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  1.  10
    A Blend of Methods of Recursion Theory and Topology.Iraj Kalantari & Larry Welch - 2003 - Annals of Pure and Applied Logic 124 (1-3):141-178.
    This paper is a culmination of our new foundations for recursive analysis through recursive topology as reported in Kalantari and Welch 125; 98 87). While in those papers we developed groundwork for an approach to point free analysis and applied recursion theory, in this paper we blend techniques of recursion theory with those of topology to establish new findings. We present several new techniques different from existing ones which yield interesting results. Incidental to our work is a unifying explanation of (...)
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  2.  9
    A Blend of Methods of Recursion Theory and Topology: A Π1 0 Tree of Shadow Points. [REVIEW]Iraj Kalantari & Larry Welch - 2004 - Archive for Mathematical Logic 43 (8):991-1008.
    This paper is a sequel to our [7]. In that paper we constructed a Π1 0 tree of avoidable points. Here we construct a Π1 0 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2<ω. One interesting fact is that while our Π1 0 tree (...)
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  3.  18
    Density and Baire Category in Recursive Topology.Iraj Kalantari & Larry Welch - 2004 - Mathematical Logic Quarterly 50 (45):381-391.
    We develop the concepts of recursively nowhere dense sets and sets that are recursively of first category and study closed sets of points in light of Baire's Category Theorem. Our theorems are primarily concerned with exdomains of recursive quantum functions and hence with avoidable points . An avoidance function is a recursive function which can be used to expel avoidable points from domains of recursive quantum functions. We define an avoidable set of points to be an arbitrary subset of the (...)
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  4.  18
    On Turing Degrees of Points in Computable Topology.Iraj Kalantari & Larry Welch - 2008 - Mathematical Logic Quarterly 54 (5):470-482.
    This paper continues our study of computable point-free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter. A function fF from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that have at (...)
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  5.  14
    On Degree-Preserving Homeomorphisms Between Trees in Computable Topology.Iraj Kalantari & Larry Welch - 2008 - Archive for Mathematical Logic 46 (7-8):679-693.
    In this paper we first give a variant of a theorem of Jockusch–Lewis– Remmel on existence of a computable, degree-preserving homeomorphism between a bounded strong ${\Pi^0_2}$ class and a bounded ${\Pi^0_1}$ class in 2 ω . Namely, we show that for mathematically common and interesting topological spaces, such as computably presented ${\mathbb{R}^n}$ , we can obtain a similar result where the homeomorphism is in fact the identity mapping. Second, we apply this finding to give a new, priority-free proof of existence (...)
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  6.  10
    When Series of Computable Functions with Varying Domains Are Computable.Iraj Kalantari & Larry Welch - 2013 - Mathematical Logic Quarterly 59 (6):471-493.