@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 = {1422-0067}, 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{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}, } @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{8538, abstract = {We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.}, author = {Akopyan, Arseniy and Schwartz, Richard and Tabachnikov, Serge}, issn = {2199-6768}, journal = {European Journal of Mathematics}, number = {4}, pages = {1313--1327}, publisher = {Springer Nature}, title = {{Billiards in ellipses revisited}}, doi = {10.1007/s40879-020-00426-9}, volume = {8}, 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}, } @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}, } @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}, } @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{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}, } @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}, } @article{15275, abstract = {In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.}, author = {Fox, Jacob and Pach, János and Suk, Andrew}, issn = {1439-6912}, journal = {Combinatorica}, keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics}, number = {6}, pages = {803--813}, publisher = {Springer Nature}, title = {{Bounded VC-dimension implies the Schur-Erdős conjecture}}, doi = {10.1007/s00493-021-4530-9}, volume = {41}, year = {2021}, } @article{10071, author = {Adams, Henry and Kourimska, Hana and Heiss, Teresa and Percival, Sarah and Ziegelmeier, Lori}, issn = {1088-9477}, journal = {Notices of the American Mathematical Society}, number = {9}, pages = {1511--1514}, publisher = {American Mathematical Society}, title = {{How to tutorial-a-thon}}, doi = {10.1090/noti2349}, volume = {68}, year = {2021}, } @inproceedings{10367, abstract = {How information is created, shared and consumed has changed rapidly in recent decades, in part thanks to new social platforms and technologies on the web. With ever-larger amounts of unstructured and limited labels, organizing and reconciling information from different sources and modalities is a central challenge in machine learning. This cutting-edge tutorial aims to introduce the multimodal entailment task, which can be useful for detecting semantic alignments when a single modality alone does not suffice for a whole content understanding. Starting with a brief overview of natural language processing, computer vision, structured data and neural graph learning, we lay the foundations for the multimodal sections to follow. We then discuss recent multimodal learning literature covering visual, audio and language streams, and explore case studies focusing on tasks which require fine-grained understanding of visual and linguistic semantics question answering, veracity and hatred classification. Finally, we introduce a new dataset for recognizing multimodal entailment, exploring it in a hands-on collaborative section. Overall, this tutorial gives an overview of multimodal learning, introduces a multimodal entailment dataset, and encourages future research in the topic.}, author = {Ilharco, Cesar and Shirazi, Afsaneh and Gopalan, Arjun and Nagrani, Arsha and Bratanič, Blaž and Bregler, Chris and Liu, Christina and Ferreira, Felipe and Barcik, Gabriek and Ilharco, Gabriel and Osang, Georg F and Bulian, Jannis and Frank, Jared and Smaira, Lucas and Cao, Qin and Marino, Ricardo and Patel, Roma and Leung, Thomas and Imbrasaite, Vaiva}, booktitle = {59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts}, isbn = {9-781-9540-8557-2}, location = {Bangkok, Thailand}, pages = {29--30}, publisher = {Association for Computational Linguistics}, title = {{Recognizing multimodal entailment}}, doi = {10.18653/v1/2021.acl-tutorials.6}, year = {2021}, } @article{7905, abstract = {We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.}, author = {Brown, Adam and Wang, Bei}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1166--1198}, publisher = {Springer Nature}, title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}}, doi = {10.1007/s00454-020-00206-y}, volume = {65}, year = {2021}, } @article{8248, abstract = {We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.}, author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {666--686}, publisher = {Springer Nature}, title = {{Local conditions for triangulating submanifolds of Euclidean space}}, doi = {10.1007/s00454-020-00233-9}, volume = {66}, year = {2021}, } @article{8338, abstract = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.}, author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {938--976}, publisher = {Springer Nature}, title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}}, doi = {10.1007/s00454-020-00240-w}, volume = {66}, year = {2021}, } @article{8773, abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.}, author = {Brown, Adam and Romanov, Anna}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, keywords = {Applied Mathematics, General Mathematics}, number = {1}, pages = {37--52}, publisher = {American Mathematical Society}, title = {{Contravariant forms on Whittaker modules}}, doi = {10.1090/proc/15205}, volume = {149}, year = {2021}, } @article{8940, abstract = {We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.}, author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Theoretical Computer Science, Computational Theory and Mathematics, Geometry and Topology, Discrete Mathematics and Combinatorics}, number = {1}, pages = {386--434}, publisher = {Springer Nature}, title = {{Triangulating submanifolds: An elementary and quantified version of Whitney’s method}}, doi = {10.1007/s00454-020-00250-8}, volume = {66}, year = {2021}, } @phdthesis{9056, abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density, and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets.}, author = {Osang, Georg F}, issn = {2663-337X}, pages = {134}, publisher = {Institute of Science and Technology Austria}, title = {{Multi-cover persistence and Delaunay mosaics}}, doi = {10.15479/AT:ISTA:9056}, year = {2021}, }