@article{13165, abstract = {A graph G=(V, E) is called fully regular if for every independent set I c V, the number of vertices in V\I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph. As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.}, author = {Fang, Lixing and Huang, Hao and Pach, János and Tardos, Gábor and Zuo, Junchi}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory. Series A}, number = {10}, publisher = {Elsevier}, title = {{Successive vertex orderings of fully regular graphs}}, doi = {10.1016/j.jcta.2023.105776}, volume = {199}, year = {2023}, } @article{14362, abstract = {Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwise weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly monotone semi(-co-)valuations. We conclude discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation of both the known intrinsic semilattice entropy and the semigroup entropy.}, author = {Castellano, Ilaria and Giordano Bruno, Anna and Zava, Nicolò}, issn = {0304-3975}, journal = {Theoretical Computer Science}, publisher = {Elsevier}, title = {{Weakly weighted generalised quasi-metric spaces and semilattices}}, doi = {10.1016/j.tcs.2023.114129}, volume = {977}, year = {2023}, } @article{13182, abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.}, author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, publisher = {Springer Nature}, title = {{Geometric characterization of the persistence of 1D maps}}, doi = {10.1007/s41468-023-00126-9}, year = {2023}, } @phdthesis{14226, abstract = {We introduce the notion of a Faustian interchange in a 1-parameter family of smooth functions to generalize the medial axis to critical points of index larger than 0. We construct and implement a general purpose algorithm for approximating such generalized medial axes.}, author = {Stephenson, Elizabeth R}, issn = {2791-4585}, pages = {43}, publisher = {Institute of Science and Technology Austria}, title = {{Generalizing medial axes with homology switches}}, doi = {10.15479/at:ista:14226}, year = {2023}, } @inproceedings{11428, abstract = {The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.}, author = {Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {38th International Symposium on Computational Geometry}, editor = {Goaoc, Xavier and Kerber, Michael}, isbn = {978-3-95977-227-3}, issn = {1868-8969}, location = {Berlin, Germany}, pages = {66:1--66:9}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{A cautionary tale: Burning the medial axis is unstable}}, doi = {10.4230/LIPIcs.SoCG.2022.66}, volume = {224}, year = {2022}, } @book{11429, abstract = {This book constitutes the refereed proceedings of the 18th International Symposium on Web and Wireless Geographical Information Systems, W2GIS 2022, held in Konstanz, Germany, in April 2022. The 7 full papers presented together with 6 short papers in the volume were carefully reviewed and selected from 16 submissions. The papers cover topics that range from mobile GIS and Location-Based Services to Spatial Information Retrieval and Wireless Sensor Networks.}, editor = {Karimipour, Farid and Storandt, Sabine}, isbn = {9783031062445}, issn = {1611-3349}, pages = {153}, publisher = {Springer Nature}, title = {{Web and Wireless Geographical Information Systems}}, doi = {10.1007/978-3-031-06245-2}, volume = {13238}, year = {2022}, } @inbook{11440, abstract = {To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation.}, author = {Bleile, Bea and Garin, Adélie and Heiss, Teresa and Maggs, Kelly and Robins, Vanessa}, booktitle = {Research in Computational Topology 2}, editor = {Gasparovic, Ellen and Robins, Vanessa and Turner, Katharine}, isbn = {9783030955182}, pages = {1--26}, publisher = {Springer Nature}, title = {{The persistent homology of dual digital image constructions}}, doi = {10.1007/978-3-030-95519-9_1}, volume = {30}, year = {2022}, } @article{12307, abstract = {Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences.}, author = {Shipman, Barbara A. and Stephenson, Elizabeth R}, issn = {1935-4053}, journal = {PRIMUS}, keywords = {Education, General Mathematics}, number = {5}, pages = {593--609}, publisher = {Taylor & Francis}, title = {{Tangible topology through the lens of limits}}, doi = {10.1080/10511970.2021.1872750}, volume = {32}, year = {2022}, } @article{11938, abstract = {A matching is compatible to two or more labeled point sets of size n with labels {1, . . . , n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled sets of n points in convex position there exists a compatible matching with ⌊√2n + 1 − 1⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ). As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(log n) copies of any set of n points are necessary and sufficient for the existence of labelings of these point sets such that any compatible matching consists only of a single edge.}, author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit}, issn = {1526-1719}, journal = {Journal of Graph Algorithms and Applications}, number = {2}, pages = {225--240}, publisher = {Brown University}, title = {{On compatible matchings}}, doi = {10.7155/jgaa.00591}, volume = {26}, year = {2022}, } @article{9649, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.}, author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics }, pages = {967--1012}, publisher = {Springer Nature}, title = {{The topological correctness of PL approximations of isomanifolds}}, doi = {10.1007/s10208-021-09520-0}, volume = {22}, year = {2022}, } @article{10413, abstract = {Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.}, author = {Dikranjan, Dikran and Giordano Bruno, Anna and Künzi, Hans Peter and Zava, Nicolò and Toller, Daniele}, issn = {0166-8641}, journal = {Topology and its Applications}, publisher = {Elsevier}, title = {{Generalized quasi-metric semilattices}}, doi = {10.1016/j.topol.2021.107916}, volume = {309}, year = {2022}, } @article{10773, abstract = {The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.}, author = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {811--842}, publisher = {Springer Nature}, title = {{Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics}}, doi = {10.1007/s00454-022-00371-2}, volume = {67}, year = {2022}, } @inproceedings{10828, abstract = {Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.