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13
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 21 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
Natural computation and nonTuring models of computation
 Theoretical Computer Science
, 2004
"... We propose certain nonTuring models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of applicatio ..."
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Cited by 18 (9 self)
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We propose certain nonTuring models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are illsuited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics. Key words: analog computation, analog computer, biocomputation, computability, computation on reals, continuous computation, formal system, hypercomputation,
Essential Components of algebraic differential equations
 J. of Symb. Comp
, 1999
"... We present an algorithm to determine the essential singular components of an algebraic differential equation. Geometrically, this corresponds to determining the singular solutions that have enveloping properties. The algorithm is practical and efficient because it is factorization free, unlike the p ..."
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Cited by 13 (4 self)
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We present an algorithm to determine the essential singular components of an algebraic differential equation. Geometrically, this corresponds to determining the singular solutions that have enveloping properties. The algorithm is practical and efficient because it is factorization free, unlike the previous such algorithm. c ○ 1999 Academic Press 1.
Identifying PredatorPrey Processes from TimeSeries
, 2000
"... this paper we apply modelfitting to ..."
Global attractors: topology and finitedimensional dynamics
 J. Dyn. Diff. Eq
, 1999
"... Proposed running head: Attractors: topology and dynamics Many a dissipative evolution equation possesses a global attractor A with finite Hausdorff dimension d. In this paper it is shown there is an embedding X of A into IR N, with N = [2d + 2], such that X is the global attractor of some finitedim ..."
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Cited by 4 (3 self)
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Proposed running head: Attractors: topology and dynamics Many a dissipative evolution equation possesses a global attractor A with finite Hausdorff dimension d. In this paper it is shown there is an embedding X of A into IR N, with N = [2d + 2], such that X is the global attractor of some finitedimensional system on IR N with trivial dynamics on X. This allows the construction of a discrete dynamical system on IR N which reproduces the dynamics of the time T map on A, and has an attractor within an arbitrarily small neighbourhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.
The nature of computing — computing in nature
, 2005
"... My goal in this report is to recontextualize the concept of computation. I review the historical roots of ChurchTuring computation to show that the theory exists in a frame of relevance, which underlies the assumptions on which it rests and the questions it is suited to answer. Although this frame ..."
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Cited by 1 (1 self)
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My goal in this report is to recontextualize the concept of computation. I review the historical roots of ChurchTuring computation to show that the theory exists in a frame of relevance, which underlies the assumptions on which it rests and the questions it is suited to answer. Although this frame of relevance is appropriate in many circumstances, there are many important applications of the idea of computation for which it is not relevant. These include natural computation (computation occurring in or inspired by nature), nanocomputation (computation based on nanoscale objects and processes), and computation based on quantum theory. As a consequence we need, not so much to abandon the ChurchTuring model of computation, as to supplement it with new models based on different assumptions and suited to answering different questions. Therefore I will discuss alternative frames of relevance more suited to the interrelated application areas of natural computation, emergent computation, and nanocomputation. Central issues include continuity, indeterminacy, and parallelism. Finally, I will argue that once we understand computation in a broader sense than the ChurchTuring model, we begin to see new possibilities for using natural processes to achieve our computational goals. These possibilities will increase in importance as we approach the limits of electronic binary logic as a basis for computation. They will also help us to understand computational processes in nature. * This report is based on an invited presentation at the workshop “Natural Processes & Models of
The UMachine: A Model of Generalized Computation
 INT. JOURN. OF UNCONVENTIONAL COMPUTING, VOL. 6, PP. 265–283
, 2010
"... ..."
Asymptotic Forms and Algebraic Differential Equations
, 1994
"... We analyse the complexity of a simple algorithm for computing asymptotic solutions of an algebraic differential equation. This analysis is based on a computation of the number of possible asymptotic monomials of a certain order, and on the study of the growth of this number as the order of the equat ..."
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Cited by 1 (1 self)
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We analyse the complexity of a simple algorithm for computing asymptotic solutions of an algebraic differential equation. This analysis is based on a computation of the number of possible asymptotic monomials of a certain order, and on the study of the growth of this number as the order of the equation grows.
Junction Conditions, Resolution of Singularities and Nonlinear Equations of Physics
, 2006
"... For large classes of systems of polynomial nonlinear PDEs necessary and sufficient conditions are given for the existence of solutions which are discontinuous across hypersurfaces. These PDEs contain the NavierStokes equations, as well as those of General Relativity and MagnetoHydrodynamics. 1. P ..."
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For large classes of systems of polynomial nonlinear PDEs necessary and sufficient conditions are given for the existence of solutions which are discontinuous across hypersurfaces. These PDEs contain the NavierStokes equations, as well as those of General Relativity and MagnetoHydrodynamics. 1. Preliminary Remarks There has for longer been an interest in finding junction condition across hypersurfaces of discontinuities for solutions of various nonlinear equations of Physics, such as in General Relativity, MagnetoHydrodynamics, and so on. Two approaches in this regard have been pursued in the literature. One of them is trying to keep the equations and introduce hard to deal with integral formulations of the respective conditions, while the other is introducing adhoc simplifying assumptions which are dictated by mathematical convenience, rather than
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.