Dr. Zhongshan Li
Professor of Mathematics
Graduate Director of Mathematics
Department of Mathematics and Statistics
Georgia State University
Atlanta, GA 30302-4110
Phone: (404) 413-6437
E-mail: zli@gsu.edu
Research
Interests
Combinatorial Matrix Theory, Sign Pattern Matrices,
Nonnegative Matrices, Integer Matrices, Oriented Matroids, Convex Polytopes,
Boolean Matrices, Matrix Theory Applications, Polynomial and Matrix Equations,
Matrix Inequalities, Algebraic Graph Theory, Neural Networks
Curriculum Vitae
Information about my professional activities can be found in my curriculum vitae.
Biography
I grew up in a poor rural
area belonging to Lanzhou, Gansu Province, China. While in middle school, I
became immensely fascinated by mathematics, and realized that my mathematics
textbooks could not satisfy my thirst for knowledge. In a region where a library
is unheard of, as a first-generation middle schooler, I had a lot of trouble
finding extracurricular study materials, but I thoroughly studied the few good
books I could get. In my last year in high school (that was in fact my second
year in high school, which consisted of two years only back then), I
transferred to a much better high school, Yongdeng County No. 2 High School,
and participated in various high school mathematics competitions, culminating
in the Top Prize in the Gansu Province Mathematics Competition in 1979. Later
that year, I entered Lanzhou
University,
a leading university in China, and obtained my B.Sc. in mathematics in 1983, at
the age of 20. Then I spent three years at Beijing Normal University, graduating with a M.Sc. in
Mathematics in 1986. After teaching briefly at Hebei Normal University, I went
to North Carolina State
University,
where I got my Ph.D. in mathematics (with concentration in algebra) in 1990 and
published two papers on matrix theory.
I have been working at GSU since 1991. Thanks to two influential
colleagues and collaborators, Drs. Frank Hall and Carolyn Eschenbach, my main
research interests shifted to combinatorial matrix theory. For further
biographic information, you may read my mini autobiography.
2023 GSU Workshop on
Matrix Theory and Applications
March 16-17, 2023 @ Room 1441, 25 Park
Place, Georgia State University, Atlanta, GA, USA
The workshop starts at 1:30 pm on March 16.
Links for Current
and Prospective Graduate Students
https://cas.gsu.edu/graduate-services/student-resources/
https://cas.gsu.edu/graduate-admissions/
https://mathstat.gsu.edu/graduate/
https://mathstat.gsu.edu/ (and click
“Students” on the top menu)
Graduate
Program Poster
Publications
D. Lin, X. Chen, Z. Li, B. Li,
X. Yang, On the existence of the exact solution of quaternion-valued neural
networks based on a sequence of approximate solutions,
IEEE Transactions on Neural Networks and Learning Systems, 9 pp. In
press.
DOI: https://doi.org/10.1109/TNNLS.2021.3129269
L. Zhang, G.-F. Mou, F. Liu, Z.
Li, Some bounds of the generalized μ-scrambling indices of primitive
digraphs with d loops. J. Inequal. Appl. (2021), Paper No. 128, 13 pp.
DOI: https://doi.org/10.1186/s13660-021-02667-y
D. Lin, X. Chen, G. Yu, Z. Li,
Y. Xia, Global exponential synchronization via nonlinear feedback control for
delayed inertial memristor-based quaternion-valued neural networks with
impulses. Appl. Math. Comput. 401 (2021), Paper No.
126093, 19 pp.
DOI:
https://doi.org/10.1016/j.amc.2021.126093
Y. Xia, X. Chen, D. Lin, Z. Li,
Some dynamical behaviors of fractional-order commutative quaternion-valued
neural networks via direct method of Lyapunov, IEEE Access 9 (2021), 693-708.
DOI: https://doi.org/10.1109/ACCESS.2020.3046842
M. Arav, F.J. Hall, Z. Li, H.
van der Holst, Two-connected signed graphs with maximum nullity at most
two, Linear Algebra Appl. 611 (2021), 82–93.
DOI: https://doi.org/10.1016/j.laa.2020.12.002
B.-L. Yu, Z. Li, S. Xu, On
the eventual exponential positivity of some tree sign patterns. Symmetry 13
(2021), 1669. DOI: https://doi.org/10.3390/sym13091669
X. Feng, W. Gao, F.J. Hall, G. Jing, Z. Li, C. Zagrodny, J. Zhou, Rank conditions for sign patterns that
allow diagonalizability, Discrete Mathematics 343 (2020), no. 5, 111798,
11 pp.
