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Multi-Dimensional Photonic Processing by Discrete-Domain Approach
Published in Le Nguyen Binh, Photonic Signal Processing, 2019
Discrete space form of a 2-D signal with predefined limits can be expressed by () a[n1,n2]=∑k1=0n1∑k2=0n2a[k1,k2]δ[n1−k1,n2−k2]
Fundamental Materials and Tools
Published in Wing C. Kwong, Guu-Chang Yang, Optical Coding Theory with Prime, 2018
For example, the two unit vectors (0,1) and (1,0), which resemble the two Cartesian coordinate axes in two-dimensional Euclidean space, form a basis of a two-dimensional vector space V2 over GF(2) with four distinct vectors: (0,0)=0(0,1)+0(1,0)(0,1)=1(0,1)+0(1,0)(1,0)=0(0,1)+1(1,0)(1,1)=1(0,1)+1(1,0)
Spectral geometry
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
Theorem 4.7.4:Let G be a spherical space form group. Let m be odd and let[(M,s,σ)]ϵM˜Jm(BG).
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
By (7), on the set Ω given in Definition 2.3 the Riemannian metric of can be expressed by with the round metric on the unit sphere . Spherically symmetric manifolds were named as generalized space forms by Katz and Kondo [13], and a standard model for such manifolds is given by the quotient manifold of the warped product equipped with the metric (10), where f satisfies the conditions in Definition 2.3, and all pairs are identified with a single point p. A space form with constant curvature K is also a spherically symmetric manifold, and in this special case we have By proposition 42 and corollary 43 of chapter 7 in [17] or subsection 3.2.3 of chapter 3 in [18], we know that the radial sectional curvature of the spherically symmetric manifold with the base point is given by Thus, Definition 2.1 is satisfied with equality in (9) and . Besides, on , the Hessian of the Riemannian distance to the base point satisfies
Evolution of aligned states within nonlinear dynamos
Published in Geophysical & Astrophysical Fluid Dynamics, 2019
I will now use the Fourier space form of the MHD equations to show why the Group A runs have a SSS. The equations are Incompressibility is used to define the projection operator which in index notation is given by . The forcing is written in terms of a real function . The general forms of and are given by and respectively with all functions being real. Substitution of and into (7) yields evolution equations for , , and . Substitution of the initial conditions of the Group A run into these equations then yields Equations (8c,d) then show that if is initially real and is initially imaginary (as is the case for Group A runs) then they will remain so.
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).