}, author = {Heiss, Teresa and Tymochko, Sarah and Story, Brittany and Garin, Adélie and Bui, Hoa and Bleile, Bea and Robins, Vanessa}, booktitle = {2021 IEEE International Conference on Big Data}, isbn = {9781665439022}, location = {Orlando, FL, United States; Virtuell}, pages = {3824--3834}, publisher = {IEEE}, title = {{The impact of changes in resolution on the persistent homology of images}}, doi = {10.1109/BigData52589.2021.9671483}, year = {2022}, } @article{11545, abstract = {We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N . We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N .}, author = {Brown, Adam and Romanov, Anna}, issn = {0021-8693}, journal = {Journal of Algebra}, keywords = {Algebra and Number Theory}, number = {11}, pages = {145--179}, publisher = {Elsevier}, title = {{Contravariant pairings between standard Whittaker modules and Verma modules}}, doi = {10.1016/j.jalgebra.2022.06.017}, volume = {609}, year = {2022}, } @article{10754, abstract = {Targeting dysregulated Ca2+ signaling in cancer cells is an emerging chemotherapy approach. We previously reported that store-operated Ca2+ entry (SOCE) blockers, such as RP4010, are promising antitumor drugs for esophageal cancer. As a tyrosine kinase inhibitor (TKI), afatinib received FDA approval to be used in targeted therapy for patients with EGFR mutation-positive cancers. While preclinical studies and clinical trials have shown that afatinib has benefits for esophageal cancer patients, it is not known whether a combination of afatinib and RP4010 could achieve better anticancer effects. Since TKI can alter intracellular Ca2+ dynamics through EGFR/phospholipase C-γ pathway, in this study, we evaluated the inhibitory effect of afatinib and RP4010 on intracellular Ca2+ oscillations in KYSE-150, a human esophageal squamous cell carcinoma cell line, using both experimental and mathematical simulations. Our mathematical simulation of Ca2+ oscillations could fit well with experimental data responding to afatinib or RP4010, both separately or in combination. Guided by simulation, we were able to identify a proper ratio of afatinib and RP4010 for combined treatment, and such a combination presented synergistic anticancer-effect evidence by experimental measurement of intracellular Ca2+ and cell proliferation. This intracellular Ca2+ dynamic-based mathematical simulation approach could be useful for a rapid and cost-effective evaluation of combined targeting therapy drugs.}, author = {Chang, Yan and Funk, Marah and Roy, Souvik and Stephenson, Elizabeth R and Choi, Sangyong and Kojouharov, Hristo V. and Chen, Benito and Pan, Zui}, issn = {14220067}, journal = {International Journal of Molecular Sciences}, number = {3}, publisher = {MDPI}, title = {{Developing a mathematical model of intracellular Calcium dynamics for evaluating combined anticancer effects of afatinib and RP4010 in esophageal cancer}}, doi = {10.3390/ijms23031763}, volume = {23}, year = {2022}, } @article{7791, abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {2199-6768}, journal = {European Journal of Mathematics}, number = {4}, pages = {1309 -- 1312}, publisher = {Springer Nature}, title = {{When different norms lead to same billiard trajectories?}}, doi = {10.1007/s40879-020-00405-0}, volume = {8}, year = {2022}, } @article{11660, abstract = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. }, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, journal = {LIPIcs}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs}}, year = {2022}, } @article{11658, abstract = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, journal = {Leibniz International Proceedings on Mathematics}, publisher = {Schloss Dagstuhl - Leibniz Zentrum für Informatik}, title = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}}, year = {2022}, } @unpublished{15090, abstract = {Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{On the size of chromatic Delaunay mosaics}}, year = {2022}, } @article{10208, abstract = {It is practical to collect a huge amount of movement data and environmental context information along with the health signals of individuals because there is the emergence of new generations of positioning and tracking technologies and rapid advancements of health sensors. The study of the relations between these datasets and their sequence similarity analysis is of interest to many applications such as health monitoring and recommender systems. However, entering all movement parameters and health signals can lead to the complexity of the problem and an increase in its computational load. In this situation, dimension reduction techniques can be used to avoid consideration of simultaneous dependent parameters in the process of similarity measurement of the trajectories. The present study provides a framework, named CaDRAW, to use spatial–temporal data and movement parameters along with independent context information in the process of measuring the similarity of trajectories. In this regard, the omission of dependent movement characteristic signals is conducted by using an unsupervised feature selection dimension reduction technique. To evaluate the effectiveness of the proposed framework, it was applied to a real contextualized movement and related health signal datasets of individuals. The results indicated the capability of the proposed framework in measuring the similarity and in decreasing the characteristic signals in such a way that the similarity results -before and after reduction of dependent characteristic signals- have small differences. The mean differences between the obtained results before and after reducing the dimension were 0.029 and 0.023 for the round path, respectively.}, author = {Goudarzi, Samira and Sharif, Mohammad and Karimipour, Farid}, issn = {1868-5145}, journal = {Journal of Ambient Intelligence and Humanized Computing}, keywords = {general computer science}, pages = {2621–2635}, publisher = {Springer Nature}, title = {{A context-aware dimension reduction framework for trajectory and health signal analyses}}, doi = {10.1007/s12652-021-03569-z}, volume = {13}, year = {2022}, }