34 found
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  1. Uniformly Convex Banach Spaces Are Reflexive—Constructively.Douglas S. Bridges, Hajime Ishihara & Maarten McKubre‐Jordens - 2013 - Mathematical Logic Quarterly 59 (4-5):352-356.
    We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.
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  2.  19
    Constructive Notions of Equicontinuity.Douglas S. Bridges - 2009 - Archive for Mathematical Logic 48 (5):437-448.
    In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.
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  3.  25
    Constructive Mathematics and Unbounded Operators — a Reply to Hellman.Douglas S. Bridges - 1995 - Journal of Philosophical Logic 24 (5):549 - 561.
    It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.
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  4.  62
    Can Constructive Mathematics Be Applied in Physics?Douglas S. Bridges - 1999 - Journal of Philosophical Logic 28 (5):439-453.
    The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.
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  5.  14
    Product a-Frames and Proximity.Douglas S. Bridges - 2008 - Mathematical Logic Quarterly 54 (1):12-26.
    Continuing the study of apartness in lattices, begun in [8], this paper deals with axioms for a product a-frame and with their consequences. This leads to a reasonable notion of proximity in an a-frame, abstracted from its counterpart in the theory of set-set apartness.
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  6.  16
    Glueing Continuous Functions Constructively.Douglas S. Bridges & Iris Loeb - 2010 - Archive for Mathematical Logic 49 (5):603-616.
    The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.
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  7.  25
    Reflections on Function Spaces.Douglas S. Bridges - 2012 - Annals of Pure and Applied Logic 163 (2):101-110.
  8.  24
    Compactness Notions for an Apartness Space.Douglas S. Bridges - 2012 - Archive for Mathematical Logic 51 (5-6):517-534.
    Two new notions of compactness, each classically equivalent to the standard classical one of sequential compactness, for apartness spaces are examined within Bishop-style constructive mathematics.
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  9.  13
    How to Construct a Product of a‐Frames.Douglas S. Bridges - 2012 - Mathematical Logic Quarterly 58 (4-5):281-293.
    It is shown how, under certain circumstances and within Bishop-style constructive mathematics, one can construct a product of two a-frames.
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  10.  19
    On Weak Operator Compactness of the Unit Ball ofL.Douglas S. Bridges - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (31-36):493-494.
  11.  27
    Geometric Intuition and Elementary Constructive Analysis.Douglas S. Bridges - 1979 - Mathematical Logic Quarterly 25 (33):521-523.
  12.  10
    Constructing Local Optima on a Compact Interval.Douglas S. Bridges - 2007 - Archive for Mathematical Logic 46 (2):149-154.
    The existence of either a maximum or a minimum for a uniformly continuous mapping f of a compact interval into ${\mathbb{R}}$ is established constructively under the hypotheses that f′ is sequentially continuous and f has at most one critical point.
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  13.  9
    Uniform Continuity Properties of Preference Relations.Douglas S. Bridges - 2008 - Notre Dame Journal of Formal Logic 49 (1):97-106.
    The anti-Specker property, a constructive version of sequential compactness, is used to prove constructively that a pointwise continuous, order-dense preference relation on a compact metric space is uniformly sequentially continuous. It is then shown that Ishihara's principle BD-ℕ implies that a uniformly sequentially continuous, order-dense preference relation on a separable metric space is uniformly continuous. Converses of these two theorems are also proved.
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  14.  7
    On Weak Operator Compactness of the Unit Ball of L(H).Douglas S. Bridges - 1978 - Mathematical Logic Quarterly 24 (31‐36):493-494.
  15.  18
    A Criterion for Compactness in Metric Spaces?Douglas S. Bridges - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (7-12):97-98.
  16.  20
    Characterising Dominated Weak-Operator Continuous Functionals on Subspaces of B.Douglas S. Bridges - 2013 - Annals of Pure and Applied Logic 164 (4):416-420.
    A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B, where H is a Hilbert space, is derived within Bishop-style constructive mathematics.
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  17.  26
    A Note on Morse's Lambda-Notation in Set Theory.Douglas S. Bridges - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (8):113-114.
