Arcs in a finite projective plane

The projective plane of order 11 is the dominant focus of this work. The motivation for working in the projective plane of order 11 is twofold. First, it is the smallest projective plane of prime power order such that the size of the largest (n, r)-arc is not known for all r ∈ {2,...,q + 1}. It is also the smallest projective plane of prime order such that the (n; 3)-arcs are not classified. Second, the number of (n, 3)-arcs is significantly higher in the projective plane of order 11 than it is in the projective plane of order 7, giving a large number of (n, 3)-arcs for study. The main application of (n, r)-arcs is to the study of linear codes. As a forerunner to the work in the projective plane of order eleven two algorithms are used to raise the lower bound on the size of the smallest complete n-arc in the projective plane of order thirty-one from 12 to 13. This work presents the classification up to projective equivalence of the complete (n, 3)- arcs in PG(2, 11) and the backtracking algorithm that is used in its construction. This algorithm is based on the algorithm used in [3]; it is adapted to work on (n, 3)-arcs as opposed to n-arcs. This algorithm yields one representative from every projectively inequivalent class of (n, 3)-arc. The equivalence classes of complete (n, 3)-arcs are then further classified according to their stabilizer group. The classification of all (n, 3)-arcs up to projective equivalence in PG(2, 11) is the foundation of an exhaustive search that takes one element from every equivalence class and determines if it can be extended to an (n′, 4)-arc. This search confirmed that in PG(2, 11) no (n, 3)-arc can be extended to a (33, 4)-arc and that subsequently m4(2, 11) = 32. This same algorithm is used to determine four projectively inequivalent complete (32, 4)-arcs, extended from complete (n, 3)-arcs. Various notions under the general title of symmetry are defined both for an (n, r)-arc and for sets of points and lines. The first of these makes the classification of incomplete (n; 3)- arcs in PG(2, 11) practical. The second establishes a symmetry based around the incidence structure of each of the four projectively inequivalent complete (32, 4)-arcs in PG(2, 11); this allows the discovery of their duals. Both notions of symmetry are used to analyze the incidence structure of n-arcs in PG(2, q), for q = 11, 13, 17, 19. The penultimate chapter demonstrates that it is possible to construct an (n, r)-arc with a stabilizer group that contains a subgroup of order p, where p is a prime, without reference to an (m < n, r)-arc, with stabilizer group isomorphic to ℤ1. This method is used to find q-arcs and (q + 1)-arcs in PG(2, q), for q = 23 and 29, supporting Conjecture 6.7. The work ends with an investigation into the effect of projectivities that are induced by a matrix of prime order p on the projective planes. This investigation looks at the points and subsets of points of order p that are closed under the right action of such matrices and their structure in the projective plane. An application of these structures is a restriction on the size of an (n, r)-arc in PG(2, q) that can be stabilized by a matrix of prime order p

Testing a Simplified Version of Einstein's Equations for Numerical Relativity

Solving dynamical problems in general relativity requires the full machinery of numerical relativity. Wilson has proposed a simpler but approximate scheme for systems near equilibrium, like binary neutron stars. We test the scheme on isolated, rapidly rotating, relativistic stars. Since these objects are in equilibrium, it is crucial that the approximation work well if we are to believe its predictions for more complicated systems like binaries. Our results are very encouraging.Comment: 9 pages (RevTeX 3.0 with 6 uuencoded figures), CRSR-107

Optimization and sensitivity studies of flight-path trajectories

The optimization of landing trajectories of the Boeing 737 is presented. The primary factor considered was the noise delivered to the population residing near an air terminal but passenger comfort, fuel consumption and time elapsed during the maneuver were also considered. A digital simulation of the aircraft, a noise model and a passenger comfort model, was completed. The digital simulation was made more efficient time-wise. A population model for an urban area was developed and the noise model was integrated into the population model. A steepest descent optimization algorithm was programmed. Some constant glide slope trajectories into an urban Airport were simulated and evaluated with respect to the performance index, and their ground track plotted

Determining crustal strain rates with a spaceborne geodynamics ranging system. 2: Station coordinate analysis

The use of a spaceborne geodynamics ranging system for determining crustal strain rates is analyzed. The use of site coordinates rather than intersite baseline distances for the strain rate determinations is emphasized. After discussing the analytical techniques which are to be employed, numerical results are presented which suggest that the use of site coordinates would result in a 20-70% improvement in the precision of the deduced values of straining. Precision of a few parts in 10 to the 9th power would be achievable with simple geometrics and a decade or two of measurements; precisions of a few parts in 10 to the 8th power would be achievable in a few years. A consideration of possible correlations among the derived target site coordinates leads to the conclusion that, with the proper choice of coordinate systems, the correlations can be made small and non-detrimental to the strain rate determinations

