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Computing The Volume Of Convex Bodies: A Case Where Randomness Provably Helps
, 1991
"... We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide som ..."
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Cited by 78 (7 self)
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We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide some applications of this latter result. Supported by NATO grant RG0088/89 y Supported by NSF grant CCR8900112 and NATO grant RG0088/89 1 Introduction The mathematical study of areas and volumes is as old as civilization itself, and has been conducted for both intellectual and practical reasons. As far back as 2000 B.C., the Egyptians 1 had methods for approximating the areas of fields (for taxation purposes) and the volumes of granaries. The exact study of areas and volumes began with Euclid 2 and was carried to a high art form by Archimedes 3 . The modern study of this subject began with the great astronomer Johann Kepler's treatise 4 Nova stereometria doliorum vinariorum, wh...
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of th ..."
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Cited by 44 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
Incompleteness Theorems for Random Reals
, 1987
"... We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property tha ..."
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Cited by 43 (0 self)
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We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin. This yields a number of powerful Godel incompletenesstype results concerning the limitations of the axiomatic method, in which entropyinformation measures are used. c fl 1987 Academic Press, Inc. 2 G. J. Chaitin 1. Introduction It is now half a century since Turing published his remarkable paper On Computable Numbers, with an Application to the Entscheidungsproblem (Turing [15]). In that paper Turing constructs a universal Turing machine that can simulate any other Turing machine. He also use...
The Garden of Knowledge as a Knowledge Manifold  A Conceptual Framework for Computer Supported Subjective Education
 CID17, TRITANAD9708, DEPARTMENT OF NUMERICAL ANALYSIS AND COMPUTING SCIENCE
, 1997
"... This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in te ..."
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Cited by 28 (18 self)
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This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols  procedures that are implemented on top of the participating subjective knowledgepatches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inversedirect limit duality of Category Theory.
Spanning Trees in Hypergraphs with Applications to Steiner Trees
, 1998
"... This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of ..."
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Cited by 25 (1 self)
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This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of finding a minimumweight spanning tree in a hypergraph. The geometric Steiner tree problem is known to be NPcomplete for the rectilinear metric, and NPhard for the Euclidean metric. The fastest exact algorithms (in practice) for these problems use two phases: First a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimal tree is constructed from this set. These phases are called FST generation and FST concatenation, respectively, and an overview of each phase is presented. FST concatenation is almost always the most expensive phase, and has traditionally been accomplished via simple backtrack search or dynamic programming.
A New Method for Establishing Conservativity of Classical Systems Over Their Intuitionistic Version
"... this paper we present such a method. Applied to I \Sigma ..."
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Cited by 22 (1 self)
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this paper we present such a method. Applied to I \Sigma
Computational Geometry and Facility Location
 Proc. International Conference on Operations Research and Management Science
, 1990
"... this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of c ..."
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Cited by 18 (3 self)
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this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as