A Simple Method for the Optimal Transportation

In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.Comment: 15 pages, to appear in Communications in Analysis and Geometr

Optimal asymptotic volume ratio for noncompact 3-manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has $k$ ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio $\limsup_{r\rightarrow\infty}\frac{\mathrm{Vol}(B(p, r))}{r}\leq4k\pi$. In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound.Comment: 21 pages. Some improvements on the main theorem: In (2) of Theorem 1.6, the assumption on the positive function f is weaken to $\int_{0}^{\infty}rf(r)dr<\infty$. Other minor changes in the expositio

Correlation of caveolin-1 expression with microlymphatic vessel density in colorectal adenocarcinoma tissues and its correlation with prognosis

AbstractObjectiveTo study the expression of caveolin-1 in colorectal adenocarcinoma tissues and its correlation with microlymphatic vessel density (LMVD), and to investigate the clinical pathological prognostic significance of caveolin-1 and LMVD in patients with colorectal cancer.MethodsThe expression of caveolin-1 and LMVD in 45 specimens of normal colorectal tissues, and 90 specimens of colorectal adenocarcinoma tissues were detected by immunohistochemistry technique. The correlation between their expression and the clinicopathologic features was analyzed. Multivariable Cox regression was used to analyze the association between the laboratory indices and overall survival time.ResultsThe positive rates of caveolin-1 in colorectal adenocarcinoma tissues were significantly higher than those in normal colorectal tissues (P < 0.01). LMVD in colorectal adenocarcinoma tissues were significantly higher than those in normal colorectal tissues (P < 0.01). Mean LMVD in group with caveolin-1 positive was significantly higher than in that with caveolin-1 negative. The median survival time was 26.7 months. Cox regression analysis showed that the caveolin-1 expression, invation depth, lymph node metastasis, TNM stage, liver metastasis and LMVD were independent risk factors of overall survival time of patients with colorectal carcinoma.ConclusionsCaveolin-1 may contribute to the lymphangiogenesis in the tumor. During the occurrence and development of colorectal adenocarcinoma, there is a close relationship between the expression of caveolin-1 and lymphatic microvessel of tumor. Caveolin-1 expression and microlymphatic vessel density are significant prognostic value of colorectal carcinoma