@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},
}
@phdthesis{6287,
abstract = {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.},
author = {Nikitenko, Anton},
pages = {86},
publisher = {IST Austria},
title = {{Discrete Morse theory for random complexes }},
doi = {10.15479/AT:ISTA:th_873},
year = {2017},
}
@article{481,
abstract = {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},
journal = {International Journal of Computational Geometry and Applications},
number = {3-4},
pages = {211 -- 229},
publisher = {World Scientific Publishing},
title = {{Planar matchings for weighted straight skeletons}},
doi = {10.1142/S0218195916600050},
volume = {26},
year = {2017},
}
@article{568,
abstract = {We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).},
author = {Franek, Peter and Krcál, Marek},
issn = {15320073},
journal = {Homology, Homotopy and Applications},
number = {2},
pages = {313 -- 342},
publisher = {International Press},
title = {{Persistence of zero sets}},
doi = {10.4310/HHA.2017.v19.n2.a16},
volume = {19},
year = {2017},
}
@inproceedings{688,
abstract = {We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. },
author = {Edelsbrunner, Herbert and Wagner, Hubert},
issn = {18688969},
location = {Brisbane, Australia},
pages = {391--3916},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Topological data analysis with Bregman divergences}},
doi = {10.4230/LIPIcs.SoCG.2017.39},
volume = {77},
year = {2017},
}
@article{521,
abstract = {Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.},
author = {Austin, Kyle and Virk, Ziga},
issn = {01668641},
journal = {Topology and its Applications},
pages = {45 -- 57},
publisher = {Elsevier},
title = {{Higson compactification and dimension raising}},
doi = {10.1016/j.topol.2016.10.005},
volume = {215},
year = {2017},
}
@inproceedings{836,
abstract = {Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.},
author = {Ethier, Marc and Jablonski, Grzegorz and Mrozek, Marian},
booktitle = {Special Sessions in Applications of Computer Algebra},
isbn = {978-331956930-7},
location = {Kalamata, Greece},
pages = {119 -- 136},
publisher = {Springer},
title = {{Finding eigenvalues of self-maps with the Kronecker canonical form}},
doi = {10.1007/978-3-319-56932-1_8},
volume = {198},
year = {2017},
}
@inbook{84,
abstract = {The advent of high-throughput technologies and the concurrent advances in information sciences have led to a data revolution in biology. This revolution is most significant in molecular biology, with an increase in the number and scale of the “omics” projects over the last decade. Genomics projects, for example, have produced impressive advances in our knowledge of the information concealed into genomes, from the many genes that encode for the proteins that are responsible for most if not all cellular functions, to the noncoding regions that are now known to provide regulatory functions. Proteomics initiatives help to decipher the role of post-translation modifications on the protein structures and provide maps of protein-protein interactions, while functional genomics is the field that attempts to make use of the data produced by these projects to understand protein functions. The biggest challenge today is to assimilate the wealth of information provided by these initiatives into a conceptual framework that will help us decipher life. For example, the current views of the relationship between protein structure and function remain fragmented. We know of their sequences, more and more about their structures, we have information on their biological activities, but we have difficulties connecting this dotted line into an informed whole. We lack the experimental and computational tools for directly studying protein structure, function, and dynamics at the molecular and supra-molecular levels. In this chapter, we review some of the current developments in building the computational tools that are needed, focusing on the role that geometry and topology play in these efforts. One of our goals is to raise the general awareness about the importance of geometric methods in elucidating the mysterious foundations of our very existence. Another goal is the broadening of what we consider a geometric algorithm. There is plenty of valuable no-man’s-land between combinatorial and numerical algorithms, and it seems opportune to explore this land with a computational-geometric frame of mind.},
author = {Edelsbrunner, Herbert and Koehl, Patrice},
booktitle = {Handbook of Discrete and Computational Geometry, Third Edition},
editor = {Toth, Csaba and O'Rourke, Joseph and Goodman, Jacob},
pages = {1709 -- 1735},
publisher = {CRC Press},
title = {{Computational topology for structural molecular biology}},
doi = {10.1201/9781315119601},
year = {2017},
}
@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{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},
}