Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

Abstract We prove that there are non-recursive r.e. sets A and C with A < _T C such that for every set . Both authors are supported by “863” and the National Science Foundation of China     

A Splitting with Infimum in the d‐c. e. Degrees

In this paper we prove that any c. e. degree is splittable with an c. e. infimum over any lesser c. e. degree in the class of d‐c. e. degrees.     

There is No Low Maximal D.C.E. Degree

We show that for any computably enumerable (c.e.) set A and any set L, if L is low and , then there is a c.e. splitting such that . In Particular, if L is low and n‐c.e., then is n‐c.e. and hence there is no low maximal n‐c.e. degree.     

Isolated maximal d.r.e. degrees

There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called $(m,n)$-cupping degree and show that there exists an isolated $(2,\omega )$-cupping degree, and there exists a proper $(2,1)$-cupping degree. It helps understanding various degree structures in the Ershov Hierarchy.     

The Isolated D. R. E. Degrees are Dense in the R. E. Degrees

In the present paper we prove that the isolated differences of r. e. degrees are dense in the r. e. degrees. Mathematics Subject Classification: 03D25.     

There is no low maximal d. c. e. degree – Corrigendum

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Z-P-S 空间中若干新的不动点

利用拓扑度的方法研究了Z-P-S空间中非线性算子的不动点问题,得到了若干新的结果.同时,推广了一些重要结论.     

Embeddings in the Strong Reducibilities Between 1 and npm

We consider the strongest (most restricted) forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities (which we call n1- and ni-reducibility) which are counterparts to 1- and i-reducibility, respectively, in the same way that nm- and npm-reducibility are counterparts to m- and pm-reducibility, respectively, we bring out the structure (under the natural relation on reducibilities strong with respect to') of the strong reducibilities. By further restricting n1- and nm-reducibility we are able to define infinite families of reducibilities which isomorphically embed the r. e. Turing degrees. Thus the many well-known results in the theory of the r. e. Turing degrees have counterparts in the theory of strong reducibilities. We are also able to positively answer the question of whether there exist distinct reducibilities ≤_y and ≤_a between ≤_e and ≤_m such that there exists a non-trivial ≤_y-contiguous ≤_z degree.     

Z—P—S空间中一类新的不动点定理

利用概率度量空间中A—proper映射拓扑度的基本性质,在投影完备的Z—P—S空间中研究了非线性映射的不动点问题,得到了一些新的结果．     

Estimates of Disjoint Sequences in Banach Lattices and R.I. Function Spaces

We introduce UDS _p -property (resp. UDT _q -property) in Banach lattices as the property that every normalized disjoint sequence has a subsequence with an upper p-estimate (resp. lower q-estimate). In the case of rearrangement invariant spaces, the relationships with Boyd indices of the space are studied. Some applications of these properties are given to the high order smoothness of Banach lattices, in the sense of the existence of differentiable bump functions     

Some New Types of KKM Theorem in H spaces with Applications

本文证明了Ｈ－空中的几个新型非空交定理．作为应用,文中得到了向量值极大极小定理和不动点定理,并研究了抽象经济平衡问题．     

The Index Set of Injectively Enumerable Classes of Recursively Enumerable Sets in ∑5-Complete

I introduce an effective enumeration of all effective enumerations of classes of r. e. sets and define with this the index set IE of injectively enumerable classes. It is easy to see that this set is ∑₅ in the Arithmetical Hierarchy and I describe a proof for the ∑₅-hardness of IE. Mathematics Subject Classification: 03D25, 03D45.     

Maximal independent sets in minimum colorings

Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G. Among all such minimum vertex-colorings of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-χ-coloring of G. The number of color classes that are dominating sets in a dominating-χ-coloring of G is defined to be the dominating-χ-color number of G. In this paper, we continue to investigate the dominating-χ-color number of a graph first defined and studied in [1].