Abstract:
Dossier/Issue: 2 - Vraisemblance et fictions contemporaines. Une nouvelle adhésion pour les héritiers du soup on - Using the examples of "Nikoski" (Nicolas Dickner, 2005) and "La kermesse" (Daniel Poliquin, 2006), this article examines the construction of a parallel verisimilitude generated by the use of a multiplicity of coincidences. This particular verisimilitude manages to impose itself in spite of the presence of a number of unlikely events and may be considered as a type of generic verisimilitude (typical of a particular genre such as the detective novel or science fiction) taking effect in a particular fictional world. — à partir des exemples de "Nikolski" (Nicolas Dickner, 2005) et de "La kermesse" (Daniel Poliquin, 2006), l’article présente la construction d’une vraisemblance parallèle générée par l’utilisation d’une multitude de co ncidences. Cette vraisemblance particulière réussit à s’imposer malgré un certain degré d’invraisemblance et pourrait s’apparenter à la vraisemblance générique (propre à un genre particulier comme le roman policier ou de science-fiction par exemple), mais s’appliquant à un univers romanesque particulier.

Abstract:
We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of order r if and only if its "matrix" $(K(A))_{m,n} := < A\phi_n,\phi_m>_{L^2(\mathbb{R})}$ is rapidly decreasing away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where applying the discrete difference operator along the diagonal decreases the order by one. Additionally, we use this result to prove an analogue of Beal's theorem for isotropic pseudodifferential operators.

Abstract:
This is a condensed form of the author's essay, which can be found at [arXiv:1105.2883]. We prove that the entropic measure constructed by von Renesse-Sturm over Wasserstein space on the unit interval (probability measures on the unit interval equipped with the 2-Wasserstein metric) does not admit generalized Ricci lower bounds in the sense of Lott-Villani-Sturm. We discuss why this is surprising, considering various heuristic arguments.

Abstract:
We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.

Abstract:
We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality. Our results require that the scalar curvature of the metric satisfies $R_{g}\geq -6$, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without $R_{g}\geq -6$ so that large centered coordinate spheres are not isoperimetric. The necessity of scalar curvature bounds is in contrast with the analogous uniqueness result proven by Bray for compact perturbations of Schwarzschild, where no such scalar curvature assumption is required. This demonstrates that from the point of view of the isoperimetric problem, mass behaves quite differently in the asymptotically hyperbolic setting compared to the asymptotically flat setting. In particular, in the asymptotically hyperbolic setting, there is an additional quantity, the "renormalized volume," which has a strong effect on the large-scale geometry of volume.

Abstract:
In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In particular, we give a detailed proof that the Lott-Villani-Sturm notion of generalized Ricci bounds agree with the classical notion on smooth manifolds. We also give the proof that generalized Ricci bounds are preserved under Gromov-Hausdorff convergence. In particular, we examine in detail the space of probability measures over the interval, $P(X)$ equipped with the Wasserstein metric $d^W$. We show that this metric space is isometric to a totally convex subset of a Hilbert space, $L^2[0,1]$, which allows for concrete calculations, contrary to the usual state of affairs in the theory of optimal transport. We prove explicitly that $(P(X),d^W)$ has vanishing Alexandrov curvature, and give an easy to work with expression for the entropy functional on this space. In addition, we examine finite dimensional Gromov-Hausdorff approximations to this space, and use these to construct a measure on the limit space, the entropic measure first considered by Von Renesse and Sturm. We examine properties of the measure, in particular explaining why one would expect it to have generalized Ricci lower bounds. We then show that this is in fact not true. We also discuss the possibility and consequences of finding a different measure which does admit generalized Ricci lower bounds.

Abstract:
The purpose of this study is to determine if the type of weekly evaluation method used in an on-line course contributes to a difference in the learning outcomes for students. Two methods for the ongoing evaluation of student learning were analyzed for differences in learning outcomes as demonstrated by mid-term and final exam test scores. Using an experimental design, students that enrolled in either a Disease Control course or an Epidemiology course were randomized into one of two sections in each course. Holding the course parameters the same except for the weekly evaluation type (homework or quiz), bivariate analysis using independent t tests supported that sections were similar in both courses with respect to test scores. A statistically significant difference did occur between final exam scores in the Disease Control course with the higher scores occurring in the quiz section. End of course student satisfaction surveys were similar for both types of evaluation methods and for both courses. Of the students that responded to the surveys, the majority felt their overall learning experience was either good or very good, regardless of whether they completed weekly interactive homework assignments or automated quizzes. This study supports the idea that learning outcomes and student satisfaction scores with a weekly automated quiz are equivalent or improved over the more interactive weekly homework assignments in select courses.

Abstract:
We show that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat $\mathbb{R}^3$ if it admits an unbounded area-minimizing surface. This answers a question of R. Schoen.