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M ay 2 00 8 Intersections of Apartments
We show that, if a building is endowed with its complete system of apartments, and if each panel is contained in at least four chambers, then the intersection of two apartments can be any convex subcomplex contained in an apartment. This combinatorial result is particularly interesting for lower dimensional convex subcomplexes of apartments, where we definitely need the assumption on the four c...
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Let H be an infinite-dimensional complex Hilbert space and let L(H) be the logic formed by all closed subspaces of H. For every natural k we denote by Gk(H) the Grassmannian consisting of k-dimensional subspaces. An orthogonal apartment of Gk(H) is the set consisting of all k-dimensional subspaces spanned by subsets of a certain orthogonal base of H. Orthogonal apartments can be characterized a...
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Buildings of types Cn and Dn are defined by rank n polar spaces. The associated building Grassmannians are polar and half-spin Grassmannians. Apartments in dual polar spaces and half-spin Grassmannians were characterized in [4]. We characterize apartments in all polar Grassmannians consisting of non-maximal singular subspaces. This characterization is a partial case of more general results conc...
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Understanding indoor scenes is an important task in computer vision. This task is typically ambiguous, so we require a strong prior, that captures the regularity of indoor environments. This is naturally expressed by a probabilistic model over 3D room layouts and geometry, reasoning over complex layouts in 3D space, including high-order spatial relations among objects. In this work, we construc...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2010
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2009.03.021