We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.
International audience We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les r ésultats fondamentaux de Zaslavsky sur le nombre de régions.
The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices.Let G be a connected graph on the vertex set [n] = {0} ∪ [n], where [n] denotes the set {1, . . . , n}. Let P denote the collection of acyclic orientations of G, and let P 0 denote the collection of acyclic orientations of G with 0 as a sink. If Ω is an orientation in P with the vertex i as a source, we can obtain a new orientation Ω with i as a sink by firing the vertex i, reorienting all the edges adjacent to i towards i. The orientations Ω and Ω agree away from i.A firing sequence from Ω to Ω in P consists of a sequence Ω = Ω 1 , . . . , Ω m+1 = Ω of orientations and a function F :, the orientation Ω i+1 is obtained from Ω i by firing the vertex F (i). We will abuse language by calling F itself a firing sequence. We make P into a preorder by writing Ω ≤ Ω if and only if there is a firing sequence from Ω to Ω . From the definition it is clear that P is reflexive and transitive. While P is only a preorder, P 0 is a poset. By finiteness, antisymmetry can be verified by showing that firing sequences in P 0 cannot be arbitrarily long. This is a consequence of the fact that neighbors of the distinguished sink 0 cannot fire. The proof depends on the following lemma. Proof. A vertex can fire only if it is a source. Firing the vertex i reverses the orientation of its edge to the vertex j. Hence the vertex i cannot fire again until the orientation is again reversed, which can only happen by firing j.As a corollary, firing sequences have bounded length, implying that P 0 is a poset.Corollary 2. The preorder P 0 of acyclic orientations with a distinguished sink is a poset. Hence firing sequences cannot be arbitrarily long, implying that P 0 is antisymmetric. *
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
Purpose This paper aims to examine how an open-source information management system was developed to manage a collection of more than 10,000 oral history interviews at the University of Kentucky Libraries’ Louie B. Nunn Center for Oral History. Design/methodology/approach Digital library architects at the University of Kentucky Libraries built an open-source information management system for oral history using the open-source tools Omeka and Blacklight. Additional open-source code was developed to facilitate interaction between these tools. Findings Information management systems that address needs of libraries and archives can be built by combining existing open-source tools in complementary ways. Originality/value This work at the University of Kentucky Libraries serves as a proof of concept for other institutions to examine as a potential model to follow or adapt for their own local needs. The SPOKEdb framework can be replicated elsewhere, as the major and minor components are open-source. SPOKEdb at its conceptual level is a unique information management system based on its tailored approach to serving the needs of oral history management at various user levels including both administrative and public.
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