Chunfeng Cui

The Second-Order Augmented Zarankiewicz Number

Liqun Qi, Chunfeng Cui

The second-order Zarankiewicz number $z_{2} (m, n)$ and the biquadratic sum-of-squares rank $BSR (m, n)$ are related by the unconditional hierarchy \[ \mathrm{BSR}(m,n)\ \ge\ z_…

A Weak Condition for Limited Augmented Zarankiewicz Numbers

Liqun Qi, Chunfeng Cui, Yi Xu

This paper introduces the weak augmented Zarankiewicz number $z_{w A} (m, n)$ and the weak limited augmented Zarankiewicz number $z_{w L} (m, n)$ , which are combinatorial extensions of t…

Three-Edges and the SOS Rank of Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Xu

We extend the augmented bipartite graph framework for biquadratic sum-of-squares (SOS) ranks by introducing $3$ -edges -- triples of cells representing squares of three-term bilinea…

A General Lower Bound for the Limited Augmented Zarankiewicz Number based upon Complete Graphs

Liqun Qi, Chunfeng Cui, Yi Xu

The limited augmented Zarankiewicz number $z_{L} (m, n)$ satisfies $BSR (m, n) \geq z_{L} (m, n) \geq z (m, n)$ , where $BSR (m, n)$ is the maximum SOS rank of $m\times…

math.OC2026

Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Chunfeng Cui, Liqun Qi, Yi Xu

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $B S R (m, n)$ . In this paper, we show that $B S R (m, n) \geq z (m, n)$ and conjecture that $BSR(m, n…

math.OC2026

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Xu

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definitenes…

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…

math.NA2025

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…

math.RA2025

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…

math.NT2025

The SOS Rank of Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Xu

In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of $\frac{3 m ( m + 1 )}{2}$ squares of quadratic forms. Ver…

math.SP2025

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…

math.NA2025

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…

math.SP2025

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…

math.RA2025

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…

Triangular Decomposition of Third Order Hermitian Tensors

Liqun Qi, Chunfeng Cui, Ziyan Luo

We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and s…

Even Order Pascal Tensors are Positive Definite

Chunfeng Cui, Liqun Qi, Yannan Chen

In this paper, we show that even order Pascal tensors are positive definite, and odd order Pascal tensors are strongly completely positive. The significance of these is that our in…

Moore Determinant of Dual Quaternion Hermitian Matrices

Chunfeng Cui, Liqun Qi, Guangjing Song +1

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian} matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion He…

math.NA2024★ 1 cited

Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

Liqun Qi, Chunfeng Cui

We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix, and discuss its application in multi-agent formation control. Suppose we have a ring, wh…

math.CO2024

Spectral Properties of Dual Unit Gain Graphs

Chunfeng Cui, Yong Lu, Liqun Qi +1

In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent…