The existence and uniqueness properties for extremal mappings with smallest weighted L p distortion between annuli and the related Grötzsch type problems are discussed. An interesting critical phase type phenomena is observed. When p < 1, apart from the identity map, minimizers never exist. When p = 1 we observe Nitsche type phenomena; minimisers exist within a range of conformal moduli determined by properties of the weight function and not otherwise. When p > 1 minimisers always exist.Interpreting the weight function as a density or "thickness profile" leads to interesting models for the deformation of highly elastic bodies and tearing type phenomena.
This paper begins an analysis of the real line using an inconsistencytolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistencyreliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.This paper has two aims. The first is to show that, by weakening the logic within which we work, so as to allow for the possibility of non-trivial inconsistency, we are still able to do everyday mathematics. We see why such practice is possible: We take classical proofs, note the obstacles from a paraconsistent viewpoint, and then reconstruct the arguments, using only inferences that are valid in nontrivial inconsistent contexts. This 'breakdown' of analysis enables a fuller understanding of the fine structure of proofs. The strategy is to reverse engineer classical results (cf. [19, sec.16.1]), eventually deriving a paraconsistent basis for analysis, and this paper is a step in that direction. A fundamental result is the Heine-Borel theorem, characterising compactness of bounded closed intervals.The second aim of the paper is to explain, in part, a basic historical fact: the original calculus of Newton and Leibniz was inconsistent-in the sense that the original definition of derivative involves a number ε which is at some points non-zero (to permit division), and at other points zero (to allow 'infinitessimal' quantities to vanish); see [6], [18].1 Yet, despite the inconsistency, early practitioners were able to draw meaningful and useful conclusions. Despite contradictions, they got the right answers. Accounting for how this is possible provides one motivation for working in a paraconsistent logic-reconstructing structures in which inconsistent reasoning actually did take place, and yet did so without falling into incoherence. 1]), again involving a different approach. By contrast, this paper is not a model-theoretic development, and so does not include similar claims to non-triviality. 1Our interest here is in compactness properties. It turns out that, to understand the Heine-Borel theorem in a paraconsistent system, detours through ideas from constructive analysis are needed; e.g. Brouwer's fan theorem plays a key role [5, §5.3]. This is something of a surprise, since paraconsistency has often been taken as a dual or opposite to constructive, paracomplete theories [16, ch. 11], [7].2 It has been quipped of constructive mathematicians that they perform mathematics with one hand tied behind their back, working without universal application of the law of excluded middle. Here we tie the other hand behind our backs, and work without disjunctive syllogism or general forms of reductio. Constructive mathematicians have developed powerful tools to overcome truth value gaps; here, by restoring excluded middle, but considering the possibility of truth-value gluts, th...
The so-called paradoxes of material implication have motivated the development of many nonclassical logics over the years [2][3][4][5]11]. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency.
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 non-existence of constructive solutions to the Navier-Stokes 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.
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