Incompleteness and jump hierarchies

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$.Comment: 11 pages. Corrects a mistake in the statements of two result

Martin's conjecture for regressive functions on the hyperarithmetic degrees

We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's conjecture, consists of showing that we can always reduce to the case of a continuous function.Comment: 12 page

Linear impulse response in hot round jets

International audienceThe linear impulse response is retrieved from a numerical solution of the spatial eigenvalue problem, which is derived from the fully compressible equations of motion. Changes in the spatiotemporal stability of heated versus isothermal jets are shown to arise solely from the effect of the baroclinic torque. By considering the full linear impulse response, the competition between jet column modes and shear layer modes is characterized. Jet column modes are only found to occur for axisymmetric disturbances. In thin shear layer jets, the jet column mode is shown to prevail at low group velocities, whereas axisymmetric and helical shear layer modes dominate at high group velocities. The absolute mode of zero group velocity is found to always be of the jet column type. Although only convectively unstable, the maximum growth rates of the shear layer modes greatly exceed those of the jet column modes in thin shear layer jets. In thick shear layer jets, axisymmetric modes of mixed jet column/shear layer type arise. The weakened maximum growth rate of mixed modes accounts for the dominance of helical modes in temporal stability studies of thick shear layer jets. © 2007 American Institute of Physics

Part 1 of Martin's Conjecture for order-preserving and measure-preserving functions

Martin's Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first classifies functions which are not above the identity and the second of classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which preserve Turing reducibility). We prove the first part of the conjecture for all order-preserving functions. We do this by introducing a class of functions on the Turing degrees which we call "measure-preserving" and proving that part 1 of Martin's Conjecture holds for all measure-preserving functions and also that all non-trivial order-preserving functions are measure-preserving. Our result on measure-preserving functions has several other consequences for Martin's Conjecture, including an equivalence between part 1 of the conjecture and a statement about the structure of the Rudin-Keisler order on ultrafilters on the Turing degrees.Comment: 44 pages; updated to correct some attributions and fix some typo

A Note on a Question of Sacks: It is Harder to Embed Height Three Partial Orders than Height Two Partial Orders

A long-standing conjecture of Sacks states that it is provable in ZFC that every locally countable partial order of size continuum embeds into the Turing degrees. We show that this holds for partial orders of height two, but provide evidence that it is hard to extend this result even to partial orders of height three. In particular, we show that the result for height two partial orders holds both in certain extensions of ZF with only limited forms of choice and in the Borel setting (where the partial orders and embeddings are required to be Borel measurable), but that the analogous result for height three partial orders fails in both of these settings. We also formulate a general obstacle to embedding partial orders into the Turing degrees, which explains why our particular proof for height two partial orders cannot be extended to height three partial orders, even in ZFC. We finish by discussing how our results connect to the theory of countable Borel equivalence relations.Comment: 19 page

Coding information into all infinite subsets of a dense set

Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is necessary in the following technical sense: if $X$ is uncomputable and $A$ is a set of positive lower density then $A$ has an infinite subset which does not compute $X$. We will show that this theorem is sharp in certain senses and also prove a quantitative version formulated in terms of Kolmogorov complexity. Our results use a modified version of Mathias forcing and build on work by Seetapun and others on the reverse math of Ramsey's theorem for pairs.Comment: 30 page

Global linear stability of a model subsonic jet

The global stability of a subsonic jet is investigated using a model base flow designed to fit experimental results for turbulent mean flows. Eigenmodes are computed for axisymmetric perturbations in order to investigate the nature of typically observed large-scale coherent oscillations ("preferred mode"). We do not find evidence that this preferred mode corre- sponds to the least damped global mode. Non-modal stability is also considered through the computation of optimal perturbations. Although non-axisymmetric perturbations (in particular for azimuthal wavenumber m = 1) are subject to larger transient growth, these reach their peak amplitude far downstream of the potential core, and therefore they are less likely to be observed

Frequency selection in globally unstable round jets

International audienceThe self-sustained formation of synchronized ring vortices in hot subsonic jets is investigated by direct numerical simulation of the axisymmetric equations of motion. The onset of global instability and the global frequency of synchronized oscillations are examined as functions of the ambient-to-jet temperature ratio and the initial jet shear layer thickness. The numerical results are found to follow the predictions from nonlinear global instability theory; global instability sets in as the unperturbed flow is absolutely unstable over a region of finite streamwise extent at the inlet, and the global frequency near the global instability threshold corresponds to the absolute frequency of the inlet profile. In strongly supercritical thin shear layer jets, however, the simulations display global frequencies well above the absolute frequency, in agreement with experimental results. The inner structure of rolled-up vortices in hot jets displays fine layers of positive and negative vorticity that are produced and maintained by the action of the baroclinic torque. © 2007 American Institute of Physics

Aerodynamic sound generation by global modes in hot jets

International audienceThe acoustic field generated by the synchronized vortex street in self-excited hot subsonic jets is investigated via direct numerical simulation of the compressible equations of motion in an axisymmetric geometry. The simulation simultaneously resolves both the aerodynamic near field and the acoustic far field. Self-sustained near-field oscillations in the present flow configurations have been described as nonlinear global modes in an earlier study. The associated acoustic far field is found to be that of a compact dipole, emanating from the location of vortex roll-up. A far-field solution of the axisymmetric Lighthill equation is derived, on the basis of the source term formulation of Lilley (AGARD-CP, vol. 131, 1974, pp. 13.1-13.12). With the near-field source distributions obtained from the direct numerical simulations, the Lighthill solution is in good agreement with the far-field simulation results. Fluctuations of the enthalpy flux within the jet are identified as the dominant aeroacoustic source. Superdirective effects are found to be negligible. © 2010 Cambridge University Press

A Unified Statistical Framework for Evaluating Predictive Methods

Predictive analytics is an important part of the business intelligence and decision support systems literature and likely to grow in importance with the emergence of big data as a discipline. Despite their importance, the accuracy of predictive methods is often not assessed using statistical hypothesis tests. Furthermore, there is no commonly agreed upon standard as to which questions should be examined when evaluating predictive methods. We fill this gap by defining three questions that involve the overall and comparative predictive accuracy of the new method. We then present a unified statistical framework for evaluating predictive methods that can be used to address all three of these questions. The framework is particularly versatile and can be applied to most problems and datasets. In addition to these practical advantages over hypotheses tests used in previous literature, the framework has the theoretical advantage that it is not necessary to assume a normal distribution