@article{692,
abstract = {We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.},
author = {Akopyan, Arseniy},
journal = {Geometriae Dedicata},
number = {1},
pages = {55 -- 64},
publisher = {Springer},
title = {{3-Webs generated by confocal conics and circles}},
doi = {10.1007/s10711-017-0265-6},
volume = {194},
year = {2018},
}
@inproceedings{187,
abstract = {Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. },
author = {Edelsbrunner, Herbert and Osang, Georg F},
location = {Budapest, Hungary},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The multi-cover persistence of Euclidean balls}},
doi = {10.4230/LIPIcs.SoCG.2018.34},
volume = {99},
year = {2018},
}
@inproceedings{188,
abstract = {Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex.},
author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
location = {Budapest, Hungary},
pages = {35:1 -- 35:13},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Smallest enclosing spheres and Chernoff points in Bregman geometry}},
doi = {10.4230/LIPIcs.SoCG.2018.35},
volume = {99},
year = {2018},
}
@inproceedings{193,
abstract = {We show attacks on five data-independent memory-hard functions (iMHF) that were submitted to the password hashing competition (PHC). Informally, an MHF is a function which cannot be evaluated on dedicated hardware, like ASICs, at significantly lower hardware and/or energy cost than evaluating a single instance on a standard single-core architecture. Data-independent means the memory access pattern of the function is independent of the input; this makes iMHFs harder to construct than data-dependent ones, but the latter can be attacked by various side-channel attacks. Following [Alwen-Blocki'16], we capture the evaluation of an iMHF as a directed acyclic graph (DAG). The cumulative parallel pebbling complexity of this DAG is a measure for the hardware cost of evaluating the iMHF on an ASIC. Ideally, one would like the complexity of a DAG underlying an iMHF to be as close to quadratic in the number of nodes of the graph as possible. Instead, we show that (the DAGs underlying) the following iMHFs are far from this bound: Rig.v2, TwoCats and Gambit each having an exponent no more than 1.75. Moreover, we show that the complexity of the iMHF modes of the PHC finalists Pomelo and Lyra2 have exponents at most 1.83 and 1.67 respectively. To show this we investigate a combinatorial property of each underlying DAG (called its depth-robustness. By establishing upper bounds on this property we are then able to apply the general technique of [Alwen-Block'16] for analyzing the hardware costs of an iMHF.},
author = {Alwen, Joel F and Gazi, Peter and Kamath Hosdurg, Chethan and Klein, Karen and Osang, Georg F and Pietrzak, Krzysztof Z and Reyzin, Lenoid and Rolinek, Michal and Rybar, Michal},
booktitle = {Proceedings of the 2018 on Asia Conference on Computer and Communication Security},
location = {Incheon, Republic of Korea},
pages = {51 -- 65},
publisher = {ACM},
title = {{On the memory hardness of data independent password hashing functions}},
doi = {10.1145/3196494.3196534},
year = {2018},
}
@article{87,
abstract = {Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
journal = {Annals of Applied Probability},
number = {5},
pages = {3215 -- 3238},
publisher = {Institute of Mathematical Statistics},
title = {{Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics}},
doi = {10.1214/18-AAP1389},
volume = {28},
year = {2018},
}
@article{106,
abstract = {The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below.},
author = {Akopyan, Arseniy and Petrunin, Anton},
journal = {Mathematical Intelligencer},
number = {3},
pages = {26 -- 31},
publisher = {Springer},
title = {{Long geodesics on convex surfaces}},
doi = {10.1007/s00283-018-9795-5},
volume = {40},
year = {2018},
}
@article{1064,
abstract = {In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.},
author = {Akopyan, Arseniy and Balitskiy, Alexey and Grigorev, Mikhail},
issn = {14320444},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {1001--1009},
publisher = {Springer},
title = {{On the circle covering theorem by A.W. Goodman and R.E. Goodman}},
doi = {10.1007/s00454-017-9883-x},
volume = {59},
year = {2018},
}
@article{312,
abstract = {Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice.},
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel},
issn = {08954801},
journal = {SIAM J Discrete Math},
number = {1},
pages = {750 -- 782},
publisher = {Society for Industrial and Applied Mathematics },
title = {{On the optimality of the FCC lattice for soft sphere packing}},
doi = {10.1137/16M1097201},
volume = {32},
year = {2018},
}
@article{718,
abstract = {Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Reitzner, Matthias},
issn = {00018678},
journal = {Advances in Applied Probability},
number = {3},
pages = {745 -- 767},
publisher = {Cambridge University Press},
title = {{Expected sizes of poisson Delaunay mosaics and their discrete Morse functions}},
doi = {10.1017/apr.2017.20},
volume = {49},
year = {2017},
}
@article{737,
abstract = {We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.},
author = {Virk, Ziga and Zastrow, Andreas},
issn = {01668641},
journal = {Topology and its Applications},
pages = {186 -- 196},
publisher = {Elsevier},
title = {{A new topology on the universal path space}},
doi = {10.1016/j.topol.2017.09.015},
volume = {231},
year = {2017},
}