DOI: https://doi.org/10.1016/j.disc.2019.111798
G.F. Mou, T.F. Wang, Z. Li, The bipartite zero forcing set for a full sign pattern
matrix,
Mathematics 8 (3) (2020), 354, 14 pp. DOI: https://doi.org/10.3390/math8030354
W. Fang, H. Yu, Y. Gao, G. Jing, Z. Li, X. Li, Minimum detour index of cactus graphs. Ars Combinatoria 144 (2019), 293–307.
X. Chen, Q. Song, and Z. Li, Design and analysis of quaternion-valued neural networks for associative memories, IEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol. 48, NO. 12, 2018, 2305-2314. DOI: https://doi.org/10.1109/TSMC.2017.2717866
W. Fang, W. Gao, Y. Gao, F. Gong, G. Jing, Z. Li, Y. Shao, L. Zhang, Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations, Linear Algebra and its Applications 558 (2018), 44-62. DOI: https://doi.org/10.1016/j.laa.2018.08.019
X. Chen, L. Li, Z. Li, Robust stability analysis of quaternion-valued neural networks via LMI approach, Advances in Difference Equations 2018 (1), Paper No.131, 20 pages.
DOI: https://doi.org/10.1186/s13662-018-1585-z
X. Chen, Q. Song, Z. Li, Z. Zhao, Y. Liu, Stability analysis of continuous-time and discrete-time quaternion-valued neural networks with linear threshold neurons, IEEE Transactions on Neural Networks and Learning Systems 29 (7) (2018), 2769-2781.
DOI: https://doi.org/10.1109/TNNLS.2017.2704286
W. Fang, H. Yu, Y. Gao, X. Li, G. Jing, Z. Li, Maximum Balaban index and sum-Balaban index of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 3, 717–742.
Z. Li, F. Zhang, X. Zhang, On the number of vertices of the stochastic tensor polytope, Linear and Multilinear Algebra 65 (2017), no. 10, 2064–2075.
DOI: https://doi.org/10.1080/03081087.2017.1310178
W. Gao, Z. Li, Path sign patterns of order n≥5 do not require Hn, Linear Algebra Appl. 532 (2017), 99–126. DOI: https://doi.org/10.1016/j.laa.2017.06.039
X. Chen, Z. Li, Q. Song, J. Hu, Y. Tan, Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties, Neural Networks 91 (2017), 55–65.
DOI: https://doi.org/10.1016/j.neunet.2017.04.006
X. Chen, Q. Song, Z. Li, Design and analysis of quaternion-valued neural networks for associative memories, IEEE Transactions on Systems, Man, and Cybernetics: Systems PP (99) (2017), 1–10. DOI: https://doi.org/10.1109/TSMC.2017.2717866
X. Chen, Q. Song, Z. Li, Z. Zhao, Y. Liu, Stability analysis of continuous-time and discrete-time quaternion-valued neural networks with linear threshold neurons, IEEE Transactions on Neural Networks and Learning Systems PP (99) (2017), 1–13.
DOI: https://doi.org/10.1109/tnnls.2017.2704286
F.J. Hall, Z. Li, C.T. Parnass, M. Rozložník, Sign patterns of J-orthogonal matrices, Special Matrices 5 (2017), 225–241. DOI: https://doi-org/10.1515/spma-2017-0016
W. Fang, Y. Gao, K. Fan, Z. Li, The second largest Balaban index (sum-Balaban index) of unicyclic graphs, J. Math. Res. Appl. 37 (2017), no. 4, 391–403.
DOI: https://doi.org/10.3770/j.issn:2095-2651.2017.04.002
W. Gao, Z. Li and L. Zhang, Characterization of
star sign patterns that require Hn, Linear Algebra and Its Applications 499 (2016), 43–65.
