1,508 research outputs found
The Brouwer Fixed Point Theorem Revisited
We revisit the investigation of the computational content of the Brouwer Fixed Point Theorem in [7], and answer the two open questions from that work. First, we show that the computational hardness is independent of the dimension, as long as it is greater than 1 (in [7] this was only established for dimension greater than 2). Second, we show that restricting the Brouwer Fixed Point Theorem to L-Lipschitz functions for any L > 1 also does not change the computational strength, which together with prior results establishes a trichotomy for L > 1, L = 1 and L < 1.SCOPUS: cp.kinfo:eu-repo/semantics/publishe
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page
Foley's Thesis, Negishi's Method, Existence Proofs and Computation
Duncan Foleyís many-faceted and outstanding contributions to macroeconomics, microeconomics, general equilibrium theory, the theory of taxation, history of economic thought, the magnificent dynamics of classical economics, classical value theory, Bayesian statistics, formal dynamics and, most recently, fascinating forays into an interpretation of economic evolution from a variety of complexity theoretic viewpoints have all left -and continue to leave - significant marks in the development and structure of economic theory. He belongs to the grand tradition of visionaries who theorise with imaginative audacity on the dynamics, evolution and contradictions of capitalist economies - a tradition that, perhaps, begins with Marx and Mill, continues with Keynes and Schumpeter, reaching new heights with the iconoclastic brilliancies of a Tsuru and a Goodwin, a Chakravarty and a Nelson, and to which Duncan Foley adds a lustre of much value. In this contribution I return to mathematical themes broached in Foleyís brilliant and pioneering Yale doctoral dissertation (Foley, 1967) and attempt to view them as a Computable Economist would.The intention is to suggest that algorithmic indeterminacies are intrinsic to the foundations of economic theory in the mathematical modeEquilibrium existence theorems, Welfare theorems, Constructive proofs, Computability
Negishi's Theorem and Method
Takashi Negishi's remarkable youthful contribution to welfare economics, general equilibrium theory and, with the benefit of hindsight, also to one strand of computable general equilibrium theory, all within the span of six pages in one article, has become one of the modern classics of general equilibrium theory and mathematical economics. Negishi's celebrated theorem and what has been called Negishi's Method have formed one foundation for computable general equilibrium theory. In this paper I investigate the computable and constructive aspects of the theorem and the methodComputable General Equilibrium, Fundamental Theorems of Welfare Economics, Negishi's Method
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