References
- Webster R. Convexity. New York: Oxford University Press; 1994.
- Marlow WH. Mathematics for operations research. New York: Dover Books on Mathematics; 2013.
- Barvinok A. A course in convexity. Providence (RI): American Mathematical Society; 2002. (Graduate studies in mathematics; 54).
- Ziegler GM. Lectures on polytopes. New York: Springer; 1995.
- Zhang F. Matrix theory: basic results and techniques. 2nd ed. New York: Springer; 2011.
- Brualdi RA, Cao L. Symmetric, Hankel-symmetric, and centrosymmetric doubly stochastic matrices. Acta Math Vietnam. 2018;43:675–700. (Special Issue for the 6th International Conference on Matrix Analysis and Applications (ICMAA), Da Nang, Vietnam, June 15–18, 2017). doi: 10.1007/s40306-018-0266-z
- Brualdi RA, Csima J. Stochastic patterns. J Combin Theory Ser A. 1975;19:1–12. doi: 10.1016/0097-3165(75)90088-6
- Cui L, Li W, Ng M-K. Birkhoff–von Neumann theorem for multistochastic tensors. SIAM J Matrix Anal Appl. 2014;35:956–973. doi: 10.1137/120896499
- Paffenholz A. Faces of Birkhoff polytopes. Electron J Combin. 2015;22(1): paper 1.67, 36 pp.
- Marshall AW, Olkin I, Arnold B. Inequalities: theory of majorization and its applications. 2nd ed.. New York: Springer; 2011.
- Brondsted A. An introduction to convex polytopes. New York: Springer; 1983.
- Wang Q-W, Zhang F. The permanent functions of tensors. Acta Math Vietnam. 2018;43(4):701–713. Special Issue for the 6th International Conference on Matrix Analysis and Applications (ICMAA). doi: 10.1007/s40306-018-0268-x
- Fischer P, Swart ER. Three dimensional line stochastic matrices and extreme points. Linear Algebra Appl. 1985;69:179–203. doi: 10.1016/0024-3795(85)90075-8
- Brualdi RA, Csima J. Extremal plane stochastic matrices of dimension three. Linear Algebra Appl. 1975;11:105–133. doi: 10.1016/0024-3795(75)90053-1
- Brualdi RA, Csima J. Small matrices of large dimension. Linear Algebra Appl. 1991;150:227–241. Proceedings of the First Conference of the International Linear Algebra Society; Provo, UT, 1989. doi: 10.1016/0024-3795(91)90171-R
- Li Z, Zhang F, Zhang X-D. On the number of vertices of the stochastic tensor polytope. Linear Multilinear Algebra. 2017;65:2064–2075. doi: 10.1080/03081087.2017.1310178
- Ding W, Wei Y. Theory and computation of tensors: multi-dimensional arrays. London: Elsevier; 2016.
- Qi L, Luo Z. Tensor analysis: spectral theory and special tensors. Philadelphia: SIAM; 2017.
- Ke R, Li W, Xiao M. Characterization of extreme points of multi-stochastic tensors. Comput Methods Appl Math. 2016;16:459–274. doi: 10.1515/cmam-2016-0005
- Jurkat WB, Ryser HJ. Extremal configurations and decomposition theorems. J Algebra. 1968;8:194–222. doi: 10.1016/0021-8693(68)90045-8
- Burkard R, Dell'Amico M, Martello S. Assignment problems: revised reprint. Philadelphia: SIAM; 2009.
- Ahmed M. Algebraic combinatorics of magic squares [Ph.D. Thesis]. Davis: University of Califorina; 2004.
- Ahmed M, De Loera J, Hemmecke R. Polyhedral cones of magic cubes and squares. In Aronov B, et al., editors. Discrete and computational geometry algorithms and combinatorics. Vol. 25. Berlin: Springer; 2003. p. 25–41.
- van Lint JH, Wilson RM. A course in combinatorics. Berlin: Cambridge University Press; 1992.
- Chang H, Paksoy VE, Zhang F. Polytopes of stochastic tensors. Ann Funct Anal. 2016;7(3):386–393. doi: 10.1215/20088752-3605195
- Gass SI. Linear programming: methods and applications. 5th ed. New York: Dover Books on Computer Science; 2003.
- Burger E. Introduction to the theory of games. New York: Prentice-Hall; 1963.
- Linial N, Luria Z. On the vertices of the d-dimensional Birkhoff polytope. Discrete Comput Geom. 2014;51(1):161–170. doi: 10.1007/s00454-013-9554-5
- McMullen P. The maximum numbers of faces of a convex polytope. Mathematika. 1970;17:179–184. doi: 10.1112/S0025579300002850
- McKay BD, Wanless IM. On the number of latin squares. Ann Comb. 2005 Oct;9(3):335–344. doi: 10.1007/s00026-005-0261-7