We give a corrected proof of an extension of the Robinson Splitting Theorem for the d.c.e. degrees. The purpose of this short paper is to clarify and correct the main result and proof contained in [1]. There we gave a simple proof that there exists no low maximal d.c.e. degree. This was obtained as an immediate corollary of the following strengthening of the Robinson Splitting Theorem (Theorem ...

A computable presentation of the linearly ordered set (ω,≤), where ω is the set of natural numbers and ≤ is the natural order on ω, is any linearly ordered set L = (ω,≤L) isomorphic to (ω,≤) such that ≤L is a computable relation. Let X be subset of ω and XL be the image of X in the linear order L under the isomorphism between (ω,≤) and L. The degree spectrum of X is the set of all Turing degree...

TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementa...

in this paper, a certain new connectedness of l-fuzzy subsets inl-topological spaces is introduced and studied by means of preclosed sets. itpreserves some fundamental properties of connected set in general topology.especially the famous k. fan’s theorem holds.

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

In this paper, we introduce λκ−closed sets and study its properties in generalized topological spaces.

Let $A$ be an abelian group. A graph $G=(V,E)$ is said to be $A$-barycentric-magic if there exists a labeling $l:E(G)longrightarrow Asetminuslbrace{0}rbrace$ such that the induced vertex set labeling $l^{+}:V(G)longrightarrow A$ defined by $l^{+}(v)=sum_{uvin E(G)}l(uv)$ is a constant map and also satisfies that $l^{+}(v)=deg(v)l(u_{v}v)$ for all $v in V$, and for some vertex $u_{v}$ adjacent t...

The purpose of this note is to establish several results, especially Theorem 1 below, relative to degree of approximation to analytic functions and their boundary values, the latter assumed to satisfy certain continuity conditions (Lipschitz conditions, etc.). The present results are closely related to, but more general than, results due to Sewell' and Elliott;2 the results have application to ...

Let S " he the set of all polynomials whose degree does not exceed n and whose all zeros are real but lie outside (-1, 1). Similarly, we say p"E Q " if p,(x) is a real polynomial whose all zeros lie outside the open disk with center at the origin and radius l. Further we will denote by H " the set of all polynomials of degree-n and of the form k=O where q,,k (x) = (1 +x)k (l-x)"-'. Elements of ...