Results and problems in the theory of doubly-stochastic matrices

  • L. Mirsky
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References

  1. [1]
    Bellman, R., and A. Hoffman: On a theorem of Ostrowski and Taussky. Arch, der Math. 5, 123–127 (1954).Google Scholar
  2. [2]
    Berge, C.: Théorie des Graphes et ses Applications. (Paris: 1958).Google Scholar
  3. [3]
    Birkhoff, G.: Tres observaciones sobre el algebra lineal. Univ. nac. Tucumán, Revista Ser. A, 5, 147–150 (1946).Google Scholar
  4. [4]
    Birkhoff, G.: Lattice Theory (revised edition; New York: 1948).Google Scholar
  5. [5]
    Bourbaki, N.: Espaces Vectoriels Topologiques (Ch. I–II). Paris: 1953.Google Scholar
  6. [6]
    Bruijn, N. G. de: Gemeenschappelijke representantensystemen van twee klassenindeelingen van een verzameling. Nieuw Arch. Wiskunde II. R. 22, 48–52 (1943).Google Scholar
  7. [7]
    Dmitriev, N., and E. Dynkin: On the characteristic numbers of a stochastic matrix. Doklady Akad. Nauk SSSR n. Ser. 49, 159–162 (1945).Google Scholar
  8. [8]
    —: Characteristic roots of stochastic matrices. Izvestija Akad. Nauk SSSR, Ser. Mat. 10, 167–184 (1946) (In Russian).Google Scholar
  9. [9]
    Dulmage, L., and I. Halperin: On a theorem of Frobenius-König and J. von Neumann's game of hide and seek. Trans. roy. Soc. Canada, Sect. III, III. Ser., 49, 23–29 (1955).Google Scholar
  10. [10]
    Eggleston, H. G.: Convexity. Cambridge: 1958.Google Scholar
  11. [11]
    Fan, K.: Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. nat. Acad. Sci. USA 37, 760–766 (1951).Google Scholar
  12. [12]
    —: A minimum property of the eigenvalues of a Hermitian transformation. Amer. math. Monthly 60, 48–50 (1953).Google Scholar
  13. [13]
    —: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Math. Z. 68, 205–216 (1957–58).Google Scholar
  14. [14]
    — Convex Sets and their Applications (Argonne National Laboratory, 1959).Google Scholar
  15. [15]
    —, and A. J. Hoffman: Some metric inequalities in the space of matrices. Proc. Amer. math. Soc. 6, 111–116 (1955).Google Scholar
  16. [16]
    —, and G. G. Lorentz: An integral inequality. Amer. math. Monthly 61, 626–631 (1954).Google Scholar
  17. [17]
    Farahat, H. K., and L. Mirsky: Permutation endomorphisms and refinement of a theorem of Birkhoff. Proc. Cambridge philos. Soc. 56, 322–328 (1960).Google Scholar
  18. [18]
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. I. New York: 1950.Google Scholar
  19. [19]
    Fuchs, L.: A new proof of an inequality of Hardy-Littlewood-Pólya. Mat. Tidsskr. B, 53–54 (1947).Google Scholar
  20. [20]
    Hall, P.: On representatives of subsets. J. London math. Soc. 10, 26–30 (1935).Google Scholar
  21. [21]
    Hammersley, J. M.: A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices. Proc. Cambridge philos. Soc. 57, 681 (1961).Google Scholar
  22. [22]
    —, and J. G. Mauldon: General principles of antithetic variates. Proc. Cambridge philos. Soc. 52, 476–481 (1956).Google Scholar
  23. [23]
    Hardy, G. H., J. E. Littlewood, and G. Pólya: Some simple inequalities satisfied by convex functions. Messenger of Math. 58, 145–152 (1929).Google Scholar
  24. [24]
    — Inequalities. Cambridge: 1934.Google Scholar
  25. [25]
    Hoffmann, A. J.: On an inequality of Hardy, Littlewood, and Pólya. (National Bureau of Standards, Report No. 2977, 1953.)Google Scholar
  26. [26]
    —, and H. W. Wielandt: The variation of the spectrum of a normal matrix. Duke math. J. 20, 37–40 (1953).Google Scholar
  27. [27]
    Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math 76, 620–630 (1954).Google Scholar
  28. [28]
    Isbell, J. R.: Birkhoff's problem 111. Proc. Amer. math. Soc. 6, 217–218 (1955).Google Scholar
  29. [29]
    Johnson, D. M., A. L. Dulmage, and N. S. Mendelsohn: On an algorithm of G. Birkhoff concerning doubly stochastic matrices. Canad. math. Bull. 3, 237–242 (1960).Google Scholar
  30. [30]
    Karamata, J.: Sur une inégalité relative aux fonctions convexes. Publ. math. Univ. Belgrade 1, 145–148 (1932).Google Scholar
  31. [31]
    Karpelevich, F. I.: On the characteristic roots of matrices with non-negative elements. Izvestija Akad. Nauk SSSR. Ser. Mat. 15, 361–383 (1951) (In Russian).Google Scholar
  32. [32]
    Kendall, D. G.: On infinite doubly-stochastic matrices and Birkhoff's problem 111. J. London math. Soc. 35, 81–84 (1960).Google Scholar
