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Generalizing Halfspaces
, 1996
"... Restrictedorientation convexity is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. We have studied the properties of restrictedorientation convex sets and demonstrated that this notion is a generalization of standard convexity. We now describ ..."
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Cited by 1 (1 self)
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describe a restrictedorientation generalization of halfspaces and explore properties of these generalized halfspaces. In particular, we establish analogs of the following properties of standard halfspaces: ffl The intersection of a halfspace with every line is empty, a ray, or a line ffl Every halfspace
Generalizing Halfspaces 1
"... Abstract Restrictedorientation convexity is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. We have studied the properties of restrictedorientation convex sets and demonstrated that this notion is a generalization of standard convexity. We no ..."
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now describe a restrictedorientation generalization of halfspaces and explore properties of these generalized halfspaces. In particular, we establish analogs of the following properties of standard halfspaces: The intersection of a halfspace with every line is empty, a ray, or a line Every halfspace
Generalized halfspaces in restrictedorientation convexity
 JOURNAL OF GEOMETRY
, 1995
"... Restrictedorientation convexity, also called Oconvexity, is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. The notion ofOconvexity generalizes standard convexity and several types of nontraditional convexity. We introduce Ohalfspaces, whic ..."
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Cited by 7 (6 self)
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Restrictedorientation convexity, also called Oconvexity, is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. The notion ofOconvexity generalizes standard convexity and several types of nontraditional convexity. We introduce Ohalfspaces
Surface deformation due to shear and tensile faults in a halfspace
, 1985
"... A complete set of closed analytical expressions is presented in a unified manner for the internal displacements and strains due to shear and tensile faults in a halfspace for both point and finite rectangular sources. These expressions are particularly compact and systematically composed of terms r ..."
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Cited by 698 (1 self)
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A complete set of closed analytical expressions is presented in a unified manner for the internal displacements and strains due to shear and tensile faults in a halfspace for both point and finite rectangular sources. These expressions are particularly compact and systematically composed of terms
Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams
 ACM Tmns. Graph
, 1985
"... The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms ar ..."
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Cited by 543 (11 self)
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to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing
Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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was the exploration of variations around a point, within the bounds imposed by the constraints, in order to help characterize solutions and portray them in terms of ‘variational principles’. Notions of perturbation, approximation and even generalized differentiability were extensively investigated. Variational theory
The strength of weak learnability
 Machine Learning
, 1990
"... Abstract. This paper addresses the problem of improving the accuracy of an hypothesis output by a learning algorithm in the distributionfree (PAC) learning model. A concept class is learnable (or strongly learnable) if, given access to a Source of examples of the unknown concept, the learner with h ..."
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Cited by 861 (24 self)
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well. In addition, the construction has some interesting theoretical consequences, including a set of general upper bounds on the complexity of any strong learning algorithm as a function of the allowed error e.
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 557 (12 self)
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mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimization problems including the general 01 integer programs, the maximum clique
Automatic Discovery of Linear Restraints Among Variables of a Program
, 1978
"... The model of abstract interpretation of programs developed by Cousot and Cousot [2nd ISOP, 1976], Cousot and Cousot [POPL 1977] and Cousot [PhD thesis 1978] is applied to the static determination of linear equality or inequality invariant relations among numerical variables of programs. ..."
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Cited by 733 (47 self)
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The model of abstract interpretation of programs developed by Cousot and Cousot [2nd ISOP, 1976], Cousot and Cousot [POPL 1977] and Cousot [PhD thesis 1978] is applied to the static determination of linear equality or inequality invariant relations among numerical variables of programs.
Results 1  10
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