scholarly journals Criterion for Unbounded Synchronous Region in Complex Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jin Zhou
Keyword(s):  
Complex Networks ◽  
Function Method ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Complex Dynamical Network ◽  
Stability Function ◽  
Coupling Matrix ◽  
Dynamical Network ◽  
General Complex ◽  
Necessary And Sufficient

Synchronization of complex networks has been extensively studied in many fields, where intensive efforts have been devoted to the understanding of its mechanisms. As for discriminating network synchronizability by Master Stability Function method, a dilemma usually encountered is that we have no prior knowledge of the network type that the synchronous region belongs to. In this paper, we investigate a sufficient condition for a general complex dynamical network in the absence of control. A main result is that, when the coupling strength is sufficiently strong, the dynamical network achieves synchronization provided that the symmetric part of the inner-coupling matrix is positive definite. According to our results, synchronous region of the network with positive definite inner-coupling matrix belongs to the unbounded one, and then the eigenvalue of the outer-coupling matrix nearest 0 can be used for judging synchronizability. Even though we cannot gain the necessary and sufficient conditions for synchronizing a network so far, our results constitute a first step toward a better understanding of network synchronization.

scholarly journals Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Function Method ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Multiple Integral ◽  
Constant Factor ◽  
Best Constant ◽  
Necessary And Sufficient ◽  
Best Constant Factor ◽  
Hilbert Type

AbstractFor the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

scholarly journals An extended Hamburger moment problem

1985 ◽  
Vol 28 (2) ◽  
pp. 167-183 ◽  
Author(s):  
Olav Njåstad
Keyword(s):  
Distribution Function ◽  
Moment Problem ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Moment Problems ◽  
Real Numbers ◽  
Orthogonal Functions ◽  
Hamburger Moment Problem ◽  
Hankel Determinants ◽  
Necessary And Sufficient

The classical Hamburger moment problem can be formulated as follows: Given a sequence {cn:n=0,1,2,…} of real numbers, find necessary and sufficient conditions for the existence of a distribution function ψ (i.e. a bounded, real-valued, non-decreasing function) on (– ∞,∞) with infinitely many points of increase, such that , n = 0,1,2, … This problem was posed and solved by Hamburger [5] in 1921. The corresponding problem for functions ψ on the interval [0,∞) had already been treated by Stieltjes [15] in 1894. The characterizations were in terms of positivity of Hankel determinants associated with the sequence {cn}, and the original proofs rested on the theory of continued fractions. Much work has since been done on questions connected with these problems, using orthogonal functions and extension of positive definite functionals associated with the sequence. Accounts of the classical moment problems with later developments can be found in [1,4,14]. Good modern accounts of the theory of orthogonal polynomials can be found in [2,3].

scholarly journals Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Wenling Zhao ◽  
Hongkui Li ◽  
Xueting Liu ◽  
Fuyi Xu
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Nonsingular Matrix ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Necessary And Sufficient ◽  
Definite Solution

We study the Hermitian positive definite solutions of the nonlinear matrix equationX+A∗X−2A=I, whereAis ann×nnonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations ofX+A∗X−2A=Iare presented while the matrix equation has a Hermitian positive definite solution.

The quadratic form positive definite on a linear manifold

1951 ◽  
Vol 47 (1) ◽  
pp. 1-6 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Real Vector Space ◽  
Real Vector ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

Topological Identification of Complex Dynamical Network with Communities and Node Delay

2013 ◽  
Vol 411-414 ◽  
pp. 2093-2097
Author(s):  
Jiang Ang Zhang ◽  
Yan Dong Chu ◽  
Wen Ju Du
Keyword(s):  
Parameter Identification ◽  
Hierarchical Structure ◽  
Invariance Principle ◽  
Topological Structure ◽  
Complex Dynamical Networks ◽  
Complex Dynamical Network ◽  
Network Systems ◽  
Dynamical Network ◽  
General Complex ◽  
Real Complex

Recently, various papers investigated the topology identification and parameter identification of uncertain general complex dynamical networks. However, in many real complex dynamical network systems, there exists community or hierarchical structure and node delay. Based on LaSalle’s invariance principle, in this letter, an adaptive controlling method is proposed to identify unknown topological structure for general weighted complex dynamical network with community and node delay. Illustrative simulations are provided to verify the correctness and effectiveness of the proposed scheme.

scholarly journals Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naglaa M. El-Shazly
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Definite Solution

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.

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