Quasi-circular Orbits for Spinning Binary Black Holes

Using an effective potential method we examine binary black holes where the individual holes carry spin. We trace out sequences of quasi-circular orbits and locate the innermost stable circular orbit as a function of spin. At large separations, the sequences of quasi-circular orbits match well with post-Newtonian expansions, although a clear signature of the simplifying assumption of conformal flatness is seen. The position of the ISCO is found to be strongly dependent on the magnitude of the spin on each black hole. At close separations of the holes, the effective potential method breaks down. In all cases where an ISCO could be determined, we found that an apparent horizon encompassing both holes forms for separations well inside the ISCO. Nevertheless, we argue that the formation of a common horizon is still associated with the breakdown of the effective potential method.Comment: 13 pages, 10 figures, submitted to PR

Use of steepest descent and various approximations for efficient computation of minimum noise aircraft landing trajectories

The following areas related to landing trajectory optimization research were discussed: (1) programming and modifying the steepest descent optimization procedure, (2) successfully iterating toward the optimum for a four-mile trajectory, (3) beginning optimization runs for a twenty-mile trajectory, and (4) adapt wind tunnel data for computer usage. Other related areas were discussed in detail in the two previous annual reports

Gravitational waves from nonspinning black hole-neutron star binaries: dependence on equations of state

We report results of a numerical-relativity simulation for the merger of a black hole-neutron star binary with a variety of equations of state (EOSs) modeled by piecewise polytropes. We focus in particular on the dependence of the gravitational waveform at the merger stage on the EOSs. The initial conditions are computed in the moving-puncture framework, assuming that the black hole is nonspinning and the neutron star has an irrotational velocity field. For a small mass ratio of the binaries (e.g., MBH/MNS = 2 where MBH and MNS are the masses of the black hole and neutron star, respectively), the neutron star is tidally disrupted before it is swallowed by the black hole irrespective of the EOS. Especially for less-compact neutron stars, the tidal disruption occurs at a more distant orbit. The tidal disruption is reflected in a cutoff frequency of the gravitational-wave spectrum, above which the spectrum amplitude exponentially decreases. A clear relation is found between the cutoff frequency of the gravitational-wave spectrum and the compactness of the neutron star. This relation also depends weakly on the stiffness of the EOS in the core region of the neutron star, suggesting that not only the compactness but also the EOS at high density is reflected in gravitational waveforms. The mass of the disk formed after the merger shows a similar correlation with the EOS, whereas the spin of the remnant black hole depends primarily on the mass ratio of the binary, and only weakly on the EOS. Properties of the remnant disks are also analyzed.Comment: 27pages, 21 figures; erratum is added on Aug 5. 201

Alternatives to standard puncture initial data for binary black hole evolution

Standard puncture initial data have been widely used for numerical binary black hole evolutions despite their shortcomings, most notably the inherent lack of gravitational radiation at the initial time that is later followed by a burst of spurious radiation. We study the evolution of three alternative initial data schemes. Two of the three alternatives are based on post-Newtonian expansions that contain realistic gravitational waves. The first scheme is based on a second-order post-Newtonian expansion in Arnowitt, Deser, and Misner transverse-traceless (ADMTT) gauge that has been resummed to approach standard puncture data at the black holes. The second scheme is based on asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to a first-order post-Newtonian expansion in ADMTT gauge away from the black holes. The final alternative is obtained through asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to a second-order post-Newtonian expansion in harmonic gauge away from the black holes. When evolved, the second scheme fails to produce quasicircular orbits (and instead leads to a nearly head-on collision). This failure can be traced back to inaccuracies in the extrinsic curvature due to low order matching. More encouraging is that the latter two alternatives lead to quasicircular orbits and show gravitational radiation from the onset of the evolution, as well as a reduction of spurious radiation. Current deficiencies compared to standard punctures data include more eccentric trajectories during the inspiral and larger constraint violations, since the alternative data sets are only approximate solutions of Einstein's equations. The eccentricity problem can be ameliorated by adjusting the initial momentum parameters.Comment: 11 pages, 11 figures, 1 appendix, typos corrected, removed duplicate reference, matches published versio

Predictions for the fracture toughness of cancellous bone of fracture neck of femur patients

Current protocol in determining if a patient is osteoporotic and their fracture risk is based on dual energy X-ray absorptiometry (DXA). DXA gives an indication of their bone mineral density (BMD) which is the product of both the porosity and density of the mineralized bone tissue; this is usually taken at the hip. The DXA results are assessed using the fracture risk assessment tool as recommended by the World Health Organization. While this provides valuable data on a person’s fracture risk advancements in medical imagining technology enables development of more robust and accurate risk assessment tools. In order to develop such tools in vitro analysis of bone is required to assess the morphological properties of bone osteoporotic bone tissue and how these pertain to the fracture toughness (Kcmax) of the tissue.Support was provided by the EPSRC (EP/K020196: Point-ofCare High Accuracy Fracture Risk Prediction), the UK Department of Transport under the BOSCOS (Bone Scanning for Occupant Safety) project, and approved by Gloucester and Cheltenham NHS Trust hospitals under ethical consent (BOSCOS – Mr. Curwen CI REC ref 01/179G)