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Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Cited by 14 (5 self)
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
Hyperbolic topology of normed linear spaces
"... Dedicated to Professor Tsugunori Nogura on his sixtieth birthday In a previous paper [6], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characte ..."
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Dedicated to Professor Tsugunori Nogura on his sixtieth birthday In a previous paper [6], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces Lp(Ω, Σ, µ) on which the hyperbolic topology induced by the norm ∥·∥p coincides with the norm topology. We show the following. (1) The hyperbolic topology and the norm topology coincide for 1 < p < ∞. (2) They coincide on L1(Ω, Σ, µ) if and only if µ(Ω) = 0 or Ω has a finite partition by atoms. (3) They coincide on L∞(Ω, Σ, µ) if and only if µ(Ω) = 0 or there is an atom in Σ.