9 found
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  1.  13
    The Arithmetical Hierarchy of Real Numbers.Xizhong Zheng & Klaus Weihrauch - 2001 - Mathematical Logic Quarterly 47 (1):51-66.
    A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left computable iff it is the supremum of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable sequences of rational numbers we introduce a non-collapsing hierarchy {Σn, Πn, Δn : n ∈ ℕ} of real numbers. We characterize the classes Σ2, Π2 and Δ2 in various ways and give (...)
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  2.  8
    Computable Metrization.Tanja Grubba, Matthias Schröder & Klaus Weihrauch - 2007 - Mathematical Logic Quarterly 53 (4‐5):381-395.
    Every second-countable regular topological space X is metrizable. For a given “computable” topological space satisfying an axiom of computable regularity M. Schröder [10] has constructed a computable metric. In this article we study whether this metric space can be considered computationally as a subspace of some computable metric space [15]. While Schröder's construction is “pointless”, i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points (...)
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  3.  10
    Representations of the Real Numbers and of the Open Subsets of the Set of Real Numbers.Klaus Weihrauch & Christoph Kreitz - 1987 - Annals of Pure and Applied Logic 35 (3):247-260.
  4.  20
    Connectivity Properties of Dimension Level Sets.Jack H. Lutz & Klaus Weihrauch - 2008 - Mathematical Logic Quarterly 54 (5):483-491.
    This paper initiates the study of sets in Euclidean spaces ℝn that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, n ], we are interested in the connectivity properties of the set DIMI, consisting of all points in ℝn whose dimensions lie in I, and of its dual DIMIstr, consisting of all points whose strong dimensions lie in I. If I is [0, 1) or.
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  5.  23
    Computable Analysis of the Abstract Cauchy Problem in a Banach Space and its Applications I.Klaus Weihrauch & Ning Zhong - 2007 - Mathematical Logic Quarterly 53 (4‐5):511-531.
    We study computability of the abstract linear Cauchy problem equation image)where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K: x ↦ u of the problem is computable. For studying computability we use the representation approach to computable analysis developed by Weihrauch and others. This approach is consistent with the model used by Pour-El/Richards.
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  6.  16
    Compactness in Constructive Analysis Revisited.Christoph Kreitz & Klaus Weihrauch - 1987 - Annals of Pure and Applied Logic 36 (1):29-38.
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  7.  14
    Editorial: Math. Log. Quart. 4–5/2007.Ker-I. Ko, Klaus Weihrauch & Xizhong Zheng - 2007 - Mathematical Logic Quarterly 53 (4‐5):325-325.
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  8.  9
    Approaches to Effective Semi‐Continuity of Real Functions.Xizhong Zheng, Vasco Brattka & Klaus Weihrauch - 1999 - Mathematical Logic Quarterly 45 (4):481-496.
    For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo is recursively enumerably open in dom × ℝ; we call f lower semi-computable of type two, if its closed epigraph Epi is recursively enumerably closed in dom × ℝ; we call f lower semi-computable of type three, if Epi is recursively closed in dom × ℝ. We show that (...)
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  9.  6
    Computable Riesz Representation for the Dual of C [0; 1].Hong Lu & Klaus Weihrauch - 2007 - Mathematical Logic Quarterly 53 (4):415-430.
    By the Riesz representation theorem for the dual of C [0; 1], if F: C [0; 1] → ℝ is a continuous linear operator, then there is a function g: [0;1] → ℝ of bounded variation such that F = ∫ f dg . The function g can be normalized such that V = ‖F ‖. In this paper we prove a computable version of this theorem. We use the framework of TTE, the representation approach to computable analysis, which allows (...)
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