1,720,966 research outputs found
The homotopy type of elliptic arrangements
Get PDFWe give combinatorial models for the homotopy type of complements of elliptic arrangements, ie certain sets of abelian subvarieties in a product of elliptic curves. We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves.Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes
Products of arithmetic matroids and quasipolynomial invariants of CW-complexes
Get PDFIn this note we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function. This allows us to answer a question by Bajo–Burdick–Chmutov [2], concerning the modified Tutte–Krushkal–Renhardy polynomials defined by these authors. Furthermore, we show that the Tutte quasi-polynomial introduced by Brändén and Moci encompasses invariants defined by Beck–Breuer–Godkin–Martin [3] and Duval–Klivans–Martin [11] and can thus be considered as a dichromate for CW complexes
Stanley–Reisner rings for symmetric simplicial complexes, -semimatroids and Abelian arrangements
Get PDFWe extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action.We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen–Macaulay property under quotients.
When the acted-upon poset is the independence complex of a semimatroid, the h-polynomial of the ring of invariants can be read off the Tutte polynomial of the associated group action. Moreover, in this case an additional condition on the action ensures that the quotient poset is Cohen–Macaulay in characteristic 0 and every characteristic that does not divide an explicitly computable number. This implies the same property for the associated Stanley–Reisner rings. In particular, this holds for independence posets and rings associated to toric, elliptic and, more generally, (p,q) -arrangements.
As a byproduct, we prove that posets of connected components (also known as posets of layers) of such arrangements are Cohen–Macaulay with the same condition on the characteristic
Corrigendum to "Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements"
Get PDFIn this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X) + 1. Below we give the complete and correct presentation of the rational cohomology ring
Many faces of symmetric edge polytopes
Get PDFSymmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics – where they are called adjacency polytopes – and to Kantorovich–Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic-combinatorial methods to investigate invariants of the associated symmetric edge polytopes
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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