Majorization, doubly stochastic matrices, and comparison of eigenvalues

Andô, Tsuyoshi

doi:10.1016/0024-3795(89)90580-6

Cited by 281 publications

(250 citation statements)

References 50 publications

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“…For any x D .x 1 ; x 2 ; :::; x m / 2 S m 1 , we define x # D .x OE1 ; x OE2 ; :::; x OEm / where x OE1 x OE2 ::: x OEm -nonincreasing rearrangement x. Recall [11,14] that for two elements x; y of the simplex S m 1 the element x is majorized by y and written x y ( or y x) if the following condition holds…”

Section: Preliminariesmentioning

confidence: 99%

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

et al. 2016

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The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finitedimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

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Section: Preliminariesmentioning

confidence: 99%

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

et al. 2016

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“…If M h = {m ∈ M | m = m * }, we recall that the spectral order ( [1]; see also [2]) on M h is defined by a ≺ b if the spectral projections satisfy…”

Section: 2mentioning

confidence: 99%

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Voiculescu

2015

J Theor Probab

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Abstract. We compute the bi-free max-convolution which is the operation on bi-variate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely-divisible laws their determination becomes in this way a classical analysis question. IntroductionThe definition and classification of free max-stable laws in [2] had been an unexpected addition to the list of free probability analogues to classical probability items. Here we take the first step in a similar direction in bi-free probability [9]. We show that there is a simple formula for computing the bi-free extremal convolution of probability measures in the plane. This corresponds to computing the distribution of (a ∨ c, b ∨ d), where (a, b) and (c, d) are two bi-free two-faced pairs of commuting hermitian operators. Like the free extremal convolution on R defined in [2], the bi-free extremal convolution in the plane reduces the question of bi-free max-stable laws in the plane to an analysis problem in the classical context which we won't pursue here.

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“…However, we cannot say that all QSOs are of Volterra-type, and the non-Volterra dynamics of QSO are still open. Basic properties of majorization and doubly stochastic matrices have been studied by Ando [14]. The limit behaviour of trajectories of QSOs was completely studied on 1D simplex by Lyubich et al [15], where it was shown that the limit of any initial values is a finite set.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Convergence Algorithm: Structural Properties with Doubly Stochastic Quadratic Operators for Multi-Agent Systems

Abdulghafor

Turaev

Zeki

et al. 2017

Journal of Artificial Intelligence and Soft Computing Research

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This paper proposes nonlinear operator of extreme doubly stochastic quadratic operator (EDSQO) for convergence algorithm aimed at solving consensus problem (CP) of discrete-time for multi-agent systems (MAS) on n-dimensional simplex. The first part undertakes systematic review of consensus problems. Convergence was generated via extreme doubly stochastic quadratic operators (EDSQOs) in the other part. However, this work was able to formulate convergence algorithms from doubly stochastic matrices, majorization theory, graph theory and stochastic analysis. We develop two algorithms: 1) the nonlinear algorithm of extreme doubly stochastic quadratic operator (NLAEDSQO) to generate all the convergent EDSQOs and 2) the nonlinear convergence algorithm (NLCA) of EDSQOs to investigate the optimal consensus for MAS. Experimental evaluation on convergent of EDSQOs yielded an optimal consensus for MAS. Comparative analysis with the convergence of EDSQOs and DeGroot model were carried out. The comparison was based on the complexity of operators, number of iterations to converge and the time required for convergences. This research proposed algorithm on convergence which is faster than the DeGroot linear model.

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Majorization, doubly stochastic matrices, and comparison of eigenvalues

Cited by 281 publications

References 50 publications

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Nonlinear Convergence Algorithm: Structural Properties with Doubly Stochastic Quadratic Operators for Multi-Agent Systems

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