@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{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{5678,
abstract = {The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer},
title = {{Poisson–Delaunay Mosaics of Order k}},
doi = {10.1007/s00454-018-0049-2},
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 = {01795376},
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{58,
abstract = {Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.},
author = {Akopyan, Arseniy and Segal Halevi, Erel},
journal = {SIAM Journal on Discrete Mathematics},
number = {3},
pages = {2242 -- 2257},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Counting blanks in polygonal arrangements}},
doi = {10.1137/16M110407X},
volume = {32},
year = {2018},
}
@article{6355,
abstract = {We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.},
author = {Akopyan, Arseniy and Avvakumov, Sergey},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}},
doi = {10.1017/fms.2018.7},
volume = {6},
year = {2018},
}
@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},
}
@article{1022,
abstract = {We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.},
author = {Pranav, Pratyush and Edelsbrunner, Herbert and Van De Weygaert, Rien and Vegter, Gert and Kerber, Michael and Jones, Bernard and Wintraecken, Mathijs},
issn = {00358711},
journal = {Monthly Notices of the Royal Astronomical Society},
number = {4},
pages = {4281 -- 4310},
publisher = {Oxford University Press},
title = {{The topology of the cosmic web in terms of persistent Betti numbers}},
doi = {10.1093/mnras/stw2862},
volume = {465},
year = {2017},
}
@article{1173,
abstract = {We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.},
author = {Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg and Nikitenko, Anton},
issn = {02099683},
journal = {Combinatorica},
number = {5},
pages = {887 -- 910},
publisher = {Springer},
title = {{The Voronoi functional is maximized by the Delaunay triangulation in the plane}},
doi = {10.1007/s00493-016-3308-y},
volume = {37},
year = {2017},
}
@article{1180,
abstract = {In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.},
author = {Akopyan, Arseniy and Bárány, Imre and Robins, Sinai},
issn = {00018708},
journal = {Advances in Mathematics},
pages = {627 -- 644},
publisher = {Academic Press},
title = {{Algebraic vertices of non-convex polyhedra}},
doi = {10.1016/j.aim.2016.12.026},
volume = {308},
year = {2017},
}
@article{1433,
abstract = {Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology.},
author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert},
issn = { 07477171},
journal = {Journal of Symbolic Computation},
pages = {76 -- 90},
publisher = {Academic Press},
title = {{Phat - Persistent homology algorithms toolbox}},
doi = {10.1016/j.jsc.2016.03.008},
volume = {78},
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},
}
@inproceedings{833,
abstract = {We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.},
author = {Heiss, Teresa and Wagner, Hubert},
editor = {Felsberg, Michael and Heyden, Anders and Krüger, Norbert},
issn = {03029743},
location = {Ystad, Sweden},
pages = {397 -- 409},
publisher = {Springer},
title = {{Streaming algorithm for Euler characteristic curves of multidimensional images}},
doi = {10.1007/978-3-319-64689-3_32},
volume = {10424},
year = {2017},
}
@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{1072,
abstract = {Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.},
author = {Bauer, Ulrich and Edelsbrunner, Herbert},
journal = {Transactions of the American Mathematical Society},
number = {5},
pages = {3741 -- 3762},
publisher = {American Mathematical Society},
title = {{The Morse theory of Čech and delaunay complexes}},
volume = {369},
year = {2017},
}
@article{1065,
abstract = {We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.},
author = {Chatterjee, Krishnendu and Osang, Georg F},
issn = {00200190},
journal = {Information Processing Letters},
pages = {25 -- 29},
publisher = {Elsevier},
title = {{Pushdown reachability with constant treewidth}},
doi = {10.1016/j.ipl.2017.02.003},
volume = {122},
year = {2017},
}
@article{707,
abstract = {We answer a question of M. Gromov on the waist of the unit ball.},
author = {Akopyan, Arseniy and Karasev, Roman},
issn = {00246093},
journal = {Bulletin of the London Mathematical Society},
number = {4},
pages = {690 -- 693},
publisher = {Wiley-Blackwell},
title = {{A tight estimate for the waist of the ball }},
doi = {10.1112/blms.12062},
volume = {49},
year = {2017},
}
@article{909,
abstract = {We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.},
author = {Akopyan, Arseniy and Vysotsky, Vladislav},
issn = {00029890},
journal = {The American Mathematical Monthly},
number = {7},
pages = {588 -- 596},
publisher = {Mathematical Association of America},
title = {{On the lengths of curves passing through boundary points of a planar convex shape}},
doi = {10.4169/amer.math.monthly.124.7.588},
volume = {124},
year = {2017},
}
@phdthesis{6287,
abstract = {The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.},
author = {Nikitenko, Anton},
pages = {86},
publisher = {IST Austria},
title = {{Discrete Morse theory for random complexes }},
doi = {10.15479/AT:ISTA:th_873},
year = {2017},
}
@article{481,
abstract = {We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.},
author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter},
journal = {International Journal of Computational Geometry and Applications},
number = {3-4},
pages = {211 -- 229},
publisher = {World Scientific Publishing},
title = {{Planar matchings for weighted straight skeletons}},
doi = {10.1142/S0218195916600050},
volume = {26},
year = {2017},
}