Despite the wide acceptance that errors incurred in numerical solutions to N-body systems grow exponentially, most research assumes that the statistical results of these systems are reliable. However, if one is to accept that the statistical results of N-body solutions are reliable, it is important to determine if there are any systematic statistical errors resulting from the incurred growth of errors in individual solutions. In this thesis we consider numerical solutions to the 3-body problem in which one of the bodies escapes the system. It is shown for a particular 3-body con guration, known as the Sitnikov problem, that the mean lifetime of the system is dependent on the accuracy of the numerical integration. To provide a theoretical explanation of the phenomenon, an approximate Poincar´e map is developed whose dynamics on a particular surface of section is shown to be similar to the dynamics of the Sitnikov Problem. In fact there is a set on which the approximate Poincar´e map is topologically equivalent, like the Sitnikov Problem, to the shift map on the set of bi-in nite sequences. The structure of the escape regions on the surface of section form a cantor set-like structure whose boundary can more easily be delineated using the approximate Poincar´e map than for the Sitnikov problem. Further it is shown that numerical errors destroy escape regions and can cause orbits to migrate to a region in which escape is faster. Finally, a relationship between the Lyapunov time, tl, and the lifetime, td, of the 3-body problem is discussed. Firstly, the Sitnikov problem and the approximate Poincar´e map of the Sitnikov problem both exhibit a two-part power law relationship beween tl and td like that for a particular case of the general 3-body problem. Further, it is demonstrated that large perturbations to the energy of the escaping body in uences the relationship between tl and td for small tl. Finally, it is shown that the approximate Poincar´e map yields a theoretical explanation of the phenomenon based on the structure of the escape regions the orbits traverse before escape.

The rhombus and parallelogram orbits are interesting families of periodic solutions, which come from celestial mechanics and the N-body problem. Variational methods with finite order symmetry group are used to construct minimizing non-collision periodic orbits. We study the question of stability or instability of periodic and symmetric periodic solutions of the rhombus and the equal mass parallelogram four body problems. We first study the stability of periodic solutions for the rhombus four body problem. An analytical description of the variational principle is used to show that the homographic solutions are the minimizers of the action functional restricted to rhombus loop space [23]. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing rhombus orbit to prove the main result, Theorem 4.2.2, which states that the reduced rhombus orbit is hyperbolic in the reduced energy manifold when it is not degenerate. We second study the stability for symmetric periodic solutions of the equal mass parallelogram four body problem. The parallelogram family is a family of Z_2× Z_4 symmetric action minimizing solutions, investigated by [7]. In this example, the minimizing solution [7] can be extended to a 4T-periodic solution using symmetries through square and collinear configurations. The Maslov index of the orbits is used to prove the main result, Theorem 5.3.1, which states that the minimizing equal mass parallelogram solution is unstable when it is non-degenerate. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-11-26 11:30:29.688

This work considers the organization and performance of computations on parallel computers of tree algorithms for the N-body problem where the number of particles is on the order of a million. The N-body problem is formulated as a set of recursive equations based on a few elementary functions, which leads to a computational structure in the form of a pyramid-like graph, where each vertex is a process, and each arc a communication link. The pyramid is mapped to three different processor configurations: (1) A pyramid of processors corresponding to the processes pyramid graph; (2) A hypercube of processors, e.g., a connection-machine like architecture; (3) A rather small array, e.g., $2 \\times 2 \\ times 2$, of processors faster than the ones considered in (1) and (2) above. Simulations of this size can be performed on any of the three architectures in reasonable time.

Cosmological simulations show that dark matter halos contain a wealth of substructure. These subhalos are assumed have a mass distribution that extends down to the smallest mass in the Cold Dark Matter (CDM) hierarchy, which lies below the current resolution limit of simulations. Substructure has important ramifications for indirect dark matter detection experiments as the signal depends sensitively on the small-scale density distribution of dark matter in the Galactic halo. A clumpy halo produces a stronger signal than halos where the density is a smooth function of radius. However, the small-scale Universe presents a daunting challenge for models of structure formation. In the CDM paradigm, structures form in a hierarchical fashion, with small-scale perturbations collapsing first to form halos that then grow via mergers. However, near the bottom of the hierarchy, dark matter structures form nearly simultaneously across a wide range of scales. To explore these small scales, I use a series of simulations of scale-free cosmological models, where the initial density power spectrum is a power-law. I can effectively examine various scales in the Universe by using the index in these artificial cosmologies as a proxy for scale. This approach is not new, but my simulations are larger than previous such simulations by a factor of 3 or more. My results call into question the often made assumption that the subhalo population is scale-free. The subhalo population does depend on the mass of the host. By combining my study with others, I construct a phenomenological model for the subhalo mass function. This model shows that the full subhalo hierarchy does not greatly boost the dark matter annihilation flux of a host halo. Thus, the enhancement of the Galactic halo signature due to substructure can not alone account the observed flux of cosmic rays produced by annihilating dark matter. Finally, I examine the nonlinear power spectrum, which is used to determine cosmological parameters based on large-scale, observational surveys. I find that in this nonlinear regime, my results are not consistent with currently used fitting formulae and present my own empirical formula. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2009-09-25 01:01:39.714

