--- _id: '2358' abstract: - lang: eng text: A study was conducted on the one-dimensional (1D) bosons in three-dimensional (3D) traps. A rigorous analysis was carried out on the parameter regions in which various types of 1D or 3D behavior occurred in the ground state. The four parameter regions include density, transverse, longitudinal dimensions and scattering length. author: - first_name: Élliott full_name: Lieb, Élliott H last_name: Lieb - first_name: Robert full_name: Robert Seiringer id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 - first_name: Jakob full_name: Yngvason, Jakob last_name: Yngvason citation: ama: Lieb É, Seiringer R, Yngvason J. One-dimensional Bosons in three-dimensional traps. Physical Review Letters. 2003;91(15):1504011-1504014. doi:10.1103/PhysRevLett.91.150401 apa: Lieb, É., Seiringer, R., & Yngvason, J. (2003). One-dimensional Bosons in three-dimensional traps. Physical Review Letters. American Physical Society. https://doi.org/10.1103/PhysRevLett.91.150401 chicago: Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “One-Dimensional Bosons in Three-Dimensional Traps.” Physical Review Letters. American Physical Society, 2003. https://doi.org/10.1103/PhysRevLett.91.150401. ieee: É. Lieb, R. Seiringer, and J. Yngvason, “One-dimensional Bosons in three-dimensional traps,” Physical Review Letters, vol. 91, no. 15. American Physical Society, pp. 1504011–1504014, 2003. ista: Lieb É, Seiringer R, Yngvason J. 2003. One-dimensional Bosons in three-dimensional traps. Physical Review Letters. 91(15), 1504011–1504014. mla: Lieb, Élliott, et al. “One-Dimensional Bosons in Three-Dimensional Traps.” Physical Review Letters, vol. 91, no. 15, American Physical Society, 2003, pp. 1504011–14, doi:10.1103/PhysRevLett.91.150401. short: É. Lieb, R. Seiringer, J. Yngvason, Physical Review Letters 91 (2003) 1504011–1504014. date_created: 2018-12-11T11:57:12Z date_published: 2003-10-10T00:00:00Z date_updated: 2021-01-12T06:57:00Z day: '10' doi: 10.1103/PhysRevLett.91.150401 extern: 1 intvolume: ' 91' issue: '15' main_file_link: - open_access: '1' url: http://arxiv.org/abs/cond-mat/0304071 month: '10' oa: 1 page: 1504011 - 1504014 publication: Physical Review Letters publication_status: published publisher: American Physical Society publist_id: '4571' quality_controlled: 0 status: public title: One-dimensional Bosons in three-dimensional traps type: journal_article volume: 91 year: '2003' ... --- _id: '2414' author: - first_name: Uli full_name: Uli Wagner id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: Wagner U. On k-Sets and Their Applications. 2003. doi:10.3929/ethz-a-004708408 apa: Wagner, U. (2003). On k-Sets and Their Applications. ETH Zurich. https://doi.org/10.3929/ethz-a-004708408 chicago: Wagner, Uli. “On K-Sets and Their Applications.” ETH Zurich, 2003. https://doi.org/10.3929/ethz-a-004708408. ieee: U. Wagner, “On k-Sets and Their Applications,” ETH Zurich, 2003. ista: Wagner U. 2003. On k-Sets and Their Applications. ETH Zurich. mla: Wagner, Uli. On K-Sets and Their Applications. ETH Zurich, 2003, doi:10.3929/ethz-a-004708408. short: U. Wagner, On K-Sets and Their Applications, ETH Zurich, 2003. date_created: 2018-12-11T11:57:31Z date_published: 2003-01-01T00:00:00Z date_updated: 2021-01-12T06:57:20Z day: '01' doi: 10.3929/ethz-a-004708408 extern: 1 month: '01' publication_status: published publisher: ETH Zurich publist_id: '4511' quality_controlled: 0 status: public title: On k-Sets and Their Applications type: dissertation year: '2003' ... --- _id: '2424' abstract: - lang: eng text: We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some (potentially very high-dimensional) Euclidean space ℝd. We assume that M is known to us only through a finite sample P ⊂ M, as it is often the case in applications. The adaptive neighborhood graph is a geometric graph on P. Its complexity is at most min{2O(k)(n, n2}, where n = |P| and k = dim M, as opposed to the n⌈d/2⌉ complexity of the Delaunay triangulation, which is often used to model manifolds. We show that we can provably correctly infer the connectivity of M and the dimension of M from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is d2O(k7 log k) for each connected component of M. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on the intrinsic dimension k, not on the ambient dimension d. This is of particular interest if the co-dimension is high, i.e., if k is much smaller than d, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in P. author: - first_name: Joachim full_name: Giesen, Joachim last_name: Giesen - first_name: Uli full_name: Uli Wagner id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Giesen J, Wagner U. Shape dimension and intrinsic metric from samples of manifolds with high co-dimension. In: ACM; 2003:329-337. doi:10.1145/777792.777841' apa: 'Giesen, J., & Wagner, U. (2003). Shape dimension and intrinsic metric from samples of manifolds with high co-dimension (pp. 329–337). Presented at the SoCG: Symposium on Computational Geometry, ACM. https://doi.org/10.1145/777792.777841' chicago: Giesen, Joachim, and Uli