Given a computably enumerable set W, there is a Turing degree which is the least jump of any set in which W is computably enumerable, namely 0′. Remarkably, this is not a phenomenon of computably enumerable sets. It is shown that for every subset A of N, there is a Turing degree, c′μ(A), which is the least degree of the jumps of all sets X for which A is ∑10(X). In addition this result provides an isomorphism invariant method for assigning Turing degrees to certain torsion‐free abelian groups.
A real is computable if its left cut, L ; is computable. If q i i is a computable sequence of rationals computably converging to ; then fq i g; the corresponding set, is always computable. A computably enumerable c.e. real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to : For example, every representation A of is Turing reducible to L : Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L necessarily contains a representation of :
Abstract. Π 0 1 classes are important to the logical analysis of many parts of mathematics. The Π 0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin's work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare's work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π 0 1 classes) forms an orbit in the lattice of Π 0 1 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π 0 1 classes. We remark that the automorphism result is proven via a ∆ 0 3 automorphism, and demonstrate that this complexity is necessary.
In this paper we describe software facilities for enabling patient positioning studies using the megavoltage imaging system developed at the Royal Marsden Hospital and Institute of Cancer Research. The study focuses on the use of the system for three purposes: patient position verification (by comparing images taken at treatment simulation with megavoltage images taken at treatment time); reproducibility studies (by analysing a set of megavoltage images); and set-up correction (by adjusting the set-up until the megavoltage image obtained at treatment registers with the simulation image). The need is discussed for suitably presented simulator images, a method of determining field boundaries and the possibility of delineating soft-tissue interfaces. Several algorithms of different types, developed specifically for the purpose of intercomparison of planar projection images, are presented. The techniques employed and their usefulness, in both the qualitative and the quantitative sense, are discussed. The results are presented of a phantom and clinical study, to evaluate the rigour and reproducibility of the algorithms. These results indicate that measurements can be made to an accuracy of about 1-2 mm, with a similar value for interobserver reproducibility for the best image comparison techniques available.
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