Computability Theory

Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Degrees of members of ∏01 classes

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2. Dispersion relations for inelastic scattering processes

The subject matter of this thesis falls into two distinct parts. Chapters II to IV are devoted to a discussion of Schwinger's action principle, and chapters V and VI are concerned with the proof of dispersion relations for inelastic meson-nucleon scattering. The material of chapter II is based on some work done in collaboration with Dr. J.C. Polkinghorne, which has been published (Kibble and Polkinghorne 1957). This work was concerned with the clarification of certain points connected with the class of permissible variations in Schwinger's principle. There are, however, substantial changes in the present treatment, principally deriving from the introduction, in section II-5, of the concept of relative phases. This chapter is restricted to the case of non-relativistic quantum theory, and the discussion is extended to relativistic quantum field theory in chapter III. These chapters are devoted to a reformulation of Schwinger's action principle, and an investigation of the consequences of the new form of the action principle. Some of this material is necessarily contained in the work of Schwinger (1951, 1953a), but the treatment differs from his in several important respects. These are discussed in greater detail in section 2. Chapter IV is devoted to a discussion of higher order spinor Lagrangians, with particular reference to the use of a two-component field satisfying a second-order equation rather than a four-component spinor satisfying a first-order equation. This procedure has been suggested by Feynman and Gell-Mann (1958) in connection with their universal Fermi interaction. The work presented in this chapter was done jointly with Dr. J.C. Polkinghorne, and has been published (Kibble and Polkinghorne 1958). Chapters V and VI are devoted to a proof of the dispersion relations for the process in which a single meson is scattered on a nucleon into a state with several mesons. The proof follows the general lines of that by Bogolyubov, Medvedev and Polivanov (1956) for the case of elastic meson-nucleon scattering, This work has also been published (Kibble 1958). The notation employed in the thesis is summarized in appendix A. Appendix B is devoted to a discussion of consistency conditions on the Lagrangian function. The chapter number is omitted in references to sections or equations, except in the case of cross references between chapters

Critical Kinetic Plasma Processes In Relativistic Astrophysics

Plasma astrophysics deals with collective plasma processes in astrophysical scenarios. As observational astronomy pushes towards unprecedented resolutions in space and time, the focus of theoretical research necessarily ventures towards a description of the plasma microphysics. On microphysical scales the plasma is pervasively collisionless and the magnetohydrodynamic approximation breaks down. Consequently theoretical concepts rely on a kinetic plasma description as the most sophisticated plasma model. The present work discusses some fundamental kinetic plasma processes in relativistic astrophysics: Fast Magnetic Reconnection (FMR) associated with discontinuities in the magnetic field topology, and the Coupled Two-Stream-Weibel instability (CTW) in the wake of collisionless shocks. Both processes are ubiquitous in astrophysical sites, prevail over competing plasma modes because of dominant growth rates, experience significant relativistic modifications, and develop essential features solely in the highly non-linear regime. The computational representation invokes the entire 6D phase space. These characteristics distinguish FMR and the CTW as distinctively critical processes. FMR and the CTW are studied here in the framework of self-consistent, relativistic and fully electromagnetic Particle-In-Cell (PIC) simulations. Typical scenarios comprise ensembles of 10^9 particles and endure for several 10^4 time steps. The computational task is challenging and completely in the realm of the massively parallelized architectures of state-of-the-art supercomputers. We present the first self-consistent 3D simulations of FMR in relativistic pair plasma. Focusing on the mechanism of particle acceleration we show that the highly dynamic evolution of the current sheet in the non-linear regime is the essential stage. Therein non-stationary acceleration zones arise in the superposition of the relativistic tearing and the relativistic drift kink mode as competing current sheet instabilities. Though the topology of electromagnetic fields is highly turbulent, the FMR process shows the remarkable quality to generate smooth and stable power-laws in the particle distribution function (PDF) out of an initial Maxwellian. The upper PDF cut-off in relativistic energy is determined by the ratio of light to Alfven velocity c/v_A. The power-law index assumes s~-1 within the reconnection X-zone irrespective of parameter variations. Intriguingly the power-law index appears as the universal characteristic of the source process. The associated synchrotron spectra provide a valid description of the extremely hard spectra and rapid variabilities of `Flat Spectrum Radio Quasars'. Conceptual Gamma-Ray Burst (GRB) synchrotron emission models depend on a plasma process which ensures efficient magnetic field generation. The CTW converts bulk-kinetic energy of counter-streaming plasma shells into Weibel magnetic fields. Pivoted by the linear analysis of the CTW, the PIC simulations confirm the correspondence between saturation magnetic fields and bulk-kinetic energy. Plasma shell collisions in GRBs are either associated with internal or external shocks. As direct consequence of the energy dependence the CTW evolves from a complex 3D topology in internal collisions towards quasi-2D, Weibel-dominated conformalizations at the higher external shock energies. The PIC results prove that the Weibel fields are sufficiently strong to sustain synchrotron emission scenarios, particularly in external shocks. By determining the first lifetime limits we show that Weibel fields are also sufficiently long-lived with respect to typical synchrotron cooling times. We further identify the stability-limiting diffusion process as of `Bohm'-type, i.e. the diffusion coefficient exhibits the T/B-dependence and herewith represents a conservative stability criterion. The CTW generates stable power-law spectra in the magnetic fields implying power-law shaped PDFs as self-similar solutions for diffusive particle scattering. This suggests a universal power-law index as the characteristic of the CTW process. Imposing a magnetic guide field of well-defined strength suppresses the Weibel contributions of the CTW in favour of the electrostatic Two-Stream instability (TSI). The pulsar magnetosphere is the paradigmatic scenario in which we discuss the mechanism of Coherent Collisionless Bremsstrahlung (CCB) triggered by the TSI. The PIC simulations show that the CCB mechanism provides a valid description of the phenomenon of `Giant Radio Pulses' as recently observed from the Crab pulsar

RELATIONSHIPS OF HAEMOSPORIDIAN PARASITES TO POPULATIONS OF THEIR AVIAN HOSTS IN EASTERN NORTH AMERICA

Avian Haemosporida are common, vector-transmitted blood parasites of birds throughout the world. During my dissertation research, I explored how multiple host species respond immunologically to natural infections in the wild (Chapter 1) and to experimental infections in the laboratory (Chapter 2). Despite their tractability as a model host-parasite system and a burgeoning literature on avian Haemosporida, little is known about how their populations interact across large areas (hereafter “regions”). I present data from parasite surveys of birds across eastern North America suggesting that continental parasite populations track host populations across the region, but also that the host breadth of a parasite can be variable across space and time (Chapter 3). Parasite lineages replace each other spatially within a host population, likely due to interspecific parasite competition mediated by host immune systems (Chapter 3). Parasite prevalence is positively related to host abundance within local assemblages (Chapter 4), but within host species across their ranges, prevalence does not vary with abundance (Chapters 3 and 4). Finally, a 12 year survey of parasites and their hosts in the Missouri Ozarks demonstrates that parasite populations vary through time, and that this variability is related to host breadth—specialist parasites (i.e., parasites infecting primarily one host) were more variable than generalist parasites (i.e., parasites infecting multiple hosts; Chapter 5). Overall my dissertation work contributes to the natural history and ecology of avian Haemosporidian parasites and their avian hosts, and to host-parasite ecological and evolutionary theory