@article{1584, abstract = {We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.}, author = {Biedl, Therese and Held, Martin and Huber, Stefan and Kaaser, Dominik and Palfrader, Peter}, journal = {Computational Geometry: Theory and Applications}, number = {5}, pages = {429 -- 442}, publisher = {Elsevier}, title = {{Reprint of: Weighted straight skeletons in the plane}}, doi = {10.1016/j.comgeo.2015.01.004}, volume = {48}, year = {2015}, } @inbook{1590, abstract = {The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.}, author = {Aichholzer, Oswin and Biedl, Therese and Hackl, Thomas and Held, Martin and Huber, Stefan and Palfrader, Peter and Vogtenhuber, Birgit}, booktitle = {Graph Drawing and Network Visualization}, isbn = {978-3-319-27260-3}, location = {Los Angeles, CA, United States}, pages = {335 -- 347}, publisher = {Springer Nature}, title = {{Representing directed trees as straight skeletons}}, doi = {10.1007/978-3-319-27261-0_28}, volume = {9411}, year = {2015}, } @article{1682, abstract = {We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K→ ℝn on a finite simplicial complex K and α > 0, it holds that each function g: K → ℝn such that ||g - f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K > 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.}, author = {Franek, Peter and Krcál, Marek}, journal = {Journal of the ACM}, number = {4}, publisher = {ACM}, title = {{Robust satisfiability of systems of equations}}, doi = {10.1145/2751524}, volume = {62}, year = {2015}, } @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{1938, abstract = {We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid Z2.}, author = {Pausinger, Florian and Steinerberger, Stefan}, journal = {Physics Letters, Section A}, number = {6}, pages = {535 -- 541}, publisher = {Elsevier}, title = {{On the distribution of local extrema in quantum chaos}}, doi = {10.1016/j.physleta.2014.12.010}, volume = {379}, year = {2015}, } @article{2035, abstract = {Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility. }, author = {Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, Marian}, journal = {Foundations of Computational Mathematics}, number = {5}, pages = {1213 -- 1244}, publisher = {Springer}, title = {{The persistent homology of a self-map}}, doi = {10.1007/s10208-014-9223-y}, volume = {15}, 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}, } @article{1793, abstract = {We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth.}, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, journal = {PLoS One}, number = {6}, publisher = {Public Library of Science}, title = {{DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots}}, doi = {10.1371/journal.pone.0127657}, volume = {10}, year = {2015}, } @article{1805, abstract = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.}, author = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André}, journal = {Computational Geometry: Theory and Applications}, number = {8}, pages = {606 -- 621}, publisher = {Elsevier}, title = {{Homological reconstruction and simplification in R3}}, doi = {10.1016/j.comgeo.2014.08.010}, volume = {48}, year = {2015}, } @article{1828, abstract = {We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.}, author = {Akopyan, Arseniy and Pirogov, Sergey and Rybko, Aleksandr}, journal = {Journal of Statistical Physics}, number = {1}, pages = {163 -- 167}, publisher = {Springer}, title = {{Invariant measures of genetic recombination process}}, doi = {10.1007/s10955-015-1238-5}, volume = {160}, year = {2015}, } @misc{9737, author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert}, publisher = {Public Library of Science}, title = {{Root traits computed by DynamicRoots for the maize root shown in fig 2}}, doi = {10.1371/journal.pone.0127657.s001}, year = {2015}, } @inbook{10817, abstract = {The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption.}, author = {Günther, David and Reininghaus, Jan and Seidel, Hans-Peter and Weinkauf, Tino}, booktitle = {Topological Methods in Data Analysis and Visualization III.}, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {135--150}, publisher = {Springer Nature}, title = {{Notes on the simplification of the Morse-Smale complex}}, doi = {10.1007/978-3-319-04099-8_9}, year = {2014}, } @inproceedings{10886, abstract = {We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Topological Methods in Data Analysis and Visualization III }, isbn = {9783319040981}, issn = {2197-666X}, pages = {249--262}, publisher = {Springer}, title = {{Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature}}, doi = {10.1007/978-3-319-04099-8_16}, year = {2014}, } @inproceedings{10892, abstract = {In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.}, author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter}, booktitle = {25th International Symposium, ISAAC 2014}, isbn = {9783319130743}, issn = {1611-3349}, location = {Jeonju, Korea}, pages = {117--127}, publisher = {Springer Nature}, title = {{Planar matchings for weighted straight skeletons}}, doi = {10.1007/978-3-319-13075-0_10}, volume = {8889}, year = {2014}, } @inbook{10893, abstract = {Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data.}, author = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik}, booktitle = {Topological Methods in Data Analysis and Visualization III }, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {55--69}, publisher = {Springer}, title = {{Toward the extraction of saddle periodic orbits}}, doi = {10.1007/978-3-319-04099-8_4}, volume = {1}, year = {2014}, } @inproceedings{10894, abstract = {PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert}, booktitle = {ICMS 2014: International Congress on Mathematical Software}, isbn = {9783662441985}, issn = {1611-3349}, location = {Seoul, South Korea}, pages = {137--143}, publisher = {Springer Berlin Heidelberg}, title = {{PHAT – Persistent Homology Algorithms Toolbox}}, doi = {10.1007/978-3-662-44199-2_24}, volume = {8592}, year = {2014}, } @article{2255, abstract = {Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.}, author = {Edelsbrunner, Herbert and Pausinger, Florian}, issn = {0924-9907}, journal = {Journal of Mathematical Imaging and Vision}, number = {1}, pages = {164 -- 177}, publisher = {Springer}, title = {{Stable length estimates of tube-like shapes}}, doi = {10.1007/s10851-013-0468-x}, volume = {50}, year = {2014}, } @inproceedings{2905, abstract = {Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions.}, author = {Edelsbrunner, Herbert and Morozovy, Dmitriy}, location = {Kraków, Poland}, pages = {31 -- 50}, publisher = {European Mathematical Society}, title = {{Persistent homology: Theory and practice}}, doi = {10.4171/120-1/3}, year = {2014}, } @inproceedings{2153, abstract = {We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).}, author = {Bauer, Ulrich and Lesnick, Michael}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {355 -- 364}, publisher = {ACM}, title = {{Induced matchings of barcodes and the algebraic stability of persistence}}, doi = {10.1145/2582112.2582168}, year = {2014}, } @inproceedings{2155, abstract = {Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).}, author = {Bauer, Ulrich and Edelsbrunner, Herbert}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {484 -- 490}, publisher = {ACM}, title = {{The morse theory of Čech and Delaunay filtrations}}, doi = {10.1145/2582112.2582167}, year = {2014}, }