TY - JOUR
AU - Timothy Browning
ID - 205
IS - 3
JF - Acta Arithmetica
TI - Counting rational points on cubic and quartic surfaces
VL - 108
ER -
TY - JOUR
AB - Let T ⊂ ℙ 4 be a non-singular threefold of degree at least four. Then we show that the number of points in T(ℚ), with height at most B, is o(B 3) or B → ∞.
AU - Timothy Browning
ID - 206
IS - 1
JF - Quarterly Journal of Mathematics
TI - A note on the distribution of rational points on threefolds
VL - 54
ER -
TY - JOUR
AU - Browning, Timothy D
ID - 207
IS - 3
JF - Mathematical Proceedings of the Cambridge Philosophical Society
TI - Sums of four biquadrates
VL - 134
ER -
TY - JOUR
AB - For any ε > 0 and any diagonal quadratic form Q ∈ ℤ[x 1, x 2, x 3, x 4] with a square-free discriminant of modulus Δ Q ≠ 0, we establish the uniform estimate ≪ε B 3/2+ε + B 2+ε/Δ Q 1/6 for the number of rational points of height at most B lying in the projective surface Q = 0.
AU - Timothy Browning
ID - 208
IS - 1
JF - Quarterly Journal of Mathematics
TI - Counting rational points on diagonal quadratic surfaces
VL - 54
ER -
TY - CONF
AU - Lieb, Élliott H
AU - Robert Seiringer
ED - Karpeshina, Yulia
ED - Weikard, Rudi
ED - Zeng, Yanni
ID - 2337
TI - Bose-Einstein condensation of dilute gases in traps
VL - 327
ER -
TY - JOUR
AB - We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the Gross-Pitaevskii (GP) limit of a dilute gas. Analysing both the absolute and the bosonic ground states of the system, we show, in particular, their different behaviour for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.
AU - Robert Seiringer
ID - 2354
IS - 37
JF - Journal of Physics A: Mathematical and Theoretical
TI - Ground state asymptotics of a dilute, rotating gas
VL - 36
ER -
TY - JOUR
AB - The classic Poincaré inequality bounds the L q-norm of a function f in a bounded domain Ω ⊂ ℝ n in terms of some L p-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Τ from Ω and concentrate our attention on Λ = Ω \ Τ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of Τ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that f has a finite L p gradient norm on the whole of Ω, not just on Λ. The important point is that the Poincaré inequality thus obtained bounds the L q-norm of f in terms of the L p gradient norm on Λ (not Ω) plus an additional term that goes to zero as the volume of Τ goes to zero. This error term depends on Τ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain Ω. In the second generalization we are given a vector field A and replace ∇ by ∇ + iA(x) (geometrically, a connection on a U(1) bundle). Unlike the A = 0 case, the infimum of ∥(∇ + iA)f∥ p over all f with a given ∥f∥ q is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Yngvason, Jakob
ID - 2357
IS - 3
JF - Annals of Mathematics
TI - Poincaré inequalities in punctured domains
VL - 158
ER -
TY - JOUR
AB - 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.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Yngvason, Jakob
ID - 2358
IS - 15
JF - Physical Review Letters
TI - One-dimensional Bosons in three-dimensional traps
VL - 91
ER -
TY - THES
AU - Uli Wagner
ID - 2414
TI - On k-Sets and Their Applications
ER -
TY - CONF
AB - 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.
AU - Uli Wagner
ID - 2422
TI - On the rectilinear crossing number of complete graphs
ER -
TY - CONF
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. [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.
AU - Matoušek, Jiří
AU - Uli Wagner
ID - 2423
TI - New constructions of weak epsilon-nets
ER -
TY - CONF
AB - 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.
AU - Giesen, Joachim
AU - Uli Wagner
ID - 2424
TI - Shape dimension and intrinsic metric from samples of manifolds with high co-dimension
ER -
TY - JOUR
AB - Male dimorphism is not genetically determined, but is induced by environmental conditions particularly decreasing temperature and density.
AU - Cremer, Sylvia
AU - Heinze, Jürgen
ID - 3917
IS - 15
JF - Blick in die Wissenschaft
TI - Zwischen Hochzeitsflug und Brudermord: reproduktive Taktiken bei Ameisenmännchen
VL - 12
ER -
TY - JOUR
AB - Unlike most social insects, many Cardiocondyla ant species have two male morphs: wingless (ergatoid) males, who remain in the natal nest, and winged males who disperse but, strangely, before leaving may also mate within the nest. Whereas ergatoid males are highly intolerant of each other and fight among themselves, they tend to tolerate their winged counterparts. This is despite the fact that these winged males, like ergatoid males, represent mating competition. Why should ergatoid males tolerate their winged rivals? We developed a mathematical model to address this question. Our model focuses on a number of factors likely toinfluence whether ergatoid males are tolerant of winged males: ergatoid male–winged male relatedness, number of virgin queens, number of winged males, and the number of ejaculates a winged male has (winged males are sperm limited, whereas ergatoid males have lifelong spermatogenesis). Surprisingly, we found that increasing the number of virgin queens favors a kill strategy, whereas an increase in the other factors favors a let-live strategy; these predictions appear true for C. obscurior and for a number of other Cardiocondyla species. Two further aspects, unequal insemination success and multiple mating in queens, were also incorporated into the model and predictions made about their effects on toleration of winged males. The model is applicable more generally in species that have dimorphic males, such as some other ants, bees, and fig wasps.
