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Lipschitz continuous ordinary differential equations are polynomialspace complete
 Comput. Complexity
, 2010
"... ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differentia ..."
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ABSTRACT. In answer to Ko’s question raised in 1983, we show that an initial value problem given by a polynomialtime computable, Lipschitz continuous function can have a polynomialspace complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomialspace computation tableaux with equally restricted feedback, and show that they are still polynomialspace complete. The same technique also settles Ko’s two later questions on Volterra integral equations.
Continuity of Operators on Continuous and Discrete Time Streams
"... its diversity and scale and, especially, in its conceptual and technical depth. His achievement stands out and defies the simple conventions of contemporary judgment and praise. As a scientist and intellectual, and a longstanding friend and collaborator of ours, he is unique. For this birthday celeb ..."
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Cited by 1 (1 self)
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its diversity and scale and, especially, in its conceptual and technical depth. His achievement stands out and defies the simple conventions of contemporary judgment and praise. As a scientist and intellectual, and a longstanding friend and collaborator of ours, he is unique. For this birthday celebration, we offer congratulations and encouragement, and record our admiration and gratitude. We consider the semantics of networks processing streams of data from a complete metric space. We consider two types of data streams: those based on continuous time (used in networks of physical components and analog devices), and those based on discrete time (used in concurrent algorithms). The networks are both governed by global clocks and together model a huge range of systems. Previously, we have investigated these two types of networks separately. Here we combine their study in a unified theory of stream transformers, given as fixed points of equations. We begin to develop this theory by using the standard mathematical techniques of topology to prove certain computationally desirable properties of these semantic functions, notably continuity, which is significant for models of a physical system, according to Hadamard’s principle. Key words and phrases: analog computing, analog networks, compact open topology, continuous stream operations, continuous time streams, discrete time streams, fixed points, Hadamard’s principle, synchronous concurrent algorithms, topological algebras. 2 1
doi:10.1093/comjnl/bxs054 A Class of Contracting Stream Operators †
"... discrete time streams. Theoret. Comput. Sci., 412, 3378–3403), Tucker and Zucker present a model for the semantics of analog networks operating on streams from topological algebras. Central to ..."
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discrete time streams. Theoret. Comput. Sci., 412, 3378–3403), Tucker and Zucker present a model for the semantics of analog networks operating on streams from topological algebras. Central to