A real is computable if its left cut, L( ); is computable. If (qi)i is a computable sequence of rationals computably converging to ; then fqig; 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. real...

We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...

We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...

We introduce a Π1-uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω1) = ω1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π1 maximal chains and thin maximal antichains in the Turing degrees. We also use the Π1-uniformization principle to study Martin’s conjectures on cones of Turing d...

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ram-sey's Theorem. An infinite set A of natural numbers is n–cohesive (respectively, n–r–cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2–coloring of the n–element sets of natural numbers. (Thus the 1–cohesive and 1–r–cohesive sets coi...

The strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) are defined in terms of Turing reductions where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We show that, for r = ibT, cl, every computably enumerable (c.e.) r-degree a > 0 has the an...

The minimum independent generalized t-degree of a graph G = (V,E) is ut = min{ IN(H H is an independent set of t vertices of G}, with N(H) = UxtH N(x). In a KI,~+I -free graph, we give an upper bound on u! in terms of r and the independence number CI of G. This generalizes already known results on u2 in KI,,+I-free graphs and on U, in KI,x-free graphs.

In this paper, we analyze the secure connectivity in Shotgun cellular systems (SCS: Wireless communication systems with randomly placed base stations) by Poisson intrinsically secure communication graph (IS-graph), i.e., a random graph which describes the connections that are secure over a network. For a base-station in SCS, a degree of secure connections is determined over two channel models: ...

Sparse Symmetric Positive Definite (SPD) Linear System Solvers (LSS) are routinely required in various Power System Analysis problems. In sparse LSS ordering of nodes is desired to reduce fill ins, which in turn reduce computations. Minimum Degree Algorithm (MDA) has established itself as a very competitive ordering algorithm, for reducing fill ins to quite often near minimum level. With the ac...

It is well known that the set of possible degree sequences for a simple graph on n vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a simple k-uniform hypergraph on n vertices is not the intersection of a lattice and a convex polytope for k > 3 and n > k + 13. We also show an analogous nonconvexity result for the set of degre...