In this article, we will give a brief survey on some recent development concerning the understanding of the structure at infinity of a complete manifold whose spectrum has a positive lower bound. Throughout this paper, we denote M to be a complete n-dimensional manifold without boundary endowed with the metric ds. We assume that the Ricci curvature of M is bounded from below by some constant. R...

PURPOSE To describe longitudinal changes in corneal curvature (CC) and axial length (AL) over 14 years, and to explore the relationship between AL and CC, and the axial length/corneal radius (AL/CR) ratio. METHODS In total 469, 6 to <12-year-old, children were enrolled in COMET. Measurements of refractive error, CC (D), CR (mm), and ocular component dimensions including AL were gathered annua...

The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scale-space, phase, orientation, energy and isophote curvature in one unified algebraic framework. The conformal monoge...

Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope to answer include “how did the universe begin?”; “how will the universe end?”; “is space finite or infinite?”. After a century of remarkable progress, cosmolo...

Maps shading terrain as if illuminated by a point light source (i.e. hill-shading) are commonly used in cartographic displays. These hill-shaded maps are especially important for representing mountainous areas, due to the rapid variations in the orientation of terrain elements. Although these hill-shaded maps reveal the coarser landforms and the finer texture of the terrain, they can be enhance...

We compare curvature combs and curvature plots. © 2016 Elsevier Ltd. All rights reserved.

1.1. Connections and derivations. Let S be a scheme, and X smooth and finite type over S throughout. We have the rank-n vector bundle ΩX/S of 1-forms on X over S. We also fix some notation in the case that S is of characteristic p: in this case, we denote by FT the absolute Frobenius map for any scheme T , by X = X×S S the p-twist of X over S, where the map used in the fiber product is FS : S →...

here, a finsler manifold $(m,f)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. certain subspaces of the tangent spaces of $m$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. it is shown that if the dimension of foliation is constant, then the distribution is involutive a...

We prove that the Atiyah-Singer Dirac operator / Dg in L 2 depends Riesz continuously on L∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g → / Dg(1 + / D 2 g) − 12 depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commu...

in this paper, the lorentzian version of beltrami-euler formula is investigated in 1n . initially,the first fundamental form and the metric coefficients of generalized timelike ruled surface are calculated and by the help of the christoffel symbols, riemann-christoffel curvatures are obtained. thus, the curvatures of spacelike and timelike tangential sections of generalized timelike ruled surf...