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Further concepts in one dimension
Published in C. Pozrikidis, The Fractional Laplacian, 2018
and i is the imaginary unit, i2 = −1. Manipulating the integral defining the function ψ(υ, k), we obtain ψ(υ,k)=(e−ikυ−eikυ)∫−∞∞f(x)eikxdx.
General Principles of Dynamics of Systems with Variable Mass
Published in L. Cveticanin, Dynamics of Machines with Variable Mass, 2022
where i=−1 is the imaginary unit. If a complex generalized force Zs exists () Zs=QX+iQY,
Stabilization of piecewise-homogeneous Markovian switching CVNNs with mode-dependent delays and incomplete transition rates
Published in Systems Science & Control Engineering, 2020
Qiang Li, Jinling Liang, Weiqiang Gong
The notation used throughout this paper is fairly standard. , , and denote, respectively, the set of n-dimensional complex vectors, n-dimensional real vectors, complex and real matrices. I is the identity matrix of appropriate dimensions. The superscript ‘T ’ denotes matrix transposition, and ‘⊗’ stands for the Kronecker product. Let be a complete probability space with a filtration satisfying the usual condition (i.e. the filtration contains all -null sets and is monotonically right continuous). The imaginary unit is denoted as , i.e. . and are the largest and smallest eigenvalues of matrix A. In a symmetric matrix, ‘’ denotes the entries induced by symmetry. and refer to and , respectively. stands for the mathematical expectation of ‘·’.
On coupled systems of nonlinear Schrödinger equations with critical exponential growth
Published in Applicable Analysis, 2018
J. M. do Ó, J. C. de Albuquerque
where i denotes the imaginary unit. For system (1), a solution of the form is called standing wave. Assuming that and , for , it is well known that is a solution of (1) if and only if (u, v) solves system (S). Such systems (1) are related to the existence of solitary wave solutions for nonlinear Schrödinger equations and Klein–Gordon equations (see [3–5]).