    No categories
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  18.  15
    Sequential, Pointwise, and Uniform Continuity: A Constructive Note.Douglas S. Bridges - 1993 - Mathematical Logic Quarterly 39 (1):55-61.
    The main result of this paper is a weak constructive version of the uniform continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.
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  19.  24
    On the Constructive Convergence of Series of Independent Functions.Douglas S. Bridges - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (3-6):93-96.
  20.  9
    Complements of Intersections in Constructive Mathematics.Douglas S. Bridges & Hajime Ishihara - 1994 - Mathematical Logic Quarterly 40 (1):35-43.
    We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and (...)
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  21.  12
    A Criterion for Compactness in Metric Spaces?Douglas S. Bridges - 1979 - Mathematical Logic Quarterly 25 (7‐12):97-98.
  22.  19
    A General Constructive Intermediate Value Theorem.Douglas S. Bridges - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):433-435.
  23.  18
    Square Roots and Powers in Constructive Banach Algebra Theory.Douglas S. Bridges & Robin S. Havea - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 68--77.
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  24.  12
    The Continuum Hypothesis Implies Excluded Middle.Douglas S. Bridges - 2016 - In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter. pp. 111-114.
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  25.  13
    Continuity Properties of Preference Relations.Marian A. Baroni & Douglas S. Bridges - 2008 - Mathematical Logic Quarterly 54 (5):454-459.
    Various types of continuity for preference relations on a metric space are examined constructively. In particular, necessary and sufficient conditions are given for an order-dense, strongly extensional preference relation on a complete metric space to be continuous. It is also shown, in the spirit of constructive reverse mathematics, that the continuity of sequentially continuous, order-dense preference relations on complete, separable metric spaces is connected to Ishihara's principleBD-ℕ, and therefore is not provable within Bishop-style constructive mathematics alone.
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  26.  16
    Church's Thesis and Bishop's Constructivism.Douglas S. Bridges - 2006 - In A. Olszewski, J. Wole'nski & R. Janusz (eds.), Church's Thesis After Seventy Years. Ontos Verlag. pp. 1--58.
  27.  15
    A Note on Morse's Lambda‐Notation in Set Theory.Douglas S. Bridges - 1978 - Mathematical Logic Quarterly 24 (8):113-114.
  28.  9
    Apartness Spaces and Uniform Neighbourhood Structures.Douglas S. Bridges - 2016 - Annals of Pure and Applied Logic 167 (9):850-864.
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  29.  12
    Constructive Complements of Unions of Two Closed Sets.Douglas S. Bridges - 2004 - Mathematical Logic Quarterly 50 (3):293.
    It is well known that in Bishop-style constructive mathematics, the closure of the union of two subsets of ℝ is ‘not’ the union of their closures. The dual situation, involving the complement of the closure of the union, is investigated constructively, using completeness of the ambient space in order to avoid any application of Markov's Principle.
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  30.  8
    A General Constructive Intermediate Value Theorem.Douglas S. Bridges - 1989 - Mathematical Logic Quarterly 35 (5):433-435.
  31.  8
    Constructive Notions of Strict Convexity.Douglas S. Bridges - 1993 - Mathematical Logic Quarterly 39 (1):295-300.
    Two classically equivalent, but constructively inequivalent, strict convexity properties of a preference relation are discussed, and conditions given under which the stronger notion is a consequence of the weaker. The last part of the paper introduces uniformly rotund preferences, and shows that uniform rotundity implies strict convexity. The paper is written from a strictly constructive point of view, in which all proofs embody algorithms. MSC: 03F60, 90A06.
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  32.  6
    Almost New Pre-Apartness From Old.Douglas S. Bridges - 2012 - Annals of Pure and Applied Logic 163 (8):1009-1015.
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  33.  6
    On the Constructive Convergence of Series of Independent Functions.Douglas S. Bridges - 1979 - Mathematical Logic Quarterly 25 (3‐6):93-96.
  34.  5
    First Steps in Constructive Game Theory.Douglas S. Bridges - 2004 - Mathematical Logic Quarterly 50 (45):501-506.
    The minimax theorem of matrix game theory is examined from a constructive point of view. It is then shown that the existence of solutions for matrix games cannot be proved constructively, but that a 2-by-2 game with at most one solution has a constructible solution.
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