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Cone (topology)

About: Cone (topology) is a research topic. Over the lifetime, 3343 publications have been published within this topic receiving 39146 citations. The topic is also known as: cone of a space & cone of a topological space.


Papers
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Book
01 Jan 1996
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: I. Hilbert Schemes and Chow Varieties.- II. Curves on Varieties.- III. The Cone Theorem and Minimal Models.- IV. Rationally Connected Varieties.- V. Fano Varieties.- VI. Appendix.- References.

1,560 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduce cone metric spaces and prove fixed point theorems of contractive mappings on these spaces, and prove some fixed point properties of the mappings.

1,171 citations

Book ChapterDOI
01 Mar 1983
TL;DR: An algorithm is presented which efficiently finds good collision-free paths for convex polygonal bodies through space littered with obstacle polygons by characterizing the volume swept by a body as it is translated and rotated as a generalized cone.
Abstract: Free space is represented as a union of (possibly overlapping) generalized cones. An algorithm is presented which efficiently finds good collision-free paths for convex polygonal bodies through space littered with obstacle polygons. The paths are good in the sense that the distance of closest approach to an obstacle over the path is usually far from minimal over the class of topologically equivalent collision-free paths. The algorithm is based on characterizing the volume swept by a body as it is translated and rotated as a generalized cone, and determining under what conditions one generalized cone is a subset of another.

657 citations

Journal ArticleDOI
TL;DR: In this article, the alternating directions method of multipliers is used to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone.
Abstract: We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) nonnegative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.

597 citations

01 Jan 2008
TL;DR: In this paper, Huang et al. showed that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normalized constant M > k, and that for non-normal cones and omitting the assumption of normality in some results of Huang and Zhang, they obtained generalizations of the results.
Abstract: Abstract Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normal constant M > k . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results.

553 citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021178
2020162
2019179
2018172
2017179