DOI: https://doi.org/10.1016/j.laa.2017.06.039
W. Fang, W. Gao, Y. Gao, F. Gong, G. Jing, Z. Li, Y. Shao, and L. Zhang, The minimum ranks of nonnegative sign patterns and convex polytopes, Czech. Math Journal 66 (141) (2016), no. 3, 895–911. DOI: https://doi.org/10.1007/s10587-016-0299-1
M. Arav, F. Hall, Z. Li, H. van der Holst, A graph minors characterization of signed graphs whose Colin de Verdière parameter ν is two, Journal of Combinatorial Theory, Series B 116 (2016), 440–455. DOI: https://doi.org/10.1016/j.jctb.2015.09.007
X. Chen, W. Fang, W. Gao, Y. Gao, G. Jing, Z. Li, Y. Shao, L. Zhang, Essential sign change numbers of full sign pattern matrices. Special Matrices 4 (2016), 247–254.
DOI: https://doi.org/10.1515/spma-2016-0023
W. Fang, Y. Gao, Y. Shao, W. Gao, G. Jing, Z. Li, Maximum Balaban index and sum-Balaban index of bicyclic Graphs, MATCH Communications in Math. and in Computer Chemistry 75 (2016), 129–156.
W. Fang, Y. Gao, Y. Shao, W. Gao, G. Jing, Z. Li, The generalized competition indices of primitive minimally strong digraphs, Linear Algebra and Its Applications 493 (2016), 206–226. DOI: https://doi.org/10.1016/j.laa.2015.11.036
W. Gao, Z. Li, L. Zhang, Sign patterns that require Hn exist for each n≥4, Linear Algebra and Its Applications 489 (2016), 15–23. DOI: https://doi.org/10.1016/j.laa.2015.09.050
M. Arav, F. Hall, Z. Li, H. van der Holst, J.H. Sinkovic, L. Zhang, Minimum ranks of sign patterns via sign vectors and duality, Electronic Journal of Linear Algebra 30 (2015), 360-371. DOI: https://doi.org/10.13001/1081-3810.3077
M. Fiedler, W. Gao, F.J. Hall, G. Jing, Z. Li, M. Stroev, Ranks of dense alternating sign matrices and their sign patterns, Linear Algebra and Its Applications 471 (2015), 109–121.
DOI: https://doi.org/10.1016/j.laa.2014.12.034
J. Luo, T-Z. Huang, H. Li, Z. Li, L. Zhang, Tree sign patterns that allow nilpotence of index 4, Linear Multilinear Algebra 63 (2015), no. 5, 1009–1025.
DOI: https://doi.org/10.1080/03081087.2014.914930
W. Gao, Z. Li, and L. Zhang, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra and Its Applications 458 (2014), 183–196.
DOI: https://doi.org/10.1016/j.laa.2014.06.009
M. Arav, F. Hall, Z. Li, W. Zhou, and L. Zhang, The minimum rank of a sign pattern with a 1-separation, Linear Algebra and Its Applications 448 (2014), 205–216.
DOI: https://doi.org/10.1016/j.laa.2014.01.019
L. Zhang, Z. Li, T.Z. Huang, Q.F. Zhu, J. Hua, L.H. Zhang, Periodic, reducible, powerful ray pattern matrices, Linear Algebra and Its Applications 444 (2014), 81–88.
DOI: https://doi.org/10.1016/j.laa.2013.11.007
L. Zhang, T.Z. Huang, Z. Li and J.Y. Zhang, Several spectrally arbitrary ray patterns, Linear and Multilinear Algebra 61 (2013), 543-564. https://doi.org/10.1080/03081087.2012.696247
Z. Li, Y. Gao, M. Arav, F. Gong, W. Gao, F.J. Hall and H. van der Holst, Sign patterns with minimum rank 2 and upper bounds on minimum ranks, Linear and Multilinear Algebra 61 (2013), 895-908. DOI: https://doi.org/10.1080/03081087.2012.716431
M. Arav, F. Hall, Z. Li and H. van der Holst, The inertia set of a signed graph, Linear Algebra and Its Applications 439 (2013), 1506–1529. DOI: https://doi.org/10.1016/j.laa.2013.04.032
X. Feng, T.Z. Huang, Z. Li, J. Luo and Y. Gao, Sign patterns that allow diagonalizability revisited, Linear and Multilinear Algebra 61 (2013), 1223-1233.
DOI: https://doi.org/10.1080/03081087.2012.746329
F. Hall and Z. Li, Sign pattern matrices, Chapter 42 in L. Hogben (Ed.), Handbook of Linear Algebra, 2nd edition, Chapman/Hall CRC Press, 2013. DOI: https://doi.org/10.1201/b16113
Y. Shao, Y. Gao and Z. Li, The m-competition indices of symmetric primitive digraphs without loop, Electronic Journal of Linear Algebra 23 (2012), 457-472.
