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Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals
 Higher Order Logic Theorem Proving and its Applications (HUG `93
, 1993
"... In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from ..."
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Cited by 33 (4 self)
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In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and cryptosystems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...
Reasoning About the Reals: the marriage of HOL and Maple
, 1993
"... . Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In ..."
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Cited by 10 (0 self)
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. Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In this paper we try to get the best of both worlds by careful exploitation of a link between a theorem prover and a computer algebra system. 1 Motivation In the HOL theorem prover[5], a theory of real numbers has been developed, using a rigorous definition in terms of Dedekind cuts [8]. It is therefore possible to apply HOL to areas traditionally within the purview of Computer Algebra Systems (CASs). This offers two main benefits. Firstly, theorem provers are designed to manipulate proofs and theorems in a coherent and structured way, with all concepts clearly defined. By contrast, most CASs have no concept of `logic' as such  they usually take an algebraic expression and return another pur...
Foundations for Relativistic Quantum Theory I: Feynman's Operator Calculus and the Dyson Conjectures
"... In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initialvalue problem and construct the Dyson series. ..."
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Cited by 2 (2 self)
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In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initialvalue problem and construct the Dyson series.
Author manuscript, published in "2009 American Control Conference (2009)" Observability Normal Forms for a class of switched systems with zeno phenomena
, 2010
"... Abstract — This paper deals with necessary and sufficient conditions to transform a class of switched systems to a particular form dedicated to observer design with and without zeno phenomena. Meanwhile, sufficient observability conditions for switched system with or without zeno phenomena are given ..."
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Abstract — This paper deals with necessary and sufficient conditions to transform a class of switched systems to a particular form dedicated to observer design with and without zeno phenomena. Meanwhile, sufficient observability conditions for switched system with or without zeno phenomena are given. In the last section, some observer structures are proposed upon two academical examples. Index Terms — Zeno phenomena, switched system, observability. I.
Observability forms for switched systems with Zeno phenomenon or high switching frequency
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2011
"... This paper deals with the observability of a class of switched systems with Zeno phenomenon or high switching frequency. Particularly, three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed. Meanwhile, sufficient and necessa ..."
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This paper deals with the observability of a class of switched systems with Zeno phenomenon or high switching frequency. Particularly, three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed. Meanwhile, sufficient and necessary conditions for the existence of a diffeomorphism to transform a class of switched systems into one of such forms are presented. Examples and simulations are given at the end to highlight the theoretical results.
Review of A garden of integrals,
, 2008
"... Executive summary: Wellwritten set of extended exercises that take the ..."
CONSTRUCTIVE REPRESENTATION THEORY FOR THE FEYNMAN OPERATOR CALCULUS
, 2007
"... 1 2 GILL AND ZACHARY Abstract. In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. We first develop an operator version of the HenstockKurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner ..."
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1 2 GILL AND ZACHARY Abstract. In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. We first develop an operator version of the HenstockKurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner originally envisioned by Feynman. After developing our timeordered operator theory we extend a few of the important theorems of semigroup theory, including the HilleYosida theorem. As an application, we unify and extend the theory of timedependent parabolic and hyperbolic evolution equations. We then develop a general perturbation theory and use it to prove that all theories generated by semigroups are asympotic in the operatorvalued sense of Poincaré. This allows us to provide a general theory for the interaction representation of relativistic quantum theory. We then show that our theory can be reformulated as a physically motivated sum over paths, and use this version to extend the Feynman path integral to include more general interactions. Our approach is independent of the space of continuous functions and thus makes the question of the existence of a measure more of a natural expectation than a death blow to the foundations for the Feynman integral. 1.
Author manuscript, published in "48th IEEE Conference on Decision and Control, CDC 2009 (2009)" Algebraic observer for a class of switched systems with Zeno phenomenon
, 2009
"... Abstract — For a large class of switched systems with zeno phenomenon, classical observer cannot be applied directly since the terms leading to zeno phenomenon are not derivable. However in this paper, by assuming that these terms are integrable in the less restrictive way, we can define a new outpu ..."
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Abstract — For a large class of switched systems with zeno phenomenon, classical observer cannot be applied directly since the terms leading to zeno phenomenon are not derivable. However in this paper, by assuming that these terms are integrable in the less restrictive way, we can define a new output, with which algebraic observer can then be adopted to estimate the states of the studied switched systems with zeno phenomenon. For simplicity, the main idea is explained via normal forms, while it can also be extended to generic switched systems. I.
ON THE DUAL SPACE OF THE HENSTOCKKURZWEIL INTEGRABLE FUNCTIONS IN N DIMENSIONS*
"... The dual space of the class of HenstockKurzweil integrable functions is well known in the onedimensional case and corresponds to the space of multipliers which, in turn, coincides with the class of functions of bounded essential variation. Comparable results in higher dimensions have been elusive. ..."
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The dual space of the class of HenstockKurzweil integrable functions is well known in the onedimensional case and corresponds to the space of multipliers which, in turn, coincides with the class of functions of bounded essential variation. Comparable results in higher dimensions have been elusive. For cases in which the partitions defining the HenstockKurzweil integrals are defined on ncells (parallelepipeds) with their sides parallel to the coordinate axes (i.e., Cartesian products of compact intervals), we prove that a function is a multiplier of the class of ndimensional (n ≥ 2) HenstockKurzweil integrable functions if and only if it is of strongly bounded essential variation as defined by Kurzweil. This result was proved earlier by T.Y. Lee, T. S. Chew, and P.Y. Lee in the twodimensional case by a different method. The sufficiency part of our proof makes use of a generalization of a method used earlier by P.Y. Lee in the onedimensional case.