In this paper, we study the behavior of the Laplacian on a sequence of manifolds {Mf} with a lower bound in Ricci curvature that converges to a metric-measure space M^. We prove that the heat kernels and Green's functions on Mf will converge to some integral kernels on M^ which can be interpreted, in different cases, as the heat kernel and Green's function on M^. We also study the Laplacian on noncollapsed metric cones; these provide a unified treatment of the asymptotic behavior of heat kernels and Green's functions on noncompact manifolds with nonnegative Ricci curvature and EucUdean volume growth. In particular, we get a unified proof of the asymptotic formulae of Colding-Minicozzi, Li and LiTam-Wang. Introduction.Assume M n is an n dimensional Riemannian manifold with a lower bound in Ricci curvature,where A > 0. By the Bishop-Gromov inequality, we have a uniform volume doubling condition,here we can take K = n if A = 0; if A > 0, we require that R is bounded from above, say, R < D for some D > 0.
Abstract. We prove the existence of nonconstant harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature, Euclidean volume growth and unique tangent cone at infinity.
This paper introduces an analysis of lightning surge propagation on a conductor without any returning current path. The conductor can be either a free-space or grounded conductor as long as the reflected surge from the ground has not arrived. Unlike a TEM transmission line, this conductor is characterized with time/position-variant surge impedance as surge current attenuates during its propagation. In this paper a simplified formula was derived. Using the unique parameterattenuation coefficient of current, an iterative method was developed to evaluate actual propagation characteristics. This method was verified numerically, and is much more efficient in calculation and easier in implementation. It is found that the surge impedance is affected by the waveform of an impulse current source, but is independent of the slope of a ramp current source. It increases quickly if the source current has a short rising time or failing time. The simplified formula generates over-estimated results, but the difference decreases with increasing distance to the source. The proposed method can be used to address surge voltage on the tower upon the arrival of a reflected surge from the ground.
We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities. YU DINGFor previous works on neck-pinching, see [1], [2], and the book [5]. In the book [4], there is a detailed treatment of nondegenerate neck-pinching in chapter 2, and a discussion of degenerate neck pinching in page 62-66.We start by reviewing some of Perelman's results in [14], [15]; for more details, see [3], [12] and [13]. In section 2 we use an estimate on Perelman's l functional to rule out noncompact ancient solutions with positive curvature that develops a type I singularity. Therefore a rescale limit of a degenerate singularity is either an eternal solution or an ancient solution that develops a type II singularity. In both cases, we need to take a further rescale limit in forward time; we treat certain issues related to this in Section 3. Then we use a theorem of Hamilton [9] to conclude that the final rescale limit is a steady soliton. Our arguments are similar to Perelman's compactness/convergence methods that were used extensively in his papers [14], [15]; for reader's convenience we will give a detailed account.Recently it comes to our attention that Gu and Zhu [7] proved the existence of type II singularity; they used Perelman's l functional argument to detect type II singularity in the radial symmetric case. See also a very recent paper [6].
Oxidation states are well-established in chemical science teaching and research. We data-mine more than 168,000 crystallographic reports to find an optimal allocation of oxidation states to each element. In doing so we uncover discrepancies between text-book chemistry and reported charge states observed in materials. We go on to show how the oxidation states we recommend can significantly facilitate materials discovery and heuristic design of novel inorganic compounds.
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