Automorphisms of the lattice of classes; perfect thin classes and anc degrees

classes are important to the logical analysis of many parts of mathematics. The 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 classes) forms an orbit in the lattice of 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 classes. We remark that the automorphism result is proven via a automorphism, and demonstrate that this complexity is necessary.

     

Index sets and parametric reductions

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholakand the first author by proving the fundamental index sets associated with a computable set A, {e : W _e ≤ _q ^u A} for q∈ {m, T} are Σ₄ ⁰ complete. We also show hat FPT(≤ _q ⁿ ), that is {e : W _e computable and ≡ _q ⁿ ?}, is Σ₄ ⁰ complete. We also look at computable presentability of these classes. Received: 13 July 1996 / Revised version: 14 April 2000 / Published online: 18 May 2001     

Sharp version of the Goldberg–Sachs theorem

We reexamine from first principles the classical Goldberg–Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)T{\Rightarrow} (algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes, we associate a natural connection, in terms of which these conditions have a very simple form.     

Stability transitions for periodic orbits in hamiltonian systems

The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h ₀. First considered is the case of a critical point (with eigenvalues ±, ±i, and >0) of the Hamiltonian at energy h ₀ with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h ₀ on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h ₀, on which the orbits are hyperbolic, involve the coexistence problem for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±±i), and >0, investigated by Henrard and Devaney.     

There is no fat orbit

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.     

An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space

J. B. Baillon [C. R. Acad. Sci. Paris Ser. A.280 (1975), 1511–1514] proved an ergodic theorem for a single nonexpansive mapping in a Hilbert space, which is a nonlinear version of von Neumann's mean ergodic theorem. In this paper, we study the ergodic behavior of a semigroup of nonexpansive mappings. We try to find a sequence of means on the semigroup, generalizing the Cesàro means on $\text{N}$, such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed-points. Our method of proof is an appropriate modification of A. Pazy's proof [Israel J. Math.26 (1977), 197–204] of Baillon's theorem.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.