@inproceedings{833,
abstract = {We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.},
author = {Heiss, Teresa and Wagner, Hubert},
editor = {Felsberg, Michael and Heyden, Anders and Krüger, Norbert},
issn = {03029743},
location = {Ystad, Sweden},
pages = {397 -- 409},
publisher = {Springer},
title = {{Streaming algorithm for Euler characteristic curves of multidimensional images}},
doi = {10.1007/978-3-319-64689-3_32},
volume = {10424},
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},
}
@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{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},
}
@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{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},
}