Available in electronic format
Available in print format		 St.Petersburg Mathematical Journal
St.Petersburg Mathematical Journal
ISSN: 1547-7371(e) ISSN: 1061-0022(p)
Previous issue | Table of contents | Next issue
Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Birkhoff's theorem for a family of probability spaces

Author(s): Y. Safarov
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 5.
Journal: St. Petersburg Math. J. 17 (2006), 797-813.
MSC (2000): Primary 05C50, 60C05
Posted: July 20, 2006
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Birkhoff's theorem on doubly stochastic matrices is extended to some countable families of discrete probability spaces with nonempty intersections.

References:

[An]
T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118 (1989), 163-248. MR 0995373 (90g:15034)

[Bi1]
G. Birkhoff, Three observations on linear algebra, Univ. Nac. Tucumán. Revista A 5 (1946), 147-151. (Spanish) MR 0020547 (8:561a)

[Bi2]
-, Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, rev. ed., Amer. Math. Soc., New York, 1948. MR 0029876 (10:673a)

[BR]
B. R. Bapat and T. E. S. Raghavan, Nonnegative matrices and applications, Encyclopedia Math. Appl., vol. 64, Cambridge Univ. Press, Cambridge, 1997. MR 1449393 (98h:15038)

[BS]
J. R. Brown and R. C. Shiflett, On extreme doubly stochastic measures, Michigan Math. J. 17 (1970), 249-254. MR 0265542 (42:451)

[CLMST]
R. M. Caron, X. Li, P. Mikusinski, H. Sherwood, and M. D. Taylor, Nonsquare ``doubly stochastic'' matrices, Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, Inst. Math. Statist., Hayward, CA, 1996, pp. 65-75. MR 1485523 (99b:15025)

[Do]
R. G. Douglas, On extremal measures and subspace density, Michigan Math. J. 11 (1964), 243-246. MR 0185427 (32:2894)

[Fe]
D. Feldman, Doubly stochastic measures: three vignettes, Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, Inst. Math. Statist., Hayward, CA, 1996, pp. 84-96. MR 1485525 (98k:28031)

[Gr]
R. Grzaslewicz, On extreme infinite doubly stochastic matrices, Illinois J. Math. 31 (1987), 529-543. MR 0909782 (88i:15046)

[Ho]
A. J. Hoffman, A special class of doubly stochastic matrices, Aequationes Math. 2 (1969), 319-326. MR 0263848 (41:8447)

[Is]
J. R. Isbell, Birkhoff's problem $ 111$, Proc. Amer. Math. Soc. 6 (1955), 217-218. MR 0068507 (16:893f)

[K]
G. Köthe, Topological vector spaces. I, Grundlehren Math. Wiss., vol. 159, Springer-Verlag New York, Inc., New York, 1969. MR 0248498 (40:1750)

[Ka]
M. Katz, On the extreme points of a certain convex polytope, J. Combin. Theory 8 (1970), 417-423. MR 0255582 (41:243)

[Ke]
D. G. Kendall, On infinite doubly-stochastic matrices and Birkhoff's problem, J. London Math. Soc. 35 (1960), 81-84. MR 0110947 (22:1815)

[KST]
A. Kaminski, H. Sherwood, and M. D. Taylor, Doubly stochastic measures with mass on the graphs of two functions, Real Anal. Exchange 28 (1987/88), no 1, 253-257. MR 0923729 (89f:28005)

[Le]
G. Letac, Représentation des mesures de probabilité sur le produit de deux espaces dénombrables, de marges données, Illinois J. Math. 10 (1966), 497-507. MR 0200957 (34:842)

[Li]
J. Lindenstrauss, A remark on extreme doubly stochastic measures, Amer. Math. Monthly 72 (1965), 379-382. MR 0181728 (31:5955)

[LLL]
J. L. Lewandowski, C. L. Liu, and J. W.-S. Liu, An algorithmic proof of a generalization of the Birkhoff-von Neumann theorem, J. Algorithms 7 (1986), no. 3, 323-330. MR 0855560 (87j:15042)

[LMST]
X. Li, P. Mikusinski, H. Sherwood, and M. D. Taylor, In quest of Birkhoff's theorem in higher dimensions, Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, Inst. Math. Statist., Hayward, CA, 1996, pp. 187-197. MR 1485531 (99b:15027)

[Mu]
H. G. Mukerjee, Supports of extremal measures with given marginals, Illinois J. Math. 29 (1985), 248-260. MR 0784522 (86j:28012)

[Ro]
J. V. Romanovsky, A simple proof of the Birkhoff-von Neumann theorem on bistochastic matrices, A Tribute to Ilya Bakelman (College Station, TX, 1993), Discourses Math. Appl., vol. 3, Texas A and M Univ., College Station, TX, 1994, pp. 51-53. MR 1423368 (98c:15067)

[RP]
B. A. Rattray and J. E. L. Peck, Infinite stochastic matrices, Trans. Roy. Soc. Canada Sect. III (3) 49 (1955), 55-57. MR 0075621 (17:778f)

[Ru]
W. H. Ruckle, Sequence spaces, Res. Notes Math., vol. 49, Pitman (Advanced Publishing Program), Boston, MA-London, 1981. MR 0634231 (83g:46012)

[Sa]
Yu. Safarov, Birkhoff's theorem and multidimensional spectra of self-adjoint operators, J. Funct. Anal. 222 (2005), 61-97. MR 2129765

[ST]
H. Schreck and G. Tinhofer, A note on certain subpolytopes of the assignment polytope associated with circulant graphs, Linear Algebra Appl. 111 (1988), 125-134. MR 0974049 (90e:05031)

[Ti]
G. Tinhofer, Strong tree-cographs are Birkhoff graphs, Discrete Appl. Math. 22 (1988/89), no. 3, 275-288. MR 0983251 (90a:05165)

[Vi]
R. A. Vitale, Parametrizing doubly stochastic measures, Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, Inst. Math. Statist., Hayward, CA, 1996, pp. 358-364. MR 1485544 (98k:28014)

Similar Articles:

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 05C50, 60C05

Retrieve articles in all Journals with MSC (2000): 05C50, 60C05

Additional Information:

Y. Safarov
Affiliation: Department of Mathematics, King's College, Strand, London, United Kingdom
Email: yuri.safarov@kcl.ac.uk

DOI: 10.1090/S1061-0022-06-00930-7
PII: S 1061-0022(06)00930-7
Keywords: Stochastic matrices, weighted graphs, Birkhoff's theorem
Received by editor(s): 10/MAR/2005
Posted: July 20, 2006
Dedicated: To the memory of O. A. Ladyzhenskaya
Copyright of article: Copyright 2006, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google