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Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 8 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Dynamical Embodiments of Computation in Cognitive Processes
"... Dynamics is not enough for cognition nor is it a substitute for information processing aspects of brain behavior. Moreover, dynamics and computation are not at odds, but are quite compatible. They can be synthesized so that any dynamical system can be analyzed in terms of its intrinsic computational ..."
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Cited by 6 (0 self)
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Dynamics is not enough for cognition nor is it a substitute for information processing aspects of brain behavior. Moreover, dynamics and computation are not at odds, but are quite compatible. They can be synthesized so that any dynamical system can be analyzed in terms of its intrinsic computational components.
Borgne. Topology and static response of interaction networks in molecular biology
 J R Soc Interface
, 2006
"... molecular biology ..."
STABLE ERGODICITY
, 2003
"... A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, cham ..."
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Cited by 3 (0 self)
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A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen
Periodic orbit spectrum in terms of Ruelle–Pollicott resonances
, 2004
"... Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory “p ” returns to its initial conditions after some fixed time τp. Our aim is to investigate the spectrum {τ1, τ2,...} of periods of the periodic orbits. An explicit formula for th ..."
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Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory “p ” returns to its initial conditions after some fixed time τp. Our aim is to investigate the spectrum {τ1, τ2,...} of periods of the periodic orbits. An explicit formula for the density ρ(τ) = ∑ p δ(τ − τp) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle–Pollicott resonances). For large periods, corrections to the well–known exponential growth of the smooth part of the density are obtained. An alternative formula for ρ(τ) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.
A computerassisted proof for Σ3chaos in the forced damped pendulum equation (in preparation
"... Abstract. The present paper is devoted to studying Hubbard’s pendulum equation ¨x + 10 −1 ˙x + sin(x) = cos(t). By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the lev ..."
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Abstract. The present paper is devoted to studying Hubbard’s pendulum equation ¨x + 10 −1 ˙x + sin(x) = cos(t). By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the level of a theorem completely proved. A distinguished family of solutions is shown to be chaotic in the sense that on consecutive time intervals (2kπ, 2(k+1)π) (k ∈ Z) individual members of the family can freely “choose ” between the following possibilities: the pendulum either crosses the bottom position exactly once clockwise or does not cross the bottom position at all or crosses the bottom position exactly once counterclockwise. The proof follows the topological index/degree approach by Mischaikow, Mrozek, and Zgliczynski. The novelty is a definition of the transition graph for which the periodic orbit lemma, the key technical result of the approach aforementioned, turns out to be a consequence of Brouwer’s fixed point theorem. The role of wholly automatic versus ‘trial and error with human overhead ’ computer procedures in detecting chaos is also discussed. Key words. forced damped pendulum, Σ3chaos, computeraided proof, transition graph, interval arithmetic
ONEDIMENSIONAL DYNAMICS IN THE NEW MILLENNIUM
"... Abstract. In the early 60’s Sarkovskii discovered his famous theorem on the coexistence of periodic orbits for interval maps. Then, in the mid 70’s, Li & Yorke rediscovered this result and somewhat later the papers by Feigenbaum and Coullet & Tresser on renormalisation and by Guckenheimer and Misiur ..."
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Abstract. In the early 60’s Sarkovskii discovered his famous theorem on the coexistence of periodic orbits for interval maps. Then, in the mid 70’s, Li & Yorke rediscovered this result and somewhat later the papers by Feigenbaum and Coullet & Tresser on renormalisation and by Guckenheimer and Misiurewicz on sensitive dependence and existence of invariant measures, kicked off one of the most exciting areas within dynamical systems: iterations in dimension one. The purpose of this paper is to survey some of the recent developments, and pose some of the challenges and questions that keep this subject so intriguing. One of the appealing aspects of the study of iterations of maps of the interval is that the situation is far from trivial, and yet the theory is remarkably complete. That it is far from trivial is for example clear from the ‘period 3 implies chaos theorem ’ and from the universality found in the periodic doubling bifurcations. It is surprising therefore that, in spite of this complexity, one can prove rather general results and that many of the natural questions have now been resolved.
Prepared using rsifpublic.cls J. R. Soc. Lond. Interface (0000)
"... Topology and static response of interaction networks in molecular biology ..."
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Topology and static response of interaction networks in molecular biology