China
Africa
Explore chapters and articles related to this topic
Hyperbolic Big Data Analytics for Dynamic Network Management and Optimization
Published in Yulei Wu, Fei Hu, Geyong Min, Albert Y. Zomaya, Big Data and Computational Intelligence in Networking, 2017
Vasileios Karyotis, Eleni Stai
The choice of hyperbolic space as the target embedding space versus other available ones, e.g., Euclidean spaces, is dictated by the properties of the hyperbolic space mentioned above and/or the potential hidden structure exhibited by the data analyzed. Regarding the first, the hyperbolic space provides nice scaling properties that address the requirements posed by big data. With respect to the second, several types of data following a hierarchical classification, exhibit tree-like dependencies (e.g., data originating in scale-free like networks). Such data graphs can be efficiently embedded in the hyperbolic space, since the hyperbolic geometry is the geometry of the trees [11].
Eigenvalue estimates for the drifting Laplacian and the p-Laplacian on submanifolds of warped products
Published in Applicable Analysis, 2021
Wei Lu, Jing Mao, Chuan-Xi Wu, Ling-Zhong Zeng
Clearly, . Schoen and Yau [3, p. 106] suggested that it is an important question to find conditions which will imply . Speaking in other words, manifolds with might have some special geometric properties. There are many interesting results supporting this. For instance, Mckean [4] showed that for an n-dimensional complete noncompact, simply connected Riemannian manifold M with sectional curvature , , and moreover, with the n-dimensional hyperbolic space of sectional curvature . Grigor'yan [5] showed that if , then M is nonparabolic.
Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms
Published in Applicable Analysis, 2022
Akram Ali, Ali H. Alkhaldi, Pişcoran Laurian-Ioan, Rifaqat Ali
Using the Wentzel–Laplacian operator on a compact submanifold with boundary in the Euclidean space, Du et al. [9] provided an estimation of the first nonzero eigenvalue. After that, using a decomposition of the Hessian on Kaehler manifolds with positive Ricci curvature, Blacker and Seto [10] proved a Lichnerowicz lower bound theorem for the first nonzero eigenvalue of the p-Laplacian on Kaehler manifold, for Dirichlet and Neumann boundary conditions. Furthermore, a submanifold can be immersed in an m-dimensional simply connected space form of constant sectional curvature c which included the Euclidean space , the unit sphere and the hyperbolic space with c = 0, 1 and respectively. Such estimation for the first nonzero eigenvalue of the Laplacian, has been proved in [1,11]. It well known, all the results obtained until now for several classes of Riemannian submanifolds in some different ambient spaces, show that, both the first nonzero eigenvalues (Dirichlet or Neumann) satisfies similar inequalities and consequently they have identical upper bounds [11,12]. Several great successes in Riemannian geometry were obtained for the theory of p-Laplacians on Riemannian submanifolds in different ambient manifolds (see [4,5,12–20]) and also through the work [3]. Inspired by this notion, our method is based on the derivation of the first eigenvalue for the p-Laplacian on C-totally real submanifold of Sasakian space form. From this point of view, using the Gauss equation and [1,14], we have been motivated the study of the first nonzero eigenvalue of the p-Laplacian on submanifold in different space forms. We proved a sharp upper bound theorem for the first eigenvalue for the p-Laplacian operator on C-totally real submanifold of Sasakian space form (cf. Theorem 3.2).