TY - JOUR AB - 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. AU - Dikranjan, Dikran AU - Giordano Bruno, Anna AU - Künzi, Hans Peter AU - Zava, Nicolò AU - Toller, Daniele ID - 10413 JF - Topology and its Applications SN - 0166-8641 TI - Generalized quasi-metric semilattices VL - 309 ER - TY - JOUR AB - 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. AU - Biswas, Ranita AU - Cultrera Di Montesano, Sebastiano AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 10773 JF - Discrete and Computational Geometry SN - 0179-5376 TI - Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics VL - 67 ER - TY - CONF AB - 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. AU - Heiss, Teresa AU - Tymochko, Sarah AU - Story, Brittany AU - Garin, Adélie AU - Bui, Hoa AU - Bleile, Bea AU - Robins, Vanessa ID - 10828 SN - 9781665439022 T2 - 2021 IEEE International Conference on Big Data TI - The impact of changes in resolution on the persistent homology of images ER - TY - JOUR AB - 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 . AU - Brown, Adam AU - Romanov, Anna ID - 11545 IS - 11 JF - Journal of Algebra KW - Algebra and Number Theory SN - 0021-8693 TI - Contravariant pairings between standard Whittaker modules and Verma modules VL - 609 ER - TY - JOUR AB - 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. AU - Chang, Yan AU - Funk, Marah AU - Roy, Souvik AU - Stephenson, Elizabeth R AU - Choi, Sangyong AU - Kojouharov, Hristo V. AU - Chen, Benito AU - Pan, Zui ID - 10754 IS - 3 JF - International Journal of Molecular Sciences SN - 16616596 TI - Developing a mathematical model of intracellular Calcium dynamics for evaluating combined anticancer effects of afatinib and RP4010 in esophageal cancer VL - 23 ER - TY - JOUR AB - Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 7791 IS - 4 JF - European Journal of Mathematics SN - 2199-675X TI - When different norms lead to same billiard trajectories? VL - 8 ER - TY - JOUR AB - 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. AU - Biswas, Ranita AU - Cultrera di Montesano, Sebastiano AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 11660 JF - LIPIcs TI - A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs ER - TY - JOUR AB - 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. AU - Biswas, Ranita AU - Cultrera di Montesano, Sebastiano AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 11658 JF - Leibniz International Proceedings on Mathematics TI - Depth in arrangements: Dehn–Sommerville–Euler relations with applications ER - TY - GEN AB - 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. AU - Biswas, Ranita AU - Cultrera di Montesano, Sebastiano AU - Draganov, Ondrej AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 15090 T2 - arXiv TI - On the size of chromatic Delaunay mosaics ER - TY - JOUR AB - 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. AU - Goudarzi, Samira AU - Sharif, Mohammad AU - Karimipour, Farid ID - 10208 JF - Journal of Ambient Intelligence and Humanized Computing KW - general computer science SN - 1868-5137 TI - A context-aware dimension reduction framework for trajectory and health signal analyses VL - 13 ER - TY - JOUR AU - Adams, Henry AU - Kourimska, Hana AU - Heiss, Teresa AU - Percival, Sarah AU - Ziegelmeier, Lori ID - 10071 IS - 9 JF - Notices of the American Mathematical Society SN - 0002-9920 TI - How to tutorial-a-thon VL - 68 ER - TY - CONF AB - 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. AU - Ilharco, Cesar AU - Shirazi, Afsaneh AU - Gopalan, Arjun AU - Nagrani, Arsha AU - Bratanič, Blaž AU - Bregler, Chris AU - Liu, Christina AU - Ferreira, Felipe AU - Barcik, Gabriek AU - Ilharco, Gabriel AU - Osang, Georg F AU - Bulian, Jannis AU - Frank, Jared AU - Smaira, Lucas AU - Cao, Qin AU - Marino, Ricardo AU - Patel, Roma AU - Leung, Thomas AU - Imbrasaite, Vaiva ID - 10367 SN - 9-781-9540-8557-2 T2 - 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, Tutorial Abstracts TI - Recognizing multimodal entailment ER - TY - JOUR AB - We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property. AU - Weighill, Thomas AU - Yamauchi, Takamitsu AU - Zava, Nicolò ID - 10608 JF - European Journal of Mathematics SN - 2199-675X TI - Coarse infinite-dimensionality of hyperspaces of finite subsets ER - TY - CONF AB - 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 convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ 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(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge. AU - Aichholzer, Oswin AU - Arroyo Guevara, Alan M AU - Masárová, Zuzana AU - Parada, Irene AU - Perz, Daniel AU - Pilz, Alexander AU - Tkadlec, Josef AU - Vogtenhuber, Birgit ID - 9296 SN - 03029743 T2 - 15th International Conference on Algorithms and Computation TI - On compatible matchings VL - 12635 ER - TY - JOUR AB - Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton AU - Osang, Georg F ID - 9465 IS - 1 JF - Journal of Geometry SN - 00472468 TI - A step in the Delaunay mosaic of order k VL - 112 ER - TY - CONF AB - Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction. AU - Edelsbrunner, Herbert AU - Heiss, Teresa AU - Kurlin , Vitaliy AU - Smith, Philip AU - Wintraecken, Mathijs ID - 9345 SN - 1868-8969 T2 - 37th International Symposium on Computational Geometry (SoCG 2021) TI - The density fingerprint of a periodic point set VL - 189 ER - TY - CONF AB - Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation. AU - Biswas, Ranita AU - Cultrera di Montesano, Sebastiano AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 9604 SN - 18688969 T2 - Leibniz International Proceedings in Informatics TI - Counting cells of order-k voronoi tessellations in ℝ3 with morse theory VL - 189 ER - TY - CONF AB - We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain. AU - Čomić, Lidija AU - Zrour, Rita AU - Largeteau-Skapin, Gaëlle AU - Biswas, Ranita AU - Andres, Eric ID - 9824 SN - 03029743 T2 - Discrete Geometry and Mathematical Morphology TI - Body centered cubic grid - coordinate system and discrete analytical plane definition VL - 12708 ER - TY - JOUR AB - When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability. AU - Aichholzer, Oswin AU - Akitaya, Hugo A. AU - Cheung, Kenneth C. AU - Demaine, Erik D. AU - Demaine, Martin L. AU - Fekete, Sándor P. AU - Kleist, Linda AU - Kostitsyna, Irina AU - Löffler, Maarten AU - Masárová, Zuzana AU - Mundilova, Klara AU - Schmidt, Christiane ID - 8317 JF - Computational Geometry: Theory and Applications SN - 09257721 TI - Folding polyominoes with holes into a cube VL - 93 ER - TY - JOUR AB - 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. AU - Brown, Adam AU - Romanov, Anna ID - 8773 IS - 1 JF - Proceedings of the American Mathematical Society KW - Applied Mathematics KW - General Mathematics SN - 0002-9939 TI - Contravariant forms on Whittaker modules VL - 149 ER -