16 found
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  1.  32
    Brouwer's Fan Theorem and Unique Existence in Constructive Analysis.Josef Berger & Hajime Ishihara - 2005 - Mathematical Logic Quarterly 51 (4):360-364.
    Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. Furthermore, (...)
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  2.  14
    The Anti-Specker Property, a Heine–Borel Property, and Uniform Continuity.Josef Berger & Douglas Bridges - 2008 - Archive for Mathematical Logic 46 (7-8):583-592.
    Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.
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  3.  16
    A Separating Hyperplane Theorem, the Fundamental Theorem of Asset Pricing, and Markov's Principle.Josef Berger & Gregor Svindland - 2016 - Annals of Pure and Applied Logic 167 (11):1161-1170.
  4.  19
    The Fan Theorem and Unique Existence of Maxima.Josef Berger, Douglas Bridges & Peter Schuster - 2006 - Journal of Symbolic Logic 71 (2):713 - 720.
    The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.
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  5.  27
    Classifying Dini's Theorem.Josef Berger & Peter Schuster - 2006 - Notre Dame Journal of Formal Logic 47 (2):253-262.
    Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in (...)
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  6.  9
    A Bizarre Property Equivalent To The -Fan Theorem.Josef Berger & Douglas Bridges - 2006 - Logic Journal of the IGPL 14 (6):867-871.
    It is shown, with intuitionistic logic, that if every locally constant function from to has a property akin to constancy, then the fan theorem for -bars holds, and conversely.
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  7.  21
    A Predicative Completion of a Uniform Space.Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2012 - Annals of Pure and Applied Logic 163 (8):975-980.
  8.  11
    Double Sequences, Almost Cauchyness and BD-N.Josef Berger, Douglas Bridges & Erik Palmgren - 2012 - Logic Journal of the IGPL 20 (1):349-354.
    It is shown that, relative to Bishop-style constructive mathematics, the boundedness principle BD-N is equivalent both to a general result about the convergence of double sequences and to a particular one about Cauchyness in a semi-metric space.
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  9.  12
    The Binary Expansion and the Intermediate Value Theorem in Constructive Reverse Mathematics.Josef Berger, Hajime Ishihara, Takayuki Kihara & Takako Nemoto - 2019 - Archive for Mathematical Logic 58 (1-2):203-217.
    We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.
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  10.  18
    The Weak König Lemma and Uniform Continuity.Josef Berger - 2008 - Journal of Symbolic Logic 73 (3):933-939.
    We prove constructively that the weak König lemma and quantifier-free number-number choice imply that every pointwise continuous function from Cantor space into Baire space has a modulus of uniform continuity.
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  11.  5
    Brouwer’s Fan Theorem and Convexity.Josef Berger & Gregor Svindland - 2018 - Journal of Symbolic Logic 83 (4):1363-1375.
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  12.  5
    Convexity and Unique Minimum Points.Josef Berger & Gregor Svindland - 2019 - Archive for Mathematical Logic 58 (1-2):27-34.
    We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.
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  13.  12
    Aligning the Weak König Lemma, the Uniform Continuity Theorem, and Brouwer’s Fan Theorem.Josef Berger - 2012 - Annals of Pure and Applied Logic 163 (8):981-985.
  14.  17
    The Genetic Code and the Origin of Life.Josef Berger - 1976 - Acta Biotheoretica 25 (4):259-263.
    The problem of the origin of life understandably counts as one of the most exciting questions in the natural sciences, but in spite of almost endless speculation on this subject, it is still far from its final solution. The complexity of the functional correlation between recent nucleic acids and proteins can e.g. give rise to the assumption that the genetic code (and life) could not originate on the Earth. It was Portelli (1975) who published the hypothesis that the genetic code (...)
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  15.  6
    Editorial: Math. Log. Quart. 1/2008.Josef Berger, Dirk Pattinson, Peter Schuster & Júlia Zappe - 2008 - Mathematical Logic Quarterly 54 (1):4-4.
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  16.  6
    Exact Calculation of Inverse Functions.Josef Berger - 2005 - Mathematical Logic Quarterly 51 (2):201-205.
    We represent continuous functions on compact intervals by sequences of functions defined on finite sets of rational numbers. We call this an exact representation. This enables us to calculate the values of the function arbitrarily exactly, without roundoff errors. As an application we develop a procedure to transfer an exact representation of an increasing function into an exact representation of the corresponding inverse function.
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