China
Africa
Explore chapters and articles related to this topic
Pertinent Properties of Euclidean Space
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The most important topological spaces are manifolds. A topological space X is called an n-dimensional manifold if X is a Hausdorff space and each x∈X has a neighborhood N(x) that is homeomorphic to Euclidean n-space ℝn. Bernhard Riemann, a former student of Carl Gauss, was the first to coin the term ”manifold” in his extensive studies of surfaces in higher dimensions. In modern physics, our universe is a manifold. Einstein's general relative theory relies on a four-dimensional spacetime manifold. The string theory model of our universe is based on a ten-dimensional manifold R×M, where R denotes the four-dimensional spacetime manifold and M denotes the six-dimensional spacial Calabi-Yau manifold. Einstein's spacetime manifold is a generalized Riemannian manifold that uses the Ricci curvature tensor [32]. We do not expect the reader to be familiar with manifold theory and associated geometries. One reason for mentioning the subject of manifolds is that even Einstein's manifold, which tells us that that space and the gravitational field are one and the same, needs the locality of ℝ4 at each point of the manifold.
Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds
Published in Applicable Analysis, 2022
Jingjing Fan, Xiaolong Qin, Bing Tan
Let be a finite dimensional differentiable manifold. The set of all tangents at is called a tangent space of at , which forms a vector space of the same dimension as and is denoted by . The tangent bundle of is denoted by , which is naturally a manifold. We denote by the scalar product on with the associated norm , where the subscript x is sometimes omitted. A differentiable manifold with a Riemannian metric is called a Riemannian manifold. Let be a piecewise differentiable curve joining to in , we can define the length of . The minimal length of all such curves joining x to y is called the Riemannian distance and it is denoted by .
H2 model reduction for negative imaginary systems
Published in International Journal of Control, 2020
Let be the tangent space of Stiefel manifold at . The matrix representation of is given by Edelman, Arias, and Smith (1998) becomes a Riemannian manifold by endowing with the following inner product: which is called the Riemannian metric. The Euclidean space is also a Riemannian manifold endowed with the standard inner product.
Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D semi-Riemannian manifold
Published in Journal of Modern Optics, 2019
In the case of the semi-Riemannian manifold vector fields and 2-forms can be described via the volume form and the Hodge star operator ★ of the manifold. Hence, divergence-free vector fields and magnetic fields are in correspondence. Therefore, for any vector field on the semi-Riemannian manifold, Lorentz equation can be given by the following formula where (19).