tends to plus and minus infinity. The difference of these two limits for a trajectory is known to be exponentially small in analytic systems. We obtain an exponent in this estimate. To this end, by means of iso-energetic reduction and canonical transformations in complexified phase space, we reduce the problem to the case of one and a half degrees of freedom, where the exponent is known. We then consider a quasi-linear Hamiltonian system with one and a half degrees of freedom. The Hamiltonian of this system differs by a small, ~ε, perturbing term from the Hamiltonian of a linear oscillatory system. We consider passage through a resonance: the frequency of the latter system slowly changes with time and passes through 0. The speed of this passage is of order of ε. We provide asymptotic formulas that describe effects of passage through a resonance with an improved accuracy O(ε3/2). A numerical verification is also provided. The problem under consideration is a model problem that describes passage through an isolated resonance in multi-frequency quasi-linear Hamiltonian systems. We also discuss a resonant phenomenon of scattering on resonances associated with discretisation arising in a numerical solving of systems with one rotating phase. Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. For arbitrarily small time step of a numerical method, discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances, that is absent in the original system.

Galaxy redshift surveys are powerful probes of cosmology. Yet, in order to fully exploit the information contained in galaxy surveys, we need to improve upon our understanding of the structure formation in the Universe. Galaxies are formed/observed at late times when the density field is no longer linear so that understanding non-linearities is essential. In this thesis, we show that, at high redshifts, we can accurately model the galaxy power spectrum in redshift space by using the standard cosmological perturbation theory. Going beyond the power spectrum, we can use the three-point function, or the bispectrum, to gain important information on the early universe as well as on the galaxy formation via measurements of primordial non-Gaussianity and galaxy bias. We show that the galaxy bispectrum is more sensitive to primordial non-Gaussianities than previously recognized, making high-redshift galaxy surveys a particularly potent probe of the physics of inflation. Weak lensing offers yet another way of probing cosmology. By cross correlating the angular position of galaxies with the shear measurement from galaxy lensing or CMB lensing, we also show that one can obtain the information on cosmological distance scale, the galaxy bias, and the primordial non Gaussianity from weak lensing method. / text

Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well conditioned L factor. We then establish relative perturbation bounds for the inverse that are entrywise and independent of the condition number. This allows us to also present relative perturbation bounds for the linear system Ax=b that are independent of the condition number. Lastly, we continue the work of Ye to provide relative perturbation bounds for the eigenvalues of symmetric indefinite matrices and non-symmetric matrices.

Analytical solutions to complex trajectory design problems are scarce, since only a few specific cases allow for closed-form solutions. The main purpose of this dissertation is to design simple algorithms for trajectory design using continuous thrust, with a focus on low-thrust applications. By “simple” here we seek to achieve algorithms that either admit an analytical solution, or require minimal input by the user and minimal computation time. The three main contributions of this dissertation are: designing Lyapunov-based closed-loop guidance laws for orbit transfers, finding semi-analytical solutions using a constant magnitude thrust, and perturbation theory for approximate solutions to low-thrust problems. The technical aspect that these problems share in common is that they all use, fully or partially, a thrusting model in which the energy of the system is kept constant. Many orbit transfer problems are shown to be solved with this thrusting protocol. / text