Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Geologic context of geodetic data across a Basin and Range normal fault, Crescent Valley, Nevada

Geodetic strain and late Quaternary faulting in the Basin and Range province is distributed over a region much wider than historic seismicity, which is localized near the margins of the province. In the relatively aseismic interior, both the magnitude and direction of geodetic strain may be inconsistent with the Holocene faulting record. We document the best example of such a disagreement across the NE striking, ~55° NW dipping Crescent normal fault, where a NW oriented, 70 km geodetic baseline records contemporary shortening of ~2 mm/yr orthogonal to the fault trace. In contrast, our geomorphic, paleoseismic, and geochronologic analyses of the Crescent fault suggest that a large extensional rupture occurred during the late Holocene epoch. An excavation across the fault at Fourmile Canyon reveals that the most recent event occurred at 2.8 ± 0.1 ka, with net vertical tectonic displacement of 4.6 ± 0.4 m at this location, corresponding to the release of ~3 m of accumulated NW-SE extension. Measured alluvial scarp profiles suggest a minimum rupture length of 30 km along the range front for the event, implying a moment magnitude M_w of at least 6.6. No prior event occurred between ~2.8 ka and ~6.4 ± 0.1 ka, the ^(14)C calender age of strata near the base of the exposed section. Assuming typical slip rates for Basin and Range faults (~0.3 mm/yr), these results imply that up to one third, or ~1 m, of the extensional strain released in the previous earthquake could have reaccumulated across the fault since ~2.8 ka. However, the contemporary shortening implies that the fault is unloading due to a transient process, whose duration is limited to between 6 years (geodetic recording time) and 2.8 ka (the age of the most recent event). These results emphasize the importance of providing accurate geologic data on the timescale of the earthquake cycle in order to evaluate geodetic measurements

Pure spinors, intrinsic torsion and curvature in even dimensions

We study the geometric properties of a $2m$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $\xi$ up to scale. The defining property of $\xi$ is that it determines an almost null structure, ie an $m$-plane distribution $\mathcal{N}_\xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $\mathcal{N}_\xi$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $\nabla$ of $g$ to be compatible with the $P$-structure. In a similar way, we examine the algebraic properties of the curvature of $\nabla$. Applications to spinorial differential equations are given. In particular, we give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. We also conjecture a Goldberg-Sachs-type theorem on the existence of a certain class of almost null structures when $(\mathcal{M},g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.Comment: v2. Cleaned up version. Typos and errors fixed. Some reordering. v3. Restructured - some material moved to an additional appendix for clarity - further typos fixed and other minor improvements v4. Presentation improved. Some material removed to be included in a future article. v5. As published: Abstract and intro rewritten. Presentation simplifie

Gridding of near vertical unrectified space photographs

Gridding of near vertical unrectified space photograph