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Constructive Aspects of the Dirichlet Problem 1
"... Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we ..."
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Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we discuss the equivalence of the existence of weak solutions of the Dirichlet Problem and the existence of minimizers for certain associated integral functionals. Our analysis pinpoints exactly what is needed to nd weak solutions of the Dirichlet Problem: namely, the computation of either the norm of a linear functional on a certain Hilbert space or, equivalently, the in mum of an associated integral functional.
Documenta Math. 973 Inheriting the AntiSpecker Property
, 2009
"... Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BDN comes into play in the discussion of products with the antiSpecker property. ..."
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Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BDN comes into play in the discussion of products with the antiSpecker property.
1 Solving the Dirichlet problem constructively
"... Abstract: The Dirichlet problem is of central importance in both applied and abstract potential theory. We prove the (perhaps surprising) result that the existence of solutions in the general case is an essentially nonconstructive proposition: there is no algorithm which will actually compute soluti ..."
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Abstract: The Dirichlet problem is of central importance in both applied and abstract potential theory. We prove the (perhaps surprising) result that the existence of solutions in the general case is an essentially nonconstructive proposition: there is no algorithm which will actually compute solutions for arbitrary domains and boundary conditions. A corollary of our results is the nonexistence of constructive solutions to the NavierStokes equations of fluid flow. But not all the news is bad: we provide reasonable conditions, omitted in the classical theory but easily satisfied, which ensure the computability of solutions.
A Constructive Study of Landau's . . .
, 2009
"... A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively. ..."
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A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively.
CCA 2008 Efficient Computation with Dedekind Reals
"... Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of D ..."
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Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of Dedekind reals by constructing them within Abstract Stone Duality (ASD), a computationally meaningful calculus for topology. This provides the theoretical background for a novel way of computing with real numbers in the style of logic programming. Real numbers are defined in terms of (lower and upper) Dedekind cuts, while programs are expressed as statements about real numbers in the language of ASD. By adapting Newton’s method to interval arithmetic we can make the computations as efficient as those based on Cauchy reals. The results reported in this talk are joint work with Paul Taylor. Keywords:
Separatedness in Constructive Topology
 DOCUMENTA MATH.
, 2003
"... We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework. ..."
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We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework.
Did Brouwer Really Believe That?
, 2007
"... This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses ..."
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This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses Brouwer (who, incidentally, was normally known not as “Jan ” but as “Bertus”, a shortening of his second name, Egbertus), 1 he says:...later in his career, he [Brouwer] became the most forceful proponent of the socalled intuitionist philosophy of mathematics, which not only forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false (thereby disallowing the method of proof by contradiction). The consequences of taking this position are dire. For instance, an intuitionist would not accept the existence of an irrational number! In fact, in his later years, Brouwer did not view the Brouwer Fixed Point Theorem as a theorem. These sentences contain a number of outdated but still common misconceptions