# Ting-Zhu Huang

## Publications

##### 9 A Simplified Higher-Order Markov Chain Model

**Authors:**
Chao Wang,
Ting-Zhu Huang,
Chen Jia

**Abstract:**

In this paper, we present a simplified higher-order Markov chain model for multiple categorical data sequences also called as simplified higher-order multivariate Markov chain model.

**Keywords:**
Higher-order multivariate Markov chain model,
Categorical data sequences,
Multivariate Markov chain

##### 8 Comparison of Two Types of Preconditioners for Stokes and Linearized Navier-Stokes Equations

**Authors:**
Ze-Jun Hu,
Ting-Zhu Huang,
Ning-Bo Tan

**Abstract:**

To solve saddle point systems efficiently, several preconditioners have been published. There are many methods for constructing preconditioners for linear systems from saddle point problems, for instance, the relaxed dimensional factorization (RDF) preconditioner and the augmented Lagrangian (AL) preconditioner are used for both steady and unsteady Navier-Stokes equations. In this paper we compare the RDF preconditioner with the modified AL (MAL) preconditioner to show which is more effective to solve Navier-Stokes equations. Numerical experiments indicate that the MAL preconditioner is more efficient and robust, especially, for moderate viscosities and stretched grids in steady problems. For unsteady cases, the convergence rate of the RDF preconditioner is slightly faster than the MAL perconditioner in some circumstances, but the parameter of the RDF preconditioner is more sensitive than the MAL preconditioner. Moreover the convergence rate of the MAL preconditioner is still quite acceptable. Therefore we conclude that the MAL preconditioner is more competitive than the RDF preconditioner. These experiments are implemented with IFISS package.

**Keywords:**
Navier-Stokes equations,
Krylov subspace method,
preconditioner,
dimensional splitting,
augmented Lagrangian preconditioner

##### 7 Sign Pattern Matrices that Admit P0 Matrices

**Authors:**
Ling Zhang,
Ting-Zhu Huang

**Abstract:**

A P0-matrix is a real square matrix all of whose principle minors are nonnegative. In this paper, we consider the class of P0-matrix. Our main aim is to determine which sign pattern matrices are admissible for this class of real matrices.

**Keywords:**
graph,
digraph,
Sign pattern matrices,
P0 matrices

##### 6 Some Results of Sign patterns Allowing Simultaneous Unitary Diagonalizability

**Authors:**
Xin-Lei Feng,
Ting-Zhu Huang

**Abstract:**

Allowing diagonalizability of sign pattern is still an open problem. In this paper, we make a carefully discussion about allowing unitary diagonalizability of two sign pattern. Some sufficient and necessary conditions of allowing unitary diagonalizability are also obtained.

**Keywords:**
eigenvalue,
Sign pattern,
unitary diagonalizability,
allowing diagonalizability

##### 5 A Note on Potentially Power-Positive Sign Patterns

**Authors:**
Ber-Lin Yu,
Ting-Zhu Huang

**Abstract:**

In this note, some properties of potentially powerpositive sign patterns are established, and all the potentially powerpositive sign patterns of order ≤ 3 are classified completely.

**Keywords:**
Sign pattern,
potentially eventually positive sign pattern,
potentially power-positive sign pattern

##### 4 A Finite-Time Consensus Protocol of the Multi-Agent Systems

**Authors:**
Xin-Lei Feng,
Ting-Zhu Huang

**Abstract:**

According to conjugate gradient algorithm, a new consensus protocol algorithm of discrete-time multi-agent systems is presented, which can achieve finite-time consensus. Finally, a numerical example is given to illustrate our theoretical result.

**Keywords:**
Multi-Agent Systems,
Graph Theory,
Conjugate Gradient algorithm,
Consensus protocols,
Finite-time

##### 3 The Partial Non-combinatorially Symmetric N10 -Matrix Completion Problem

**Authors:**
Gu-Fang Mou,
Ting-Zhu Huang

**Abstract:**

An n×n matrix is called an N1 0 -matrix if all principal minors are non-positive and each entry is non-positive. In this paper, we study the partial non-combinatorially symmetric N1 0 -matrix completion problems if the graph of its specified entries is a transitive tournament or a double cycle. In general, these digraphs do not have N1 0 -completion. Therefore, we have given sufficient conditions that guarantee the existence of the N1 0 -completion for these digraphs.

**Keywords:**
digraph,
matrix completion,
cycle,
N10 -matrix,
non-combinatorially symmetric

##### 2 Minimal Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 3

**Authors:**
Ber-Lin Yu,
Ting-Zhu Huang

**Abstract:**

If there exists a nonempty, proper subset S of the set of all (n + 1)(n + 2)/2 inertias such that S Ôèå i(A) is sufficient for any n × n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In [3], Kim, Olesky and Driessche identified all minimal critical sets of inertias for 2 × 2 zero-nonzero patterns. Identifying all minimal critical sets of inertias for n × n zero-nonzero patterns with n ≥ 3 is posed as an open question in [3]. In this paper, all minimal critical sets of inertias for 3 × 3 zero-nonzero patterns are identified. It is shown that the sets {(0, 0, 3), (3, 0, 0)}, {(0, 0, 3), (0, 3, 0)}, {(0, 0, 3), (0, 1, 2)}, {(0, 0, 3), (1, 0, 2)}, {(0, 0, 3), (2, 0, 1)} and {(0, 0, 3), (0, 2, 1)} are the only minimal critical sets of inertias for 3 × 3 irreducible zerononzero patterns.

**Keywords:**
Zero-nonzero pattern,
critical set of inertias,
inertially arbitrary,
Permutation digraph,
irreducible pattern

##### 1 A Note on the Minimum Cardinality of Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 4

**Authors:**
Ber-Lin Yu,
Ting-Zhu Huang

**Abstract:**

If there exists a nonempty, proper subset S of the set of all (n+1)(n+2)/2 inertias such that S Ôèå i(A) is sufficient for any n×n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In [Kim, Olesky and Driessche, Critical sets of inertias for matrix patterns, Linear and Multilinear Algebra, 57 (3) (2009) 293-306], identifying all minimal critical sets of inertias for n×n zero-nonzero patterns with n ≥ 3 and the minimum cardinality of such a set are posed as two open questions by Kim, Olesky and Driessche. In this note, the minimum cardinality of all critical sets of inertias for 4 × 4 irreducible zero-nonzero patterns is identified.

**Keywords:**
Zero-nonzero pattern,
inertia,
critical set of inertias,
inertially arbitrary