We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex.As a corollary ...of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.
Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular ...space, the search for a family of spaces whose homologies have properties analogous to intersection homology has developed. For some stratified spaces, M. Banagl has introduced such a family by using a topological truncation: the original link is replaced by a truncation of its homological Moore resolution. In this work, we study the dual approach in the Eckmann-Hilton sense : we consider the stratified space obtained by replacing the original link by a Postnikov approximation. The main result is that our construction restores the space constructed by Gajer to establish an intersection Dold-Thom theorem. We are conducting this study within the general framework of Quinn's homotopically stratified spaces.
Simplicial closure and higher-order link prediction Benson, Austin R.; Abebe, Rediet; Schaub, Michael T. ...
Proceedings of the National Academy of Sciences - PNAS,
11/2018, Letnik:
115, Številka:
48
Journal Article
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Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place ...among more than two nodes at once—for example, communication within a group rather than person to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental difference from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.
For a finite set of balls of radius r , the k -fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is ...called the k -fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k , it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k -fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the k -fold filtrations for several values of k , with the same size and complexity bounds.
In this corrigendum, we explain and correct a mistake in our article "Curved Koszul duality theory". Our definitions of morphisms between semi-augmented properads and between curved coproperads have ...to be modified.
Dans ce corrigendum, nous expliquons et corrigeons une erreur dans notre article "Curved Koszul duality theory". Nos définitions de morphisme entre propérades semi-augmentées et entre coopérades courbées ont été modifiées.
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