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Sum of Squares Decompositions for Structured Biquadratic Forms
Yi Xu, Chunfeng Cui, Liqun Qi
This paper studies sum-of-squares (SOS) representations for structured biquadratic forms. We prove that diagonally dominated symmetric biquadratic tensors are always SOS. For the s…
The Power Method for Non-Hermitian Dual Quaternion Matrices
Hao Yang, Liqun Qi, Chunfeng Cui
This paper proposes a power method for computing the dominant eigenvalues of a non-Hermitian dual quaternion matrix (DQM). Although the algorithmic framework parallels the Hermitia…
Structured Symmetric Tensors
Liqun Qi, Chunfeng Cui, Yi Xu
In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of square…
Completely Positive Biquadratic Tensors
Liqun Qi, Chunfeng Cui, Haibin Chen +1
In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares te…
A Dual Quaternion Control Law for Formation Control of Multiple 3-D Rigid Bodies
Chunfeng Cui, Liqun Qi, Hao Chen +1
This paper studies the integrated position and attitude control problem for multi-agent systems of 3D rigid bodies. While the state-of-the-art method in [Olfati-Saber and Murray, 2…
Postive Semidefinite and Sum of Squares Biquadratic Polynomials
Chunfeng Cui, Liqun Qi, Yi Xu
Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic…
The SOS Rank of Biquadratic Forms
Liqun Qi, Chunfeng Cui, Yi Xu
In 1973, Calderón proved that an positive semidefinite (psd) biquadratic form can always be expressed as the sum of squares of quadratic forms. Ver…
Quasi-Irreducibility of Nonnegative Biquadratic Tensors
Liqun Qi, Chunfeng Cui, Yi Xu
While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irr…
Nonnegative Biquadratic Tensors
Chunfeng Cui, Liqun Qi
An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors…
Biquadratic Tensors: Eigenvalues and Structured Tensors
Liqun Qi, Chunfeng Cui
The covariance tensors in statistics{, elasticity tensor in solid mechanics, Riemann curvature tensor in relativity theory are all biquadratic tensors that are weakly symmetric, bu…
Strongly Positive Semi-Definite Tensors and Strongly SOS Tensors
Liqun Qi, Chunfeng Cui
We introduce {odd-order} strongly PSD (positive semi-definite) tensors which map real vectors to nonnegative vectors. We then introduce odd-order strongly SOS (sum-of-squares) tens…