TY - JOUR
AB - 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.
AU - Akopyan, Arseniy
AU - Bárány, Imre
AU - Robins, Sinai
ID - 1180
JF - Advances in Mathematics
SN - 00018708
TI - Algebraic vertices of non-convex polyhedra
VL - 308
ER -
TY - JOUR
AB - 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.
AU - Bauer, Ulrich
AU - Kerber, Michael
AU - Reininghaus, Jan
AU - Wagner, Hubert
ID - 1433
JF - Journal of Symbolic Computation
SN - 07477171
TI - Phat - Persistent homology algorithms toolbox
VL - 78
ER -
TY - JOUR
AB - 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.
AU - Pranav, Pratyush
AU - Edelsbrunner, Herbert
AU - Van De Weygaert, Rien
AU - Vegter, Gert
AU - Kerber, Michael
AU - Jones, Bernard
AU - Wintraecken, Mathijs
ID - 1022
IS - 4
JF - Monthly Notices of the Royal Astronomical Society
SN - 00358711
TI - The topology of the cosmic web in terms of persistent Betti numbers
VL - 465
ER -
TY - JOUR
AB - 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.
AU - Chatterjee, Krishnendu
AU - Osang, Georg F
ID - 1065
JF - Information Processing Letters
SN - 00200190
TI - Pushdown reachability with constant treewidth
VL - 122
ER -
TY - JOUR
AB - 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.
AU - Bauer, Ulrich
AU - Edelsbrunner, Herbert
ID - 1072
IS - 5
JF - Transactions of the American Mathematical Society
TI - The Morse theory of Čech and delaunay complexes
VL - 369
ER -
TY - JOUR
AB - 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.
AU - Edelsbrunner, Herbert
AU - Glazyrin, Alexey
AU - Musin, Oleg
AU - Nikitenko, Anton
ID - 1173
IS - 5
JF - Combinatorica
SN - 02099683
TI - The Voronoi functional is maximized by the Delaunay triangulation in the plane
VL - 37
ER -
TY - JOUR
AB - We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]d is generated for fixed m∈ℕ by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 1/10 d/N1/2+1/2d ≤EDN∗(P)≤ √d(log N) 1/2/N1/2+1/2d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points.
AU - Pausinger, Florian
AU - Steinerberger, Stefan
ID - 1617
JF - Journal of Complexity
TI - On the discrepancy of jittered sampling
VL - 33
ER -
TY - JOUR
AB - We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.
AU - Edelsbrunner, Herbert
AU - Pausinger, Florian
ID - 1662
JF - Advances in Mathematics
TI - Approximation and convergence of the intrinsic volume
VL - 287
ER -
TY - JOUR
AB - A framework fo r extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ2.
AU - Kasten, Jens
AU - Reininghaus, Jan
AU - Hotz, Ingrid
AU - Hege, Hans
AU - Noack, Bernd
AU - Daviller, Guillaume
AU - Morzyński, Marek
ID - 1216
IS - 1
JF - Archives of Mechanics
TI - Acceleration feature points of unsteady shear flows
VL - 68
ER -
TY - JOUR
AB - We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.
AU - Musin, Oleg
AU - Nikitenko, Anton
ID - 1222
IS - 1
JF - Discrete & Computational Geometry
TI - Optimal packings of congruent circles on a square flat torus
VL - 55
ER -