DOI: https://doi.org/10.13001/1081-3810.1532
X. Feng, Z. Li and T.Z. Huang, Is every nonsingular matrix diagonally equivalent to a matrix with all distinct eigenvalues? Linear Algebra and Its Applications 436 (2012), 120-125.
L. Zhang, T.Z. Huang, and Z. Li, Primitive zero-symmetric sign pattern matrices with zero diagonal attaining the maximum base, Journal of Applied Math 2012 (2012), Article ID 276386, 28 pages, doi 10.1155/2012/276386.
M. Arav, F. Hall, Z. Li, K. Kaphle and N. Manzagol, Spectrally arbitrary tree sign patterns of order 4, Electronic Journal of Linear Algebra 20 (2010), 180-197.
M. Arav, F. Hall, Z. Li, A. Merid and Y. Gao, Sign patterns that require almost unique rank, Linear Algebra and Its Applications 430 (2009), 7-16.
M. Arav, F. Hall, Z. Li and B. Rao, ZPC matrices and zero cycles, International Journal of Combinatorics 2009 (2009), 1-5.
M. Arav, F. Hall, Z. Li and B. Rao, Rational solutions of certain matrix equations, Linear Algebra and Its Applications 430 (2009), 660-663.
M. Arav, F. Hall and Z. Li, A Cauchy-Schwarz inequality for triples of vectors, Mathematical Inequalities and Applications 11 (2008), 629-634.
Y. Gao, Z. Li and Y. Shao, A note on spectrally arbitrary sign patterns, JP Journal of Algebra, Number Theory and Applications 11 (2008), 15-35
T. Huang, G. Mou, G. Tian, Z. Li, D. Wang, Subdirect sums of P (P0)-matrices
and totally nonnegative matrices, Linear Algebra and Its Applications 429 (2008), 1730-1743.
Y. Gao, Z. Li and Y Shao,
Sign patterns allowing nilpotence of index 3, Linear Algebra and Its
Applications 424 (2007), 55-70.
F. Hall and Z. Li, Sign pattern matrices, Chapter 33 in L. Hogben (Ed.), Handbook of Linear Algebra, Chapman and Hall/CRC Press, 2007.
M. Arav, F. Hall, S. Koyuncu, Z. Li and B. Rao, Rational realizations of the minimum rank of a sign pattern matrix, Linear Algebra and Its Applications 411 (2005), 111-125.
Z. Li, F. Hall and J. Stuart, Reducible powerful ray pattern matrices, Linear Algebra and Its Applications 399 (2005), 125-140.
G. Chen, G. Davis, F. Hall, Z. Li, K. Patel, M. Stewart, An interlacing result on normalized Laplacians, SIAM Journal on Discrete Mathematics 18 (2004), 353-361.
F. Hall, Z. Li and B. Rao, From Boolean to sign pattern matrices, Linear Algebra and Its Applications 393 (2004), 233-51.
G. Chen, F. Hall, Z. Li and B. Wei, On ranks of matrices associated with trees, Graphs and Combinatorics 19 (2003), 323-334.
F. Hall and Z. Li, Sign Patterns, Inverses and generalized inverses — a brief survey, Numerical Mathematics: A Journal of Chinese Universities 12 (Supplement) (2003),
8-11.
G. Chen, F. Hall, Z. Li and B. Wei, On ranks of matrices associated with trees,
Graphs and Combinatorics 19 (2003), no. 3, 323–334.
Z. Li and L. Harris, Sign patterns that require all distinct eigenvalues, JP Journal of Algebra and Number Theory and Applications 2 (2) (2002), 161-179.
Z. Li, F.
Hall and J. Stuart, Irreducible powerful ray pattern matrices, Linear
Algebra and its Applications 342 (2002), 47-58.
F. Hall and Z. Li, Inertia sets of sign patterns, Numerical Mathematics: A Journal of Chinese Universities 10 (2001), 226-240.
F. Hall, Z.
Li and Di Wang, Symmetric sign pattern matrices that require unique
inertia, Linear Algebra and Its Applications 338 (2001),
153-169.
