We discuss the problem of computing the volume of a convex body K in IR n . We review worst-case 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 CCR-8900112 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...

An exposition of results of Yoccoz, Branner, Hubbard and Douady concerning polynomial Julia sets. The contents are as follows: 1. Local connectivity of quadratic Julia sets (following Yoccoz) . 2 2. Polynomials for which all but one of the critical orbits escape (following Branner and Hubbard) . . . . . . . . . . . . 21 3. An innitely renormalizable non locally connected Julia set (following Douady and Hubbard) . . . . . . . . . . . . 34 Appendix: Length-area-modulus inequalities . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . 47 Introduction The following notes provide an introduction to work of Branner, Hubbard, Douady, and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets. (Compare [Hu3].) It presents only the \easy" part of his theory, in the sense that it considers...

We obtain some dramatic results using statistical mechanics--thermodynamics 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 incompleteness-type results concerning the limitations of the axiomatic method, in which entropy--information 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...

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.

This work presents a unied pattern-based 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 knowledge-patches. 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 inverse-direct limit duality of Category Theory.

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements. 1. Introduction This paper deals with moduli spaces of planar linkages. An abstract linkage (L; `) is a graph L with a positive real number `(e) assigned to each edge e. We assume that we have chosen a distinguished oriented edge e = [v 1 v 2 ] in L. The moduli space M(L) of planar realizations of L := (L; `; e ) is the set 1 of maps OE from the vertex set of L into the Euclidean plane R 2 (which will be identified with the complex plane C ) such that ffl jOE(v) \Gamma OE(w)j 2 = (`[vw]) 2 for each edge [vw] of L. ffl OE(v 1 ) = (0; 0). ffl OE(v 2 ) = (`(e ); 0). Clearly these conditions give M(L) a natural structure of a real-algebraic set in R 2r where r is the number of vertices...

This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimum-length 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 minimum-weight spanning tree in a hypergraph. The geometric Steiner tree problem is known to be NP-complete for the rectilinear metric, and NP-hard 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.

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

Many motion planning algorithms use Morse functions to characterize the free space. Specifically, these algorithms look at the critical points of a Morse function to denote the topological changes in the free space. This paper introduces methods to sense critical points and ensure all critical points are "seen" by a coverage algorithm. Experimental results performed on a mobile robot are also presented.

Different researchers use "the philosophy of automated theorem proving " to cover different concepts, indeed, different levels of concepts. Some would count such issues as how to efficiently index databases as part of the philosophy of automated theorem proving. Others wonder about whether formulas should be represented as strings or as trees or as lists, and call this part of the philosophy of automated theorem proving. Yet others concern themselves with what kind of search should be embodied in any automated theorem prover, or to what degree any automated theorem prover should resemble Prolog. Still others debate whether natural deduction or semantic tableaux or resolution is "better", and call this a part of the philosophy of automated theorem proving. Some people wonder whether automated theorem proving should be "human oriented " or