Completeness in supergravity constructions

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.Comment: 29 page

Applications of a completeness lemma in minimal surface theory to various classes of surfaces

We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a useful machinery for the study of completeness in these classes of surfaces. In particular, we show that a constant mean curvature one (i.e. CMC-1) surface in de Sitter 3-space is complete if and only if it is weakly complete, the singular set is compact and all the ends are conformally equivalent to a puntured disk.Comment: 9 page

Stream Fusion, to Completeness

Stream processing is mainstream (again): Widely-used stream libraries are now available for virtually all modern OO and functional languages, from Java to C# to Scala to OCaml to Haskell. Yet expressivity and performance are still lacking. For instance, the popular, well-optimized Java 8 streams do not support the zip operator and are still an order of magnitude slower than hand-written loops. We present the first approach that represents the full generality of stream processing and eliminates overheads, via the use of staging. It is based on an unusually rich semantic model of stream interaction. We support any combination of zipping, nesting (or flat-mapping), sub-ranging, filtering, mapping-of finite or infinite streams. Our model captures idiosyncrasies that a programmer uses in optimizing stream pipelines, such as rate differences and the choice of a "for" vs. "while" loops. Our approach delivers hand-written-like code, but automatically. It explicitly avoids the reliance on black-box optimizers and sufficiently-smart compilers, offering highest, guaranteed and portable performance. Our approach relies on high-level concepts that are then readily mapped into an implementation. Accordingly, we have two distinct implementations: an OCaml stream library, staged via MetaOCaml, and a Scala library for the JVM, staged via LMS. In both cases, we derive libraries richer and simultaneously many tens of times faster than past work. We greatly exceed in performance the standard stream libraries available in Java, Scala and OCaml, including the well-optimized Java 8 streams