AU - Anderson, Carl
AU - Cremer, Sylvia
AU - Heinze, Jürgen
ID - 3921
IS - 1
JF - Behavioral Ecology
TI - Live and let die: Why fighter males of the ant Cardiocondyla kill each other but tolerate their winged rivals
VL - 14
ER -
TY - JOUR
AB - Dispersal is advantageous, but, at the same time, it implies high costs and risks. Due to these counteracting selection pressures, many species evolved dispersal polymorphisms, which, in ants, are typically restricted to the female sex (queens). Male polymorphism is presently only known from a few genera, such as Cardiocondyla, in which winged dispersing males coexist with wingless fighter males that mate exclusively inside their maternal nests. We studied the developmental mechanisms underlying these alternative male morphs and found that, first, male dimorphism is not genetically determined, but is induced by environmental conditions (decreasing temperature and density). Second, male morph is not yet fixed at the egg stage, but it differentiates during larval development. This flexible developmental pattern of male morphs allows Cardiocondyla ant colonies to react quickly to changes in their environment. Under good conditions, they invest exclusively in philopatric wingless males. But, when environmental conditions turn bad, colonies start to produce winged dispersal males, even though these males require a many times higher investment by the colony than their much smaller wingless counterparts. Cardiocondyla ants share this potential of optimal resource allocation with other colonial animals and some seed dimorphic plants.
AU - Cremer, Sylvia
AU - Heinze, Jürgen
ID - 3922
IS - 3
JF - Current Biology
TI - Stress grows wings: Environmental induction of winged dispersal males in Cardiocondyla ants
VL - 13
ER -
TY - CHAP
AB - We give analytic inclusion-exclusion formulas for the area and perimeter derivatives of a union of finitely many disks in the plane.
AU - Cheng, Ho-Lun
AU - Herbert Edelsbrunner
ID - 3991
T2 - Computer Science in Perspective: Essays Dedicated to Thomas Ottmann
TI - Area and perimeter derivatives of a union of disks
VL - 2598
ER -
TY - JOUR
AB - Computing the volume occupied by individual atoms in macromolecular structures has been the subject of research for several decades. This interest has grown in the recent years, because weighted volumes are widely used in implicit solvent models. Applications of the latter in molecular mechanics simulations require that the derivatives of these weighted volumes be known. In this article, we give a formula for the volume derivative of a molecule modeled as a space-filling diagram made up of balls in motion. The formula is given in terms of the weights, radii, and distances between the centers as well as the sizes of the facets of the power diagram restricted to the space-filling diagram. Special attention is given to the detection and treatment of singularities as well as discontinuities of the derivative.
AU - Herbert Edelsbrunner
AU - Koehl, Patrice
ID - 3992
IS - 5
JF - PNAS
TI - The weighted-volume derivative of a space-filling diagram
VL - 100
ER -
TY - JOUR
AB - We present algorithms for constructing a hierarchy of increasingly coarse Morse-Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linearcategory by ensuring structural integrity and simulating differentiability. We then simplify Morse-Smale complexes by canceling pairs of critical points in order of increasing persistence.
AU - Herbert Edelsbrunner
AU - Harer, John
AU - Zomorodian, Afra
ID - 3993
IS - 1
JF - Discrete & Computational Geometry
TI - Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds
VL - 30
ER -
TY - JOUR
AB - The body defined by a finite collection of disks is a subset of the plane bounded by a tangent continuous curve, which we call the skin. We give analytic formulas for the area, the perimeter, the area derivative, and the perimeter derivative of the body. Given the filtrations of the Delaunay triangulation and the Voronoi diagram of the disks, all formulas can be evaluated in time proportional to the number of disks.
AU - Cheng, Ho-Lun
AU - Herbert Edelsbrunner
ID - 3994
IS - 2
JF - Computational Geometry: Theory and Applications
TI - Area, perimeter and derivatives of a skin curve
VL - 26
ER -
TY - CONF
AB - We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multi-resolution representations.
AU - Bremer, Peer-Timo
AU - Herbert Edelsbrunner
AU - Hamann, Bernd
AU - Pascucci, Valerio
ID - 3997
TI - A multi-resolution data structure for two-dimensional Morse-Smale functions
ER -