<div><div><div><p>Several socially and economically important real-world systems comprise large numbers of interacting constituent entities. Examples include the World Wide Web and Online Social Networks (OSNs). Developing the capability to forecast the macroscopic behavior of such systems based on the microscopic interactions of the constituent parts is of considerable economic importance.</p><p>Previous researchers have investigated phenomenological forecasting models in such contexts as the spread of diseases in the real world and the diffusion of innovations in the OSNs. The previous forecasting models work well in predicting future states of a system that are at equilibrium or near equilibrium. However, forecasting non-equilibrium states – such as the transient emergence of hotspots in web traffic – remains a challenging problem. In this thesis we investigate a hypothesis, rooted in Ludwig Boltzmann's celebrated H-theorem, that the evolutionary dynamics of a large system – such as the World Wide Web – is driven by the system's innate tendency to evolve towards a state of maximum entropy.</p><p>Whereas closed systems may be expected to evolve towards a state of maximum entropy, most real-world systems are not closed. However, the stipulation that if a system is closed then it should asymptotically approach a state of maximum entropy provides a strong constraint on the inverse problem of formulating the microscopic interaction rules that give rise to the observed macroscopic behavior. We make the constraint stronger by insisting that, if closed, a system should evolve monotonically towards a state of maximum entropy and formulate microscopic interaction rules consistent with the stronger constraint.</p><p>We test the microscopic interaction rules that we formulate by applying them to two real world phenomena: the flow of web traffic in the gaming forums on Reddit and the spread of Covid-19 virus. We show that our hypothesis leads to a statistically significant improvement over the existing models in predicting the traffic flow in gaming forums on Reddit. Our interaction rules are also able to qualitatively reproduce the heterogeneity in the number of COVID-19 cases across the cities around the globe. The above experiments provide supporting evidence for our hypothesis, suggesting that our approach is worthy of further investigation.</p><p>In addition to the above stochastic model, we also study a deterministic model of attention flow over a network and establish sufficient conditions that, when met, signal imminent parabolic accretion of attention at a node<br></p></div></div></div>

This thesis explores topics related to the formation and development of the large-scale structure in the Universe, with the focus being to compute properties of the evolved non-linear density field in an approximate way. The first three chapters form an introduction: Chapter 1 contains the theoretical basis of modern cosmology, Chapter 2 discusses the role of N-body simulations in the study of structure formation and Chapter 3 considers the phenomenological halo model. In Chapter 4 a novel method of computing the matter power spectrum is developed. This method uses the halo model directly to make accurate predictions for the matter spectrum. This is achieved by fitting parameters of the model to spectra from accurate simulations. The final predictions are good to 5% up to k = 10 hMpc-1 across a range of cosmological models at z = 0, however accuracy degrades at higher redshift and at quasi-linear scales. Chapter 5 is dedicated to a new method of rescaling a halo catalogue that has previously been generated from a simulation of a specific cosmological model to a different model; a gross rescaling of the simulation box size and redshift label takes place, then individual halo positions are modified in accord with the large scale displacement field and their internal structure is altered. The final power spectrum of haloes can be matched at the 5% level up to k = 1 hMpc-1, as can the spectrum of particles within haloes reconstituted directly from the rescaled catalogues. Chapter 6 applies the methods of the previous two chapters to modified gravity models. This is done in as general a way possible but tests are restricted to f(R) type models, which have a scale-dependent linear growth rate as well as having 'chameleon screening' - by which modifications to gravity are screened within some haloes. Taking these effects into account leads to predictions of the matter spectrum at the 5% level and rescaled halo distributions that are accurate to 5% in both real and redshift space. For the spectrum of halo particles it is demonstrated that accurate results may be obtained by taking the enhanced gravity in some haloes into account.

We use Bayesian statistics and Markov chain Monte Carlo (MCMC) techniques to construct dynamical models for the spiral galaxy NGC 6503. The constraints include surface brightness profiles which display a Freeman Type II structure; HI and ionized gas rotation curves; the stellar rotation, which is nearly coincident with the ionized gas curve; and the line of sight stellar dispersion, which displays a $\sigma-$drop at the centre. The galaxy models consist of a S\'rsic bulge, an exponential disc with an optional inner truncation and a cosmologically motivated dark halo. The Bayesian/MCMC technique yields the joint posterior probability distribution function for the input parameters, allowing constraints on model parameters such as the halo cusp strength, structural parameters for the disc and bulge, and mass-to-light ratios. We examine several interpretations of the data: the Type II surface brightness profile may be due to dust extinction, to an inner truncated disc or to a ring of bright stars; and we test separate fits to the gas and stellar rotation curves to determine if the gas traces the gravitational potential. We test each of these scenarios for bar stability, ruling out dust extinction. We also find that the gas cannot trace the gravitational potential, as the asymmetric drift is then too large to reproduce the stellar rotation. The disc is well fit by an inner-truncated profile, but the possibility of ring formation by a bar to reproduce the Type II profile is also a realistic model. We further find that the halo must have a cuspy profile with $\gamma \gtrsim 1$; the bulge has a lower $M/L$ than the disc, suggesting a star forming component in the centre of the galaxy; and the bulge, as expected for this late type galaxy, has a low S\'{e}rsic index with $n_b\sim1-2$, suggesting a formation history dominated by secular evolution. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2010-01-10 00:11:41.946

