Lightweight MPI Communicators with Applications to Perfectly Balanced Quicksort

MPI uses the concept of communicators to connect groups of processes. It provides nonblocking collective operations on communicators to overlap communication and computation. Flexible algorithms demand flexible communicators. E.g., a process can work on different subproblems within different process groups simultaneously, new process groups can be created, or the members of a process group can change. Depending on the number of communicators, the time for communicator creation can drastically increase the running time of the algorithm. Furthermore, a new communicator synchronizes all processes as communicator creation routines are blocking collective operations. We present RBC, a communication library based on MPI, that creates range-based communicators in constant time without communication. These RBC communicators support (non)blocking point-to-point communication as well as (non)blocking collective operations. Our experiments show that the library reduces the time to create a new communicator by a factor of more than 400 whereas the running time of collective operations remains about the same. We propose Janus Quicksort, a distributed sorting algorithm that avoids any load imbalances. We improved the performance of this algorithm by a factor of 15 for moderate inputs by using RBC communicators. Finally, we discuss different approaches to bring nonblocking (local) communicator creation of lightweight (range-based) communicators into MPI

Late Medieval Mediterranean Apocalypticism: Joachimist Ideas in Ramon Llull’s Crusade Treatises

The thirteenth century witnessed dramatic changes that transformed the medieval world and remain important today. The violent changes caused by the War of the Sicilian Vespers and Spiritual Franciscan movement popularized the apocalyptic ideas of the twelfth-century Italian abbot, Joachim of Fiore. The abbot\u27s historical paradigms of biblical history influenced many southern Europeans, including the medieval mystic, missionary, and philosopher Ramon Llull (c. 1232-1316). Llull dedicated his life to converting the world to Catholic Christianity using a variety of means, including evangelical missions, Neoplatonic philosophy, and crusades. Llull\u27s crusade treatises, the Tractatus de modo convertendi infideles (1292), Liber de fine (1305), and Liber de acquisitione Terrae Sanctae (1309), offer the best summaries of his entire conversion program as well as many examples of his apocalyptic ideas. Many scholars have criticized Llull as a utopian, a man disconnected from his time. Yet examining how Llull carefully incorporates Joachimist ideas into his crusade treatises shows the supposed utopian actually paid great attention to the world around him. In short, Llull used the apocalypticism which permeated the late medieval Mediterranean world as a rhetorical device to promote his conversion program and bring the world into a bonum statum (good age)

Minimal Surfaces, Hyperbolic 3-manifolds, and Related Deformation Spaces.

Given a closed, oriented, smooth surface $Sigma$ of negative Euler characteristic, the relationships between three deformation spaces of geometric structures are compared: the space of minimal hyperbolic germs $mathcal{H},$ the space of representations $mathcal{R}(pi_1(Sigma),text{PSL}_2(mathbb{C})),$ and the space $mathcal{M}(X)$ of solutions to the self-duality equations on a rank-2 complex vector bundle over a Riemann surface $Xsimeq Sigma.$ Inside both $mathcal{H}$ and $mathcal{R}(pi_1(Sigma),text{PSL}_2(mathbb{C}))$ lies the space $mathcal{AF}$ of almost-Fuchsian manifolds comprised of quasi-Fuchsian 3-manifolds $Msimeq Sigma times mathbb{R}$ which contain an immersed closed minimal surface whose principal curvatures lie in the interval $(-1,1).$ The structure of these manifolds is explored through a study of the domain of discontinuity of the associated almost-Fuchsian holonomy group. It is proved that there are no doubly degenerate geometric limits of almost-Fuchsian manifolds. Next, the space $mathcal{H}$ is studied through an analysis of a smooth real valued function which records the topological entropy of the geodesic flow arising from a minimal hyperbolic germ. Estimates on this function are obtained which culminate in a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. As a corollary we obtain a new proof of Bowen's theorem on quasi-circles: a quasi-Fuchsian group is Fuchsian if and only if the Hausdorff dimension of its limit set is equal to $1.$ Lastly, we recall a construction of Donaldson which shows how each minimal hyperbolic germ gives rise to a solution of the self-duality equations. In this context, we compare various deformations of a Fuchsian representation $pi_1(Sigma)rightarrow text{PSL}_2(mathbb{R}),$ finally obtaining an explicit formula for a deformation arising from minimal surfaces in terms of Fuchsian and bending deformations. Interestingly, the hyperk"{a}hler structure on the moduli space $mathcal{M}$ of solutions to the self-duality equations makes an appearance here