Sacks [Sa1966a] asks if the metarecursively enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O(ω) or, equivalently, that of the truth set of LωCK 1 .

Abstract We study subfields of quotient division algebras of domains of finite GK dimension and introduce a combinatorial property we call the straightening property. We show that many classes of algebras have this straightening property and show that if A is a domain of GK dimension d with this property that is not PI, then the maximal subfields of the quotient division algebra of A have trans...

It is shown that the lower irredundance number and secure domination number of an n vertex tree T with maximum degree 3, are bounded below by 2(n+ 1)/(2 + 3) (T = K1, ) and ( n+ − 1)/(3 − 1), respectively. The bounds are sharp and extremal trees are exhibited. © 2006 Elsevier B.V. All rights reserved. MSC: 05C69

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). A function f(p) is defined, and it is shown that γ×2(G) = min f(p), where the minimum is taken over the n-dimensional cube C = {p = (p1, . . ....

Let Z ⊂ P, r ≥ 4, be a closed subscheme with dim(Z) ≤ r−4. Fix integers c > 0 and gi ≥ 0, i = 1, . . . , c. We prove that the general union of Z and c smooth curves Yi ⊂ P with genus gi and deg(Yi) ≥ r + gr as maximal rank (i.e. the expected postulation) if deg(Y1) + · · ·+ deg(Yc) 0.

We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump by given c.e. operators.

For a positive integer k, a k-subdominating function of a graph G=(V; E) is a function f :V →{−1; 1} such that ∑u∈NG [v] f(u)¿ 1 for at least k vertices v of G. The ksubdomination number of G, denoted by ks(G), is the minimum of ∑ v∈V f(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We al...

We show that the identity bounded Turing degrees of computably enumerable sets are not dense.