Wagner. “Shape Dimension and Intrinsic Metric from Samples of Manifolds with High Co-Dimension,” 329–37. ACM, 2003. https://doi.org/10.1145/777792.777841. ieee: 'J. Giesen and U. Wagner, “Shape dimension and intrinsic metric from samples of manifolds with high co-dimension,” presented at the SoCG: Symposium on Computational Geometry, 2003, pp. 329–337.' ista: 'Giesen J, Wagner U. 2003. Shape dimension and intrinsic metric from samples of manifolds with high co-dimension. SoCG: Symposium on Computational Geometry, 329–337.' mla: Giesen, Joachim, and Uli Wagner. Shape Dimension and Intrinsic Metric from Samples of Manifolds with High Co-Dimension. ACM, 2003, pp. 329–37, doi:10.1145/777792.777841. short: J. Giesen, U. Wagner, in:, ACM, 2003, pp. 329–337. conference: name: 'SoCG: Symposium on Computational Geometry' date_created: 2018-12-11T11:57:35Z date_published: 2003-06-01T00:00:00Z date_updated: 2021-01-12T06:57:24Z day: '01' doi: 10.1145/777792.777841 extern: 1 month: '06' page: 329 - 337 publication_status: published publisher: ACM publist_id: '4501' quality_controlled: 0 status: public title: Shape dimension and intrinsic metric from samples of manifolds with high co-dimension type: conference year: '2003' ... --- _id: '2423' abstract: - lang: eng text: A finite set N ⊃ Rd is a weak ε-net for an n-point set X ⊃ Rd (with respect to convex sets) if N intersects every convex set K with |K ∩ X| ≥ εn. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [7], that every point set X in Rd admits a weak ε-net of cardinality O(ε-d polylog(1/ε)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak ε-nets in time O(n ln(1/ε)). We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d - 1)-dimensional sphere. author: - first_name: Jiří full_name: Matoušek, Jiří last_name: Matoušek - first_name: Uli full_name: Uli Wagner id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Matoušek J, Wagner U. New constructions of weak epsilon-nets. In: ACM; 2003:129-135. doi:10.1145/777792.777813' apa: 'Matoušek, J., & Wagner, U. (2003). New constructions of weak epsilon-nets (pp. 129–135). Presented at the SoCG: Symposium on Computational Geometry, ACM. https://doi.org/10.1145/777792.777813' chicago: Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak Epsilon-Nets,” 129–35. ACM, 2003. https://doi.org/10.1145/777792.777813. ieee: 'J. Matoušek and U. Wagner, “New constructions of weak epsilon-nets,” presented at the SoCG: Symposium on Computational Geometry, 2003, pp. 129–135.' ista: 'Matoušek J, Wagner U. 2003. New constructions of weak epsilon-nets. SoCG: Symposium on Computational Geometry, 129–135.' mla: Matoušek, Jiří, and Uli Wagner. New Constructions of Weak Epsilon-Nets. ACM, 2003, pp. 129–35, doi:10.1145/777792.777813. short: J. Matoušek, U. Wagner, in:, ACM, 2003, pp. 129–135. conference: name: 'SoCG: Symposium on Computational Geometry' date_created: 2018-12-11T11:57:34Z date_published: 2003-06-01T00:00:00Z date_updated: 2021-01-12T06:57:24Z day: '01' doi: 10.1145/777792.777813 extern: 1 month: '06' page: 129 - 135 publication_status: published publisher: ACM publist_id: '4502' quality_controlled: 0 status: public title: New constructions of weak epsilon-nets type: conference year: '2003' ... --- _id: '2422' abstract: - lang: eng text: We prove a lower bound of 0.3288(4 n) for the rectilinear crossing number cr̄(Kn) of a complete graph on n vertices, or in other words, for the minimum number of convex quadrilaterals in any set of n points in general position in the Euclidean plane. As we see it, the main contribution of this paper is not so much the concrete numerical improvement over earlier bounds, as the novel method of proof, which is not based on bounding cr̄(Kn) for some small n. author: - first_name: Uli full_name: Uli Wagner id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Wagner U. On the rectilinear crossing number of complete graphs. In: SIAM; 2003:583-588.' apa: 'Wagner, U. (2003). On the rectilinear crossing number of complete graphs (pp. 583–588). Presented at the SODA: Symposium on Discrete Algorithms, SIAM.' chicago: Wagner, Uli. “On the Rectilinear Crossing Number of Complete Graphs,” 583–88. SIAM, 2003. ieee: 'U. Wagner, “On the rectilinear crossing number of complete graphs,” presented at the SODA: Symposium on Discrete Algorithms, 2003, pp. 583–588.' ista: 'Wagner U. 2003. On the rectilinear crossing number of complete graphs. SODA: Symposium on Discrete Algorithms, 583–588.' mla: Wagner, Uli. On the Rectilinear Crossing Number of Complete Graphs. SIAM, 2003, pp. 583–88. short: U. Wagner, in:, SIAM, 2003, pp. 583–588. conference: name: 'SODA: Symposium on Discrete Algorithms' date_created: 2018-12-11T11:57:34Z date_published: 2003-01-01T00:00:00Z date_updated: 2021-01-12T06:57:24Z day: '01' extern: 1 main_file_link: - open_access: '0' url: http://dl.acm.org/citation.cfm?id=644206 month: '01' page: 583 - 588 publication_status: published publisher: SIAM publist_id: '4503' quality_controlled: 0 status: public title: On the rectilinear crossing number of complete graphs type: conference year: '2003' ...