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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 exposition

In this paper, we introduce a notion, called generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounded below, which gives a unified proof of smooth fibration theorems in many previous works (including the ones proved by Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration theorem is the transformation technique for almost splitting maps, which originates from Cheeger-Naber (\cite{CN}) and Cheeger-Jiang-Naber (\cite{CJN21}). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in \cite{CJN21}) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper., Comment: to appear in Adv. Math

In physical-layer security, one of the most fundamental issues is the secrecy capacity. The objective of this paper is to determine the secrecy capacity for an indoor visible light communication system consisting of a transmitter, a legitimate receiver and an eavesdropping receiver. In such a system, both signal-independent and signal-dependent Gaussian noises are considered. Under non-negativity and average optical intensity constraints, lower and upper secrecy capacity bounds are first derived by the variational method, the dual expression of the secrecy capacity, and the concept of "the optimal input distribution that escapes to infinity". Numerical results show that the secrecy capacity upper and lower bounds are tight. By an asymptotic analysis at large optical intensity, there is a small performance gap between the asymptotic upper and lower bounds. Then, by adding a peak optical intensity constraint, we further analyze the exact and asymptotic secrecy capacity bounds. Finally, the tightness of the derived bounds is verified by numerical results., Comment: Submitted to IEEE Transactions on Information Theory

We prove that on an essentially non-branching $\mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants., Comment: Submitted. 18 pages

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $\alpha=\lim_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain $\lim_{k\rightarrow\infty}k^{1-n}h_{k}=\frac{2\alpha}{(n-1)!\omega_{n}}$. These results are sharp, as they recover the corresponding well-known properties of $h_{k}(\mathbb{R}^{n})$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth., Comment: Comments are welcome!

Channel capacity bounds are derived for a point-to-point indoor visible light communications (VLC) system with signal-dependent Gaussian noise. Considering both illumination and communication, the non-negative input of VLC is constrained by peak and average optical intensity constraints. Two scenarios are taken into account: one scenario has both average and peak optical intensity constraints, and the other scenario has only average optical intensity constraint. For both two scenarios, we derive closed-from expressions of capacity lower and upper bounds. Specifically, the capacity lower bound is derived by using the variational method and the property that the output entropy is invariably larger than the input entropy. The capacity upper bound is obtained by utilizing the dual expression of capacity and the principle of "capacity-achieving source distributions that escape to infinity". Moreover, the asymptotic analysis shows that the asymptotic performance gap between the capacity lower and upper bounds approaches zero. Finally, all derived capacity bounds are confirmed using numerical results., Comment: 32 pages, 9 figures, accepted by IEEE Transactions on Wireless Communications, 2020

When reporting the results of clinical studies, some researchers may choose the five-number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation, particularly for skewed data. For these studies, when included in a meta-analysis, it is often desired to convert the five-number summary back to the sample mean and standard deviation. For this purpose, several methods have been proposed in the recent literature and they are increasingly used nowadays. In this paper, we propose to further advance the literature by developing a smoothly weighted estimator for the sample standard deviation that fully utilizes the sample size information. For ease of implementation, we also derive an approximation formula for the optimal weight, as well as a shortcut formula for the sample standard deviation. Numerical results show that our new estimator provides a more accurate estimate for normal data and also performs favorably for non-normal data. Together with the optimal sample mean estimator in Luo et al., our new methods have dramatically improved the existing methods for data transformation, and they are capable to serve as "rules of thumb" in meta-analysis for studies reported with the five-number summary. Finally for practical use, an Excel spreadsheet and an online calculator are also provided for implementing our optimal estimators., Comment: 30 pages and 4 figures and 2 tables. arXiv admin note: substantial text overlap with arXiv:1801.01267

In some clinical studies, researchers may report the five number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation. To conduct meta-analysis for pooling studies, one needs to first estimate the sample mean and standard deviation from the five number summary. A number of studies have been proposed in the recent literature to solve this problem. However, none of the existing estimators for the standard deviation is satisfactory for practical use. After a brief review of the existing literature, we point out that Wan et al.'s method (BMC Med Res Methodol 14:135, 2014) has a serious limitation in estimating the standard deviation from the five number summary. To improve it, we propose a smoothly weighted estimator by incorporating the sample size information and derive the optimal weight for the new estimator. For ease of implementation, we also provide an approximation formula of the optimal weight and a shortcut formula for estimating the standard deviation from the five number summary. The performance of the proposed estimator is evaluated through two simulation studies. In comparison with Wan et al.'s estimator, our new estimator provides a more accurate estimate for normal data and performs favorably for non-normal data. In real data analysis, our new method is also able to provide a more accurate estimate of the true sample standard deviation than the existing method. In this paper, we propose an optimal estimator of the standard deviation from the five number summary. Together with the optimal mean estimator in Luo et al. (Stat Methods Med Res, in press, 2017), our new methods have improved the existing literature and will make a solid contribution to meta-analysis and evidence-based medicine., Comment: 18 pages, 5 figures, 2 tables

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Secondly, we apply this almost rigidity theorem to study noncompact RCD(0, N) spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence of harmonic functions with polynomial growth on such RCD(0,N) spaces., Comment: 40 pages

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that $h_{d}(M)$ is finite. Later on, there are many researches which give better estimates of $h_{d}(M)$. We study the behavior of $h_{d}(M)$ when $d$ is large in this paper. More precisely, suppose that $(M^{n},g)$ has maximal volume growth and has a unique tangent cone at infinity, then when $d$ is sufficiently large, we obtain some estimates of $h_{d}(M)$ in terms of the growth order $d$, the dimension $n$ and the the asymptotic volume ratio $\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}$. When $\alpha=\omega_{n}$, i.e., $(M^{n},g)$ is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of $h_{d}(\mathbb{R}^{n})$., Comment: Comments are welcome

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth condition. Another main result in this paper is a splitting theorem at the noncompact end for a RCD(0, N) space with strongly minimal volume growth. These results generalize some theorems on noncompact manifolds with nonnegative Ricci curvature to non-smooth settings., Comment: In this version, we add a new result on the growth of the diameter of level sets of Busemann functions on noncompact RCD(0, N) spaces. In addition, we drop the non-branching assumption on the splitting type theorem. There are some adjustment on the structure of the paper. All comments are welcome!

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds., Comment: 25 pages

Wang and Yau [10] introduced a quasi-local mass, which is a hyperbolic background generalization of Liu-Yau's expression [7] [8], and proved its positivity. In this note, we prove that the positivity of this quasi-local mass is still valid under weaker assumptions on the boundary hypersurface in general dimensions. The method we used is similar to that used by Eichmair, Miao and Wang in [4]., Comment: 8 pages; some typos are corrected; all comments are welcome!

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 Geometry

Recent studies demonstrate that chlorophyll a (chl a) concentrations in the surface ocean can be significantly enhanced due to typhoons. The present study investigated chl a concentrations in the middle of the South China Sea (SCS) from 1997-2007. Only the Category1 (minimal) Typhoon Hagibis (2007) had a notable effect on the chl a concentrations. Typhoon Hagibis had a strong upwelling potential due to its location near the equator, and the forcing time of the typhoon (>82 h) was much longer than the geostrophic adjustment time (~63 h). The higher upwelling velocity and the longer forcing time increased the depth of the mixed-layer, which consequently induced a strong phytoplankton bloom that accounted for about 30% of the total annual chl a concentration in the middle of the SCS. The implication is that the forcing time of a typhoon should be long enough to establish a strong upwelling and consequently for the induction of significant upper ocean responses., Comment: Typhoon-Ocean Environment interactions