In this dissertation, we explore parallel algorithms for general N-Body problems in high dimensions, and their applications in machine learning and image analysis on distributed infrastructures. In the first part of this work, we proposed and developed a set of basic tools built on top of Message Passing Interface and OpenMP for massively parallel nearest neighbors search. In particular, we present a distributed tree structure to index data in arbitrary number of dimensions, and a novel algorithm that eliminate the need for collective coordinate exchanges during tree construction. To the best of our knowledge, our nearest neighbors package is the first attempt that scales to millions of cores in up to a thousand dimensions. Based on our nearest neighbors search algorithms, we present "ASKIT", a parallel fast kernel summation tree code with a new near-far field decomposition and a new compact representation for the far field. Specially our algorithm is kernel independent. The efficiency of new near far decomposition depends only on the intrinsic dimensionality of data, and the new far field representation only relies on the rand of sub-blocks of the kernel matrix. In the second part, we developed a Bayesian inference framework and a variational formulation for a MAP estimation of the label field for medical image segmentation. In particular, we propose new representations for both likelihood probability and prior probability functions, as well as their fast calculation. Then a parallel matrix free optimization algorithm is given to solve the MAP estimation. Our new prior function is suitable for lots of spatial inverse problems. Experimental results show our framework is robust to noise, variations of shapes and artifacts.

We consider the curved N-body problem, N > 2, on a surface of constant Gaussian curvature κ ≠ 0; i.e., on spheres S2κ, for κ > 0, and on hyperbolic manifolds H2κ, for κ < 0. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant during the motion. We find new relative equilibria in the curved N-body problem for N = 4, and see whether bifurcations occur when passing through κ = 0. After obtaining a criterion for the existence of quadrilateral configurations on the equator of the sphere, we study two restricted 4-body problems: One in which two bodies are massless , and the second in which only one body is massless. In the former we prove the evidence for square-like relative equilibria, whereas in the latter we discuss the existence of kite-shaped relative equilibria. We will further study the 5-body problem on surfaces of constant curvature. Four of the masses arranged at the vertices of a square, and the fifth mass at the north pole of S2κ, when the curvature is positive, it is shown that relative equilibria exists when the four masses at the vertices of the square are either equal or two of them are infinitesimal, such that they do not affect the motion of the remaining three masses. In the hyperbolic case H2κ, κ < 0, there exist two values for the angular velocity which produce negative elliptic relative equilibria when the masses at the vertices of the square are equal. We also show that the square pyramidal relative equilibria with non-equal masses do not exist in H2κ. Based on the work of Florin Diacu on the existence of relative equilibria for 3-body problem on the equator of S2κ, we investigate the motion of more than three bodies. Furthermore, we study the motion of the negative curved 2-and 3-centre problems on the Poincaré upper semi-plane model. Using this model, we prove that the 2-centre problem is integrable, and we study the dynamics around the equilibrium point. Further, we analyze the singularities of the 3- centre problem due to the collision; i.e., the configurations for which at least two bodies have identical coordinates. / Graduate

>Magister Scientiae - MSc / We quantitatively examine the effects of merger and environment within a cosmological hydrodynamic simulation. We show that our simulation model broadly reproduces the observed scatter in H I at a given stellar mass as quantified by the HI mass function in bins of stellar mass, as well as the H I richness versus local galaxy density. The predicted H I fluctuations and environmental effects are roughly consistent with data, though some discrepancies are present at group scales. For satellite galaxies in & 1012Mhalos, the H I richness distribution is bimodal and drops towards the largest halo masses. The depletion rate of H I once a galaxy enters a more massive halo is more rapid at higher halo mass, in contrast to the specific star formation rate which shows much less variation in the attenuation rate versus halo mass. This suggests that, up to halo mass scales probed here (. 1014M), star formation is mainly attenuated by starvation, but H I is additionally removed by stripping once a hot gaseous halo is present. In low mass halos, the H I richness of satellites is independent of radius, while in high mass halos they become gas-poor towards the center, confirming the increasing strength of the stripping with halo mass. By tracking the progenitors of galaxies, we show that the gas fraction of satellite and central galaxiesdecreases from z =5 ! 0, tracking each other until z⇠1 after which the satellites’ H I content drops much more quickly, particularly for the highest halo masses. Mergers somewhat increase the H I richness and its scatter about the mean relation, but these variations are consistent with arising form inflow fluctuations, unlike in the case of star formation where mergers boost it above that expected from inflow fluctuations. In short, our simulations suggest that the H I content in galaxies is determined by their ability to accrete gas from their surroundings, with stripping effects playing a driving role once a hot gaseous halo is present.