Abstract. In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem. §0 Introduction. The structure of a manifold with positive or nonnegative scalar curvature has been studied extensively. There are many beautiful results for compact manifolds without boundary, see [L,. For example, in [L], Lichnerowicz found that some compact manifolds admit no Riemannian metric with positive scalar curvature. In [SY1-2] Schoen and Yau proved that every torus T n with n ≤ 7 admits no metric with positive scalar curvature, and admits no non-flat metric with nonnegative scalar curvature. This is also proved later by Gromov and Lawson [GW3] for n > 7.For complete noncompact manifolds, the most famous result is the Positive Mass Theorem (PMT), first proved by Schoen and Yau [SY3-4] and later by Witten [Wi] using spinors, see also [PT,B1]. One of their results is as follow: Suppose (M, g) is an asymptotically flat manifold such that g behaves like Euclidean at infinity near each end, and suppose its scalar curvature is nonnegative, then (M, g) is actually flat if the ADM mass of one of the ends is zero.It is natural to ask what we can say about manifolds with boundary and with nonnegative scalar curvature. In a recent work of Yau [Y], it was proved that if Ω is a noncompact complete three manifold with boundary and with scalar curvature not less than −3/2c 2 . Suppose one of the component of ∂Ω has nonpositive Euler number and mean curvature

We have demonstrated that ICCC is a consistent method to assess observer variation when a continuous scoring system is used, compared with kappa statistics, which depends on a categorical system. Given the importance of accurate assessment of protein expression in diagnostic and experimental medicine, we suggest raising thresholds for observer variation: ICCC of 0.7 should be regarded as the minimum acceptable standard, ICCC of 0.8 as good and ICCC of > or = 0.9 as excellent.

Abstract. We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kähler manifolds with nonnegative bisectional or sectional curvature.

Abstract. Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flat manifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.

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The main cause of prostate cancer-related mortality is the development of hormone-refractory disease. Circulating serum levels of IL-6 are raised in hormone-refractory prostate cancer patients and evidence from cell line studies suggests that the IL-6R/JAK/STAT3 pathway may be involved in development of this disease. In the current study we investigate if expression levels of these family members are implicated in the development of hormone-refractory prostate cancer. Immunohistochemistry using IL-6R, JAK1, STAT3, pSTAT3 Tyr705 and pSTAT3 Ser727 antibodies was performed on 50 matched hormone-sensitive and hormone-refractory tumours pairs. An increase in expression of cytoplasmic IL-6 receptor, with the development of hormone-refractory prostate cancer was associated with reduced time to relapse (P ¼ 0.0074) while an increase in expression of cytoplasmic pSTAT3 Tyr705 was associated with reduced patient survival (P ¼ 0.0003). In addition, those patients with high expression of cytoplasmic pSTAT3 Tyr705 in their hormone-refractory tumours had significantly shorter time to death from biochemical relapse and overall survival in comparison to those patients with low expression of cytoplasmic pSTAT3 Tyr705 (P ¼ 0.002 and P ¼ 0.0027, respectively). Activation of STAT3, via phosphorylation is associated with reduced patient survival, suggesting that activation of the IL-6R/JAK/STAT3 pathway is involved with development of hormone-refractory prostate cancer.

These preliminary data support the reliability and validity of the Chinese version of the CARE Measure in primary care in Hong Kong.