@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{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{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},
}