  33. [33]
    König, D.: über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77, 435–465 (1916).Google Scholar
  34. [34]
    Lorentz, G. G.: An inequality for rearrangements. Amer. math. Monthly 60, 176–179 (1953).Google Scholar
  35. [35]
    Marcus, M.: Convex functions of quadratic forms. Duke math. J. 24, 321–326 (1957).Google Scholar
  36. [36]
    —: Some properties and applications of doubly stochastic matrices. Amer. math. Monthly 67, 215–222 (1960).Google Scholar
  37. [37]
    Marcus, M., H. Minc, and B. Moyls: Some results on non-negative matrices. J. Res. nat. Bur. Standards 65 B, 205–209 (1961).Google Scholar
  38. [38]
    —, and M. Newman: On the minimum of the permanent of a doubly stochastic matrix. Duke math. J. 26, 61–72 (1959).Google Scholar
  39. [39]
    — —: Inequalities for the permanent function. Ann. of Math. II. Ser., 75, 47–62 (1962).Google Scholar
  40. [40]
    —, and R. Ree: Diagonals of doubly stochastic matrices. Quart. J. Math., Oxford II. Ser., 10, 296–302 (1959).Google Scholar
  41. [41]
    Mauldon, J. G.: Extreme points of convex sets of doubly-stochastic matrices. Not yet published.Google Scholar
  42. [42]
    Mendelsohn, N. S., and A. L. Dulmage: The convex hull of sub-permutation matrices. Proc. Amer. math. Soc. 9, 253–254 (1958).Google Scholar
  43. [43]
    Mirsky, L.: Proofs of two theorems on doubly-stochastic matrices. Proc. Amer. math. Soc. 9, 371–374 (1958).Google Scholar
  44. [44]
    —: Matrices with prescribed characteristic roots and diagonal elements. J. London math. Soc. 33, 14–21 (1958).Google Scholar
  45. [45]
    —: On a convex set of matrices. Arch. der Math. 10, 88–92 (1959).Google Scholar
  46. [46]
    —: Inequalities for certain classes of convex functions. Proc. Edinburgh math. Soc. 11, 231–235 (1959).Google Scholar
  47. [47]
    —: Symmetric gauge functions and unitarily invariant norms. Quart. J. Math., Oxford II. Ser., 11, 50–59 (1960).Google Scholar
  48. [48]
    —: An existence theorem for infinite matrices. Amer. math. Monthly 68, 465–469 (1961).Google Scholar
  49. [49]
    —: Even doubly-stochastic matrices. Math. Ann. 144, 418–421 (1961).Google Scholar
  50. [50]
    Neumann, J. von: A certain zero-sum two-person game equivalent to the optimal assignment problem. Contributions to the Theory of Games, vol. II (Princeton, 1953), 5–12.Google Scholar
  51. [51]
    Ostrowski, A.: Sur quelques applications des fonctions convexes et concaves au sens de I. Schur. J. Math. pur. appl. IX. Sér., 31, 253–292 (1952).Google Scholar
  52. [52]
    Pólya, G.: Remark on Weyl's note: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. nat. Acad. Sci. USA 36, 49–51 (1950).Google Scholar
  53. [53]
    Rado, R.: An inequality. J. London math. Soc. 27, 1–6 (1952).Google Scholar
  54. [54]
    Rattray, B. A., and J. E. L. Peck: Infinite stochastic matrices. Trans. roy. Soc. Canada, Sect. III, III. Ser., 49, 55–57 (1955).Google Scholar
  55. [55]
    Révész, P.: Seminar on Random Ergodic Theory. (Mathematical Institute, University of Aarhus, 1961.)Google Scholar
  56. [56]
    Schreiber, S.: On a result of S. Sherman concerning doubly stochastic matrices. Proc. Amer. math. Soc. 9, 350–353 (1958).Google Scholar
  57. [57]
    Schur, I.: über eine Klasse von Mittelbildungen mit Anwendung auf die Determinantentheorie. S.-Ber. Berliner math. Ges. 22, 9–20 (1923).Google Scholar
  58. [58]
    Sherman, S.: On a conjecture concerning doubly stochastic matrices. Proc. Amer. math. Soc. 3, 511–513 (1952).Google Scholar
  59. [59]
    —: A correction to ‘On a conjecture concerning doubly stochastic matrices’. Proc. Amer. math. Soc. 5, 998–999 (1954).Google Scholar
  60. [60]
    —: Doubly stochastic matrices and complex vector spaces. Amer. J. Math. 77, 245–246 (1955).Google Scholar
  61. [61]
    Vogel, W.: Lineare Programme und allgemeine Vertretersysteme. Math. Z. 76, 103–115 (1961).Google Scholar
  62. [62]
    Waerden, B. L. van der: Aufgabe 45. J.-Ber. Deutsch. Math.-Verein. 35, 117 (1926).Google Scholar

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© Springer-Verlag 1963

Authors and Affiliations

  • L. Mirsky
    • 1
  1. 1.Department of Pure MathematicsThe UniversitySheffield 10England

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