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凡得瓦登猜想

維基百科,自由的百科全書

線性代數中,Van der Waerden猜想是一個關於積和式的命題,其具體內容如下:

對於任意的雙轉移矩陣,其積和式的值大於等於

注意到該下界在所有元素均為時成立。該猜想由Bartel Leendert van der Waerden英語Bartel Leendert van der Waerden在1926年提出[1],在1980年由B. Gyires[2],1981年由G. P. Egorychev[3]和D. I. Falikman[4]獨立證明,其中Egorychev的證明用到了Alexandrov–Fenchel不等式英語Alexandrov–Fenchel inequality[5]由於這項工作,Egorychev和Falikman贏得了1982年的Fulkerson獎英語Fulkerson Prize[6]

參考

[編輯]
  1. ^ van der Waerden, B. L., Aufgabe 45, Jber. Deutsch. Math.-Verein., 1926, 35: 117 .
  2. ^ Gyires, B., The common source of several inequalities concerning doubly stochastic matrices, Publicationes Mathematicae Institutum Mathematicum Universitatis Debreceniensis, 1980, 27 (3-4): 291–304, MR 0604006 .
  3. ^ Egoryčev, G. P., Reshenie problemy van-der-Vardena dlya permanentov, Krasnoyarsk: Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz.: 12, 1980, MR 0602332 (俄語) . Egorychev, G. P., Proof of the van der Waerden conjecture for permanents, Akademiya Nauk SSSR, 1981, 22 (6): 65–71, 225, MR 0638007 (俄語) . Egorychev, G. P., The solution of van der Waerden's problem for permanents, Advances in Mathematics, 1981, 42 (3): 299–305, MR 0642395, doi:10.1016/0001-8708(81)90044-X .
  4. ^ Falikman, D. I., Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Akademiya Nauk Soyuza SSR, 1981, 29 (6): 931–938, 957, MR 0625097 (俄語) .
  5. ^ Brualdi (2006) p.487
  6. ^ Fulkerson Prize頁面存檔備份,存於互聯網檔案館), Mathematical Optimization Society, retrieved 2012-08-19.