@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{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},
}
@article{1272,
abstract = {We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.},
author = {Held, Martin and Huber, Stefan and Palfrader, Peter},
journal = {Computer-Aided Design and Applications},
number = {5},
pages = {712 -- 721},
publisher = {Taylor and Francis},
title = {{Generalized offsetting of planar structures using skeletons}},
doi = {10.1080/16864360.2016.1150718},
volume = {13},
year = {2016},
}
@article{1289,
abstract = {Aiming at the automatic diagnosis of tumors using narrow band imaging (NBI) magnifying endoscopic (ME) images of the stomach, we combine methods from image processing, topology, geometry, and machine learning to classify patterns into three classes: oval, tubular and irregular. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.},
author = {Dunaeva, Olga and Edelsbrunner, Herbert and Lukyanov, Anton and Machin, Michael and Malkova, Daria and Kuvaev, Roman and Kashin, Sergey},
journal = {Pattern Recognition Letters},
number = {1},
pages = {13 -- 22},
publisher = {Elsevier},
title = {{The classification of endoscopy images with persistent homology}},
doi = {10.1016/j.patrec.2015.12.012},
volume = {83},
year = {2016},
}
@article{1292,
abstract = {We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.},
author = {Durst, Sebastian and Kegel, Marc and Klukas, Mirko D},
journal = {Acta Mathematica Hungarica},
number = {2},
pages = {441 -- 455},
publisher = {Springer},
title = {{Computing the Thurston–Bennequin invariant in open books}},
doi = {10.1007/s10474-016-0648-4},
volume = {150},
year = {2016},
}
@article{1295,
abstract = {Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls.},
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel},
journal = {Electronic Notes in Discrete Mathematics},
pages = {169 -- 174},
publisher = {Elsevier},
title = {{Multiple covers with balls II: Weighted averages}},
doi = {10.1016/j.endm.2016.09.030},
volume = {54},
year = {2016},
}
@article{1330,
abstract = {In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.},
author = {Akopyan, Arseniy and Balitskiy, Alexey},
journal = {Israel Journal of Mathematics},
number = {2},
pages = {833 -- 845},
publisher = {Springer},
title = {{Billiards in convex bodies with acute angles}},
doi = {10.1007/s11856-016-1429-z},
volume = {216},
year = {2016},
}
@article{1360,
abstract = {We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. },
author = {Akopyan, Arseniy and Balitskiy, Alexey and Karasev, Roman and Sharipova, Anastasia},
journal = {Proceedings of the American Mathematical Society},
number = {10},
pages = {4501 -- 4513},
publisher = {American Mathematical Society},
title = {{Elementary approach to closed billiard trajectories in asymmetric normed spaces}},
doi = {10.1090/proc/13062},
volume = {144},
year = {2016},
}
@article{1408,
abstract = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.},
author = {Franek, Peter and Krcál, Marek},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {126 -- 164},
publisher = {Springer},
title = {{On computability and triviality of well groups}},
doi = {10.1007/s00454-016-9794-2},
volume = {56},
year = {2016},
}
@article{1617,
abstract = {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.},
author = {Pausinger, Florian and Steinerberger, Stefan},
journal = {Journal of Complexity},
pages = {199 -- 216},
publisher = {Academic Press},
title = {{On the discrepancy of jittered sampling}},
doi = {10.1016/j.jco.2015.11.003},
volume = {33},
year = {2016},
}
@article{1662,
abstract = {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.},
author = {Edelsbrunner, Herbert and Pausinger, Florian},
journal = {Advances in Mathematics},
pages = {674 -- 703},
publisher = {Academic Press},
title = {{Approximation and convergence of the intrinsic volume}},
doi = {10.1016/j.aim.2015.10.004},
volume = {287},
year = {2016},
}
@article{1149,
abstract = {We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.},
