We analyze the torus fixed loci of Mixed Spin P fields moduli, and deduce its localization formulas with explicit factors. An algorithm toward evaluating quintic's Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants is derived.

Let <i>M</i> be a compact connected (topological) manifold of finite- or infinite-dimension <i>n</i>. Let 0 <i>r</i> 1 be arbitrary but fixed. We construct in this paper a space-filling curve <i>f</i> from [0,1] onto <i>M</i>, under which <i>M</i> is the image of a compact set <i>A</i> of Hausdorff dimension <i>r</i>. Moreover, the restriction of f to <i>A</i> is one-to-one over the image of a dense subset provided that 0 <i>r</i> log|2^<i>n</i>/log(2^<i>n</i> + 2). The proof is based on the special case where <i>M</i> is the Hilbert cube [0,1].

We consider surfaces of geometric genus 3 with the property that their transcendental cohomology splits into 3 pieces, each piece coming from a K3 surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on an n-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.

Statistical estimation and inference for marginal hazard models with varying coefficients for multivariate failure time data are important subjects in survival analysis. A local pseudo-partial likelihood procedure is proposed for estimating the unknown coefficient functions. A weighted average estimator is also proposed in an attempt to improve the efficiency of the estimator. The consistency and asymptotic normality of the proposed estimators are established and standard error formulas for the estimated coefficients are derived and empirically tested. To reduce the computational burden of the maximum local pseudo-partial likelihood estimator, a simple and useful one-step estimator is proposed. Statistical properties of the one-step estimator are established and simulation studies are conducted to compare the performance of the one-step estimator to that of the maximum local pseudo-partial likelihood estimator. The