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176
On The Sample Complexity Of PacLearning Using Random And Chosen Examples
 IN PROCEEDINGS OF THE 1990 WORKSHOP ON COMPUTATIONAL LEARNING THEORY
, 1991
"... Two protocols used for learning under the paclearning 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 paclearn using random examples. ..."
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Cited by 26 (0 self)
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Two protocols used for learning under the paclearning 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 paclearn 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 paclearn a concept class C already known to be paclearnable using just random examples. We focus on concept classes that are dense in themselves, such as haftspaces of R ', rectangles in the plane, and the class Z = {[0, a]: 0 _ a < 1} of initial segments of [0, 1]. The main
Principal vertex operator representations for toroidal Lie algebras, preprint hepth/9703002
"... 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 KacMoody algebras. Two of ..."
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Cited by 25 (6 self)
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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 KacMoody algebras. Two of
Symmetry Breaking in Graphs
 Electronic Journal of Combinatorics
, 1996
"... A labeling of the vertices of a graph G, OE : V (G) ! f1; : : : ; rg, is said to be rdistinguishing 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 rdistinguishing labeling. T ..."
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Cited by 22 (4 self)
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A labeling of the vertices of a graph G, OE : V (G) ! f1; : : : ; rg, is said to be rdistinguishing 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 rdistinguishing 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
Yhaping multidimensional signal spacesPart 11: shelladdressed constellations," submitted to
 IEEE Trans. Inform. Theory. A. K. Khandani and
"... degree of Doctor of PhilosophyTo my mother and to the memory of my fatherAcknowledgement ..."
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Cited by 20 (8 self)
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degree of Doctor of PhilosophyTo my mother and to the memory of my fatherAcknowledgement
The Cell Structures of Certain Lattices
, 1991
"... . 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 * ..."
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Cited by 19 (8 self)
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. 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 spacefilling polytopes in dimensions 6 and 7. 1. Introduction The CoxeterDynkin 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., SpringerVerlag, NY, 1991, pp. 71107. (**) From the English version AutodaFe(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...
Drawing Stressed Planar Graphs in Three Dimensions
 In
, 1995
"... There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3connected planar graph as ..."
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Cited by 16 (0 self)
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There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3connected 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 3connected 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 3connected planar graphs can be drawn as convex polyhedra. However, no practical algorithms exist to draw general 3connected planar graphs as convex polyhedra. The twodimensional (2D) drawing in Figure 1 is 3connected and planar, and the corresponding polyhedron is drawn in Figure 2 as three different views. The 2D ...
Multitriangulations as complexes of starpolygons
"... Abstract. Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of s ..."
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Cited by 16 (7 self)
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Abstract. Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of ktriangulations, as well as some new results. This interpretation also opensup new avenues of research, that we briefly explore in the last section. 1.
Creating Polyhedral Models by Computer
 Journal of Computers in Mathematics and Science Teaching
, 1997
"... 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 domainenriche ..."
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Cited by 15 (7 self)
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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 domainenriched programming environment based on the Scheme language; the application is thus designed to be accessible to students of geometry while providing challenging projects for longterm 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 middleschool 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...
A Constructive Enumeration of Fullerenes
"... In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regul ..."
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Cited by 15 (2 self)
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In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regular spherical maps with no face having more than 6 edges in its boundary.