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A computational model for multivariable differential calculus
 Proc. FoSSaCS 2005, LNCS
, 2005
"... Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the pro ..."
Abstract

Cited by 10 (6 self)
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Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domaintheoretic notion of line integral is used to extend Greenâ€™s theorem to intervalvalued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n + 1 rational step functions, assuming the integrability of the ntuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multivariable differential calculus. 1
A continuous derivative for realvalued functions
 New Computational Paradigms, Changing Conceptions of What is Computable
, 2008
"... We develop a notion of derivative of a realvalued function on a Banach space, called the Lderivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the Lderivative of a function are nonempty weak * compact and conv ..."
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Cited by 3 (3 self)
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We develop a notion of derivative of a realvalued function on a Banach space, called the Lderivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the Lderivative of a function are nonempty weak * compact and convex subsets of the dual of the Banach space. The Lderivative, however, is shown to be upper semi continuous, a result which is not known to hold for the Clarke gradient. We also formulate the notion of primitive maps dual to the Lderivative, an extension of Fundamental Theorem of Calculus for the Lderivative and a domain for computation of realvalued functions on a Banach space with a corresponding notion of effectivity. For realvalued functions on finite dimensional Euclidean spaces, the Lderivative can be obtained within an effectively given continuous domain. We also show that in finite dimensions the Lderivative and the Clarke gradient coincide thus providing a computable representation for the latter in this case. This paper is dedicated to the historical memory of Sharaf aldin Tusi (d. 1213), the Iranian mathematician who was the first to use the derivative systematically to solve for roots of cubic polynomials and find their maxima. 1
Differential operator and weak topology for Lipschitz maps
, 2009
"... We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function spac ..."
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Cited by 1 (1 self)
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We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of nonempty weak * compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the Lderivative and the Clarke gradient coincide, we provide a simple characterisation of the basic open subsets of the Ltopology in terms of ties or primitive maps of functions. We use this to verify that the Ltopology is strictly coarser than the wellknown Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the Ltopology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of nonempty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the Lderivative. We finally show that in dimension one the Lderivative operator is onto and that it is a computable functional.
Weak topology and a differentiable operator for Lipschitz maps
"... We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function spac ..."
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We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of nonempty weak * compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the Lderivative and the Clarke gradient coincide, we provide a simple characterisation of the basic open subsets of the Ltopology in terms of ties or primitive maps of functions. We use this to verify that the Ltopology is strictly coarser than the wellknown Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of nonempty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the Lderivative.