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 -