@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},
  issn         = {1088-6826},
  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{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{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{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},
}

@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},
}

@inproceedings{1483,
  abstract     = {Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.},
  author       = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland},
  location     = {Boston, MA, USA},
  pages        = {4741 -- 4748},
  publisher    = {IEEE},
  title        = {{A stable multi-scale kernel for topological machine learning}},
  doi          = {10.1109/CVPR.2015.7299106},
  year         = {2015},
}

@inproceedings{1495,
  abstract     = {Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations. },
  author       = {Edelsbrunner, Herbert and Iglesias Ham, Mabel and Kurlin, Vitaliy},
  booktitle    = {Proceedings of the 27th Canadian Conference on Computational Geometry},
  location     = {Ontario, Canada},
  pages        = {128--135},
  publisher    = {Queen's University},
  title        = {{Relaxed disk packing}},
  volume       = {2015-August},
  year         = {2015},
}

@inproceedings{1510,
  abstract     = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. 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 R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n 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},
  location     = {Eindhoven, Netherlands},
  pages        = {842 -- 856},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{On computability and triviality of well groups}},
  doi          = {10.4230/LIPIcs.SOCG.2015.842},
  volume       = {34},
  year         = {2015},
}

@inbook{1531,
  abstract     = {The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.},
  author       = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid},
  booktitle    = {Visualization and Processing of Higher Order Descriptors for Multi-Valued Data},
  editor       = {Hotz, Ingrid and Schultz, Thomas},
  isbn         = {978-3-319-15089-5},
  pages        = {257 -- 267},
  publisher    = {Springer},
  title        = {{Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature}},
  doi          = {10.1007/978-3-319-15090-1_13},
  volume       = {40},
  year         = {2015},
}

@article{1555,
  abstract     = {We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.},
  author       = {Knipl, Diána and Pilarczyk, Pawel and Röst, Gergely},
  issn         = {1536-0040},
  journal      = {SIAM Journal on Applied Dynamical Systems},
  number       = {2},
  pages        = {980 -- 1017},
  publisher    = {Society for Industrial and Applied Mathematics },
  title        = {{Rich bifurcation structure in a two patch vaccination model}},
  doi          = {10.1137/140993934},
  volume       = {14},
  year         = {2015},
}