Two protocols used for learning under the pac-learning model introduced by Valiant are learning from random examples and learning from membership queries. Membership queries have also been used to efficiently and exactly learn a concept class that is too difficult to pac-learn using random examples. We ask whether using membership queries-- in conjunction with or instead of random examples serve a new purpose: helping to reduce the total number of examples needed to pac-learn a concept class C already known to be pac-learnable using just random examples. We focus on concept classes that are dense in themselves, such as haft-spaces of R ', rectangles in the plane, and the class Z = {[0, a]: 0 _ a < 1} of initial segments of [0, 1]. The main

A labeling of the vertices of a graph G, OE : V (G) ! f1; : : : ; rg, is said to be r-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any group \Gamma, there is a graph G such that Aut(G) = \Gamma and D(G) = 2; (ii) D(G) = O(log(jAut(G)j)); (iii) if Aut(G) is abelian, then D(G) 2; (iv) if Aut(G) is dihedral, then D(G) 3; and (v) If Aut(G) = S 4 , then either D(G) = 2 or D(G) = 4. Mathematics Subject Classification 05C,20B,20F,68R 1

. The most important lattices in Euclidean space of dimension n 8 are the lattices A n (n ³ 2), D n (n ³ 4), E n (n = 6 , 7 , 8) and their duals. In this paper we determine the cell structures of all these lattices and their Voronoi and Delaunay polytopes in a uniform manner. The results for E 6 * and E 7 * simplify recent work of Worley, and also provide what may be new space-filling polytopes in dimensions 6 and 7. 1. Introduction The Coxeter-Dynkin diagrams of types A n , D n , E 6 , E 7 and E 8 arise in surprisingly different parts of mathematics -- see the discussions by Arnold [1] and Hazewinkel et al. [30]. In the present paper we study __________________ * This paper appeared in {\m Miscellanea mathematica}, P. Hilton, F. Hirzebruch, and R. Remmert, Eds., Springer-Verlag, NY, 1991, pp. 71--107. (**) From the English version Auto-da-Fe(Continuum, New York, p. 385) as translated by C. V. Wedgwood: "You have but to know an object by its proper name for it to lose its dange...

There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples. Key words: graph, stressed graph, convex polyhedron, reciprocal polyhedron 1 Introduction It is well known that 3-connected planar graphs can be drawn as convex polyhedra. However, no practical algorithms exist to draw general 3-connected planar graphs as convex polyhedra. The two-dimensional (2D) drawing in Figure 1 is 3-connected and planar, and the corresponding polyhedron is drawn in Figure 2 as three different views. The 2D ...

Abstract not found

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of

This paper describes a computer application named HyperGami that permits users to design, explore, decorate, and study a rich variety of paper polyhedral models. In structure, HyperGami is a "programmable design environment", including both a direct manipulation interface as well as a domain-enriched programming environment based on the Scheme language; the application is thus designed to be accessible to students of geometry while providing challenging projects for long-term or expert users (such as professional mathematicians and designers). In the course of this paper, we describe the HyperGami interface and language; illustrate the construction of "customized polyhedra" of various sorts; discuss the results of our initial experiences using the system in working with middle-school students; and argue for the utility of embedding programming languages in educational design environments such as this one. 1. Introduction Over the centuries, human beings have been fascinated by polyhe...

: We use Mathematica to construct zonotopes and display zonohedra. *Work supported in part by NSF grant CCR-9258355 and by matching funds from Xerox corp. Introduction A zonotope is a set of points in d-dimensional space constructed by the sum of scaled vectors a[[i]] v[[i]] where a[[i]] is a scalar between 0 and 1 and v[[i]] is a d-dimensional vector. Alternately it can be viewed as a Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope. A zonohedron is just a three-dimensional zonotope. This notebook contains code for constructing zonotopes and displaying zonohedra. There is some confusion in the definition of zonotopes; Wells [W91] requires the generating vectors to be in general position (all d-tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others [BEG95,Z95] do not make this ...

In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a top-down approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGI-workstation. The method described can easily be generalised for 3-regular spherical maps with no face having more than 6 edges in its boundary.

This paper deals with the problem of obtaining a rough estimate of three dimensional object position and orientation from a single two dimensional camera image. Such an estimate is required by most 3-D to 2-D registration and tracking methods that can efficiently refine an initial value by numerical optimization to precisely recover 3-D pose. However, the analytic computation of an initial pose guess requires the solution of an extremely complex correspondence problem that is due to the large number of topologically distinct aspects that arise when a three dimensional opaque object is imaged by a camera. Hence general analytic methods fail to achieve real-time performance and most tracking and registration systems are initialized interactively or by ad hoc heuristics. To overcome these limitations we present a novel method for approximate object pose estimation that is based on a neural net and that can easily be implemented in real-time. A modification of Kohonen's self-organizing fe...