TY - JOUR AB - 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. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton AU - Reitzner, Matthias ID - 718 IS - 3 JF - Advances in Applied Probability SN - 00018678 TI - Expected sizes of poisson Delaunay mosaics and their discrete Morse functions VL - 49 ER - TY - THES AB - 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. AU - Nikitenko, Anton ID - 6287 SN - 2663-337X TI - Discrete Morse theory for random complexes 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 - 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 - 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 -