TY - JOUR
AB - An optical lattice model developed that is similar to the Bose-Hubbard model to describe the transition between Bose-Einstein condensation (BEC) and a Mott insulator state was analyzed. It was found that the system was a hard core lattice gas at half of the maximum density and the optical lattice was modeled by a periodic potential of strength λ. It was also observed that the interparticle interaction was essential for this transition that occurred even in the ground state. The results show that all the essential features could be proved rigorously such as the existence of BEC for small λ and its suppression for a large λ.
AU - Aizenman, Michael
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Solovej, Jan P
AU - Yngvason, Jakob
ID - 2360
IS - 2
JF - Physical Review A - Atomic, Molecular, and Optical Physics
TI - Bose-Einstein quantum phase transition in an optical lattice model
VL - 70
ER -
TY - CHAP
AU - Lovász, László
AU - Vesztergombi, Katalin
AU - Uli Wagner
AU - Welzl, Emo
ED - Pach, János
ID - 2417
T2 - Towards a Theory of Geometric Graphs
TI - Convex quadrilaterals and k-sets
VL - 342
ER -
TY - JOUR
AB - 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., that every point set X in Rd admits a weak ε-net of cardinality O(ε-dpolylog(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)).
AU - Matoušek, Jiří
AU - Uli Wagner
ID - 2425
IS - 2
JF - Discrete & Computational Geometry
TI - New constructions of weak ε-nets
VL - 32
ER -
TY - JOUR
AB - We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some Euclidean space ℝ d. We assume that M is known to us only through a finite sample P ⊂ M, as 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 prove that we can correctly infer the connected components 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 d20(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.
AU - Giesen, Joachim
AU - Uli Wagner
ID - 2426
IS - 2
JF - Discrete & Computational Geometry
TI - Shape dimension and intrinsic metric from samples of manifolds
VL - 32
ER -
TY - GEN
AU - Sauer, Michael
AU - Friml, Jirí
ID - 2461
IS - 23
T2 - Development
TI - The Matryoshka dolls of plant polarity
VL - 131
ER -