G. Chen, F. Hall, Z. Li and H. Zhou, Isomorphisms involving reversing arcs of di-graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 36 (2001), 155--160.
C. Eschenbach, F. Hall, R. Hemasinha, S.
Kirkland, Z. Li, B. Shader, J.
Stuart and J. Weaver, Properties of almost regular tournaments, American Mathematical Monthly 107 (no.10) (2000), 881-892.
C. Eschenbach, F. Hall and Z. Li, Eigenvalue distribution of certain ray patterns, Czechoslovak Mathematical Journal 50(125) (2000), 749-762.
C. Eschenbach, F. Hall, R. Hemasinha, S. Kirkland, Z. Li, B. Shader, J. Stuart and J. Weaver, On almost regular tournament matrices, Linear Algebra and Its Applications 306 (2000), 103--121.
C. Eschenbach and Z. Li, Potentially nilpotent sign pattern matrices, Linear Algebra and its Applications 299 (1999), 81-99.
F. Hall and Z. Li, Sign patterns of idempotent matrices, Journal of Korean Mathematical Society 36 (1999), 469-487.
C. Eschenbach, F. Hall, D. Harrell and Z. Li, When does the inverse have the same sign
pattern as the transpose?, Czechoslovak Mathematical Journal 49(124) (1999), 255-275.
C. Eschenbach, F. Hall and Z. Li, From real to complex sign pattern matrices,
Bulletin of the Australian Mathematical Society 57 (1998), 159-172.
C. Eschenbach, F. Hall, C. Johnson and Z. Li, The graphs of unambiguous entries
in the product of two (+,-) sign patterns, Linear Algebra and Its Applications
260 (1997), 95-118.
C. Eschenbach and Z. Li, How many negative entries can A2 have?,
Linear
Algebra and Its Applications 254 (1997), 99-117.
Z. Li, F. Hall and F. Zhang, Sign patterns of nonnegative normal
matrices, Linear
Algebra and Its Applications 254 (1997), 335-354.
C. Eschenbach, F. Hall and Z. Li, Some sign patterns that allow an inverse pair
B and B-1, Linear Algebra and Its Applications 252(1997), 299-321.
Z. Li, F. Hall and C. Eschenbach, The structure of nonnegative cyclic matrices,
Linear and Multilinear Algebra 41 (1996), 23-33.
Z. Li, F. Hall and C. Eschenbach, On the period and base of a sign pattern matrix,
Linear Algebra and Its Applications 212 (1994), 101-120.
C. Eschenbach, F. Hall and Z. Li, Eigenvalue frequency and consistent sign
pattern matrices, Czechoslovak Mathematical Journal 44 (1994), 461-479.
C. Eschenbach, F. Hall and Z. Li, Sign pattern matrices and generalized
inverses,
Linear Algebra and Its Applications 211 (1994), 53-66.
Z. Li, A determinantal description of GCD-closed sets and k-sets, Linear
and
Multilinear Algebra 31 (1992), 245-250.
Z. Li, The determinants of GCD matrices, Linear Algebra and Its
Applications
134 (1990), 137-143.
Editorial Activity
I am
a member of the Editorial Board of JP Journal of
Algebra, Number Theory and Applications.
Guest
editor of Mathematics.
Combinatorial Matrix
Theory Seminar
1:30 pm – 3:30 pm on Fridays, via Webex
Courses Recently Taught
Math 2211 Calculus of One Variable I
Math 2212 Calculus of One Variable II
Math 2215 Multivariate Calculus
Math 2420 Discrete Mathematics
Math 2641 Linear Algebra I
Math 3000 Bridge to Higher
Mathematics
Math 4435/6435 Linear Algebra II
Math 4441/6441 Modern Algebra I
Math 4450/6450 Theory of Numbers
Math 4661/6661 Analysis I
Math 4662/6662 Analysis II
Math 8200 Advanced Matrix Analysis
Math 8201 Combinatorial Matrix Theory
Math 8210 Topics in Applied
Matrix Analysis
Math 8220 Abstract Algebra I
Math 8535 Applied Matrix Algebra
Family Photo
Gallery
Angela and Andrew with Grandpa in Dunhuang, 2008
Ying and Zhongshan at Qinghai Lake, 2009
Ying and Zhongshan in Chengdu, 2008
Ying and Zhongshan in Hangzhou, 2008
Ying and Zhongshan in Shanghai, 2008
