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The Principle of Open Induction on Cantor space and the Approximate-Fan Theorem
The paper is a contribution to intuitionistic reverse mathematics. We work in
a weak formal system for intuitionistic analysis. The Principle of Open
Induction on Cantor space is the statement that every open subset of Cantor
space that is progressive with respect to the lexicographical ordering of
Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an
extension of the Fan Theorem that follows from Brouwer's principle of induction
on bars in Baire space and implies the Principle of Open Induction on Cantor
space. The Principle of Open Induction in Cantor space implies the Fan Theorem,
but, conversely the Fan Theorem does not prove the Principle of Open Induction
on Cantor space. We list a number of equivalents of the Principle of Open
Induction on Cantor space and also a number of equivalents of the
Approximate-Fan Theorem
A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators
First, we reconstruct Wim Veldman's result that Open Induction on Cantor
space can be derived from Double-negation Shift and Markov's Principle. In
doing this, we notice that one has to use a countable choice axiom in the proof
and that Markov's Principle is replaceable by slightly strengthening the
Double-negation Shift schema. We show that this strengthened version of
Double-negation Shift can nonetheless be derived in a constructive intermediate
logic based on delimited control operators, extended with axioms for
higher-type Heyting Arithmetic. We formalize the argument and thus obtain a
proof term that directly derives Open Induction on Cantor space by the shift
and reset delimited control operators of Danvy and Filinski
Single phase matrix converter for radio frequency induction heating
Conventional converters for radio frequency induction heating usually follow an AC-DC-AC structure, which can exhibit non-unity power factor and introduce large harmonic currents into the utility supply. The need for a direct converter for radio frequency induction heating, featuring unity power factor, and sinusoidal input current, has motivated the development of a single phase matrix converter as an induction heater. A novel commutation strategy is therefore required to ensure smooth operation of the converter whilst creating a high frequency output under soft switching conditions. The operating principle and features of the proposed converter are described here, and experimentally verifie
Eisenstein integrals and induction of relations
I give a survey of joint work with Henrik Schlichtkrull on the induction of
certain relations among (partial) Eisenstein integrals for the minimal
principal series of a reductive symmetric space. I explain the application of
this principle of induction to the proofs of a Fourier inversion formula and a
Paley-Wiener theorem. Finally, the relation with the Plancherel decomposition
is discussed.Comment: Latex2e, 22 pp, Proc. Conf. `Analyse Harmonique Non Commutative
(colloque en l'honneur de Jacques Carmona)' CIRM, Luminy, 20-24 Mai, 200
A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics
Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory -.i.e., a computable approximation to Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. >From these points of view I suggest, first, a recasting of Jevons's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics.
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