this paper, we assume not only that the target concept is chosen from a particular class, but that the concepts of this class are encoded using a particular representation language, and we allow the time and the number of examples required by our prediction algorithms to grow polynomially in the length of the target representation, which we take to be a measure of the complexity of the hidden concept. A more formal definition of the model (which was introduced in [HLW88] and [PW90]) will be given in the following section. Since any set of points on a sphere can be shattered by the class of convex polytopes, this class has infinite Vapnik-Chervonenkis (VC) dimension

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.

In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G, compute a representation of the function H = F ffi G: We examine this problem in detail for the case when F and G are given in either B'ezier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric modeling problems can be solved with a small amount of additional software. This paper was published in TOG, April 1993, pg 113-135 Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, and object representations; J.6 [...

Every transformation of a perfectly nested loop consisting of a combination of loop interchanging, loop skewing and loop reversal can be modeled by a linear transformation represented by a unimodular matrix. This modeling offers more flexibility than the traditional step-wise application of loop transformations because we can directly construct a unimodular matrix for a particular goal. In this paper, we present implementation issues arising when this framework is incorporated in a compiler. 1 Introduction Inherent to the application of program transformations in an optimizing or restructuring compiler is the so-called `phase ordering problem', i.e. the problem of finding an effective order in which particular transformations must be applied. This problem is still an important research topic [WS90]. An important step forwards in solving the phase ordering problem has been accomplished by the observation that any combination of the iteration-level loop transformations loop interchangin...

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.

Let P be a d-dimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report

Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial

dedicated to Alberto Verjovsky on his 60 th birthday Abstract. In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in C n which are invariant with respect to the natural action of the real torus (S 1) n onto C n. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.

We begin by defining and studying tightness and the two-piece property for smooth and polyhedral surfaces in three-dimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations to the case of smooth and polyhedral submanifolds of higher dimension and codimension, in particular highly connected submanifolds. Twenty-six open

ION-BASED PROBABILISTIC PLANNING In the framework of decision-theoretic planning, uncertainty in the state of the world and in the effects of actions are represented with probabilities, and the planner's goals, as well as tradeoffs among them, are represented with utilities. 9 Given this representation, the objective is to find an optimal plan or policy, where optimality is defined as maximizing expected utility. In most of the existing decision-theoretic planning approaches, the world is represented with a probability distribution over the state space, and actions are represented as stochastic mappings among the states [14,5,1,26]. Given this framing of the problem, all probabilistic and decision-theoretic planners face the burden of computational complexity in seeking an optimal or near-optimal solution. One popular way to address this problem is to use abstraction techniques to guide the search through the plan space and to reduce the cost of plan evaluation. This concept has bee...