Let M be a compact Criemannian manifold of nonpositive curvature1 and with fundamental group . It is well known [8, p. 102] that M is a K (, 1) and thus completely determined up to homotopy type by . In light of this fact it is natural to ask to what extent the riemannian structure of M is determined by the structure of , and the intent of this paper is to demonstrate that rather strong implications of this sort exist. In the case that M has strictly negative curvature, the group is known to be highly noncommutative. Every abelian, in fact, every solvable, subgroup of is cyclic [3]. It is therefore a plausible conjecture that in the nonpositive curvature case, will possess large amounts of commutativity only under special geometric circumstances. We shall show that this is true, that indeed those properties of which involve commutativity have a dramatic reflection in the riemannian structure of M.

We study five-dimensional superconformal field theories and their holographic dual, matter-coupled Romans supergravity. On the one hand, some recently derived formulae allow us to extract the central charges from deformations of the supersymmetric five-sphere partition function, whose large N expansion can be computed using matrix model techniques. On the other hand, the conformal and flavor central charges can be extracted from the six-dimensional supergravity action, by carefully analyzing its embedding into type I’ string theory. The results match on the two sides of the holographic duality. Our results also provide analytic evidence for the symmetry enhancement in five-dimensional superconformal field theories.

We examine the effect of errors in covariates in nonparametric function estimation. These functions include densities, regressions and conditional quantiles. To estimate these functions, we use the idea of deconvoluting kernels in conjunction with the ordinary kernel methods. We also discuss a new class of function estimators based on local polynomials.

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact K ahler manifold with positive orthogonal bisectional curvature must be biholomorphic to \mathbb {P}^ n .