author = {Miyaji, Tomoyuki and Pilarczyk, Pawel and Gameiro, Marcio and Kokubu, Hiroshi and Mischaikow, Konstantin},
journal = {Applied Numerical Mathematics},
pages = {34 -- 47},
publisher = {Elsevier},
title = {{A study of rigorous ODE integrators for multi scale set oriented computations}},
doi = {10.1016/j.apnum.2016.04.005},
volume = {107},
year = {2016},
}
@article{1216,
abstract = {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.},
author = {Kasten, Jens and Reininghaus, Jan and Hotz, Ingrid and Hege, Hans and Noack, Bernd and Daviller, Guillaume and Morzyński, Marek},
journal = {Archives of Mechanics},
number = {1},
pages = {55 -- 80},
publisher = {Polish Academy of Sciences Publishing House},
title = {{Acceleration feature points of unsteady shear flows}},
volume = {68},
year = {2016},
}
@article{1222,
abstract = {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.},
author = {Musin, Oleg and Nikitenko, Anton},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {1 -- 20},
publisher = {Springer},
title = {{Optimal packings of congruent circles on a square flat torus}},
doi = {10.1007/s00454-015-9742-6},
volume = {55},
year = {2016},
}
@inproceedings{1237,
abstract = {Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.},
author = {Krcál, Marek and Pilarczyk, Pawel},
location = {Marseille, France},
pages = {140 -- 151},
publisher = {Springer},
title = {{Computation of cubical Steenrod squares}},
doi = {10.1007/978-3-319-39441-1_13},
volume = {9667},
year = {2016},
}
@article{1252,
abstract = {We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.},
author = {Harker, Shaun and Kokubu, Hiroshi and Mischaikow, Konstantin and Pilarczyk, Pawel},
journal = {Proceedings of the American Mathematical Society},
number = {4},
pages = {1787 -- 1801},
publisher = {American Mathematical Society},
title = {{Inducing a map on homology from a correspondence}},
doi = {10.1090/proc/12812},
volume = {144},
year = {2016},
}
@article{1254,
abstract = {We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.},
author = {Golmakani, Ali and Luzzatto, Stefano and Pilarczyk, Pawel},
journal = {Experimental Mathematics},
number = {2},
pages = {116 -- 124},
publisher = {Taylor and Francis},
title = {{Uniform expansivity outside a critical neighborhood in the quadratic family}},
doi = {10.1080/10586458.2015.1048011},
volume = {25},
year = {2016},
}
@article{1710,
abstract = {We consider the hollow on the half-plane {(x, y) : y ≤ 0} ⊂ ℝ2 defined by a function u : (-1, 1) → ℝ, u(x) < 0, and a vertical flow of point particles incident on the hollow. It is assumed that u satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph(u) and goes away without hitting the graph of u anymore. We solve the problem: find the function u minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the y-axis. Assuming that the resistance of u ≡ 0 equals 1, we show that the minimal resistance equals π/2 - 2arctan(1/2) ≈ 0.6435. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730-2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space {(x1,...,xd,y) : y ≤ 0} ⊂ ℝd+1 is defined by a radial function U satisfying the SIC, U(x) = u(|x|), with x = (x1,...,xd), u(ξ) < 0 for 0 ≤ ξ < 1, and u(ξ) = 0 for ξ ≥ 1, and the flow is parallel to the y-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as d → ∞.},
author = {Akopyan, Arseniy and Plakhov, Alexander},
journal = {Society for Industrial and Applied Mathematics},
number = {4},
pages = {2754 -- 2769},
publisher = {SIAM},
title = {{Minimal resistance of curves under the single impact assumption}},
doi = {10.1137/140993843},
volume = {47},
year = {2015},
}
@article{1792,
abstract = {Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.},
author = {Pausinger, Florian and Svane, Anne},
journal = {Journal of Complexity},
number = {6},
pages = {773 -- 797},
publisher = {Academic Press},
title = {{A Koksma-Hlawka inequality for general discrepancy systems}},
doi = {10.1016/j.jco.2015.06.002},
volume = {31},
year = {2015},
}