Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dubna/2015/notes/musin-slides.pdf Дата изменения: Tue Jul 28 18:30:38 2015 Дата индексирования: Sun Apr 10 13:13:22 2016 Кодировка: Поисковые слова: п п п п п п п п п п п п

:

. .
UTB

­2015






. . , : . : , 1989 . . , , : , 1999.




(1910, 1912)

. f : B n B n x , . . f (x ) = x . B n n- .




(1928)

( ) n- ,





1

2 1 2 3 2 1 3 2

1

1 3

.: 2-




KKM (1929)

KKM (Knaster­Kuratowski­Mazurkiewicz) , d - d A1 , A2 , . . . , Ad +1 d + 1 {Ci } i I := {1, 2, . . . , d + 1}, Ik I d Ai , i Ik , Ci , i Ik . Ci . , C1 , C2 , C3 Ai Ci Ai Aj Ci Cj , Ci .




­ (1930)

, F1 , . . . , Fn+1 Sn n + 1 . Fi , (x , -x ). , Fi (-Fi ) = . ( .)




­ (1933)

(1) f : Sn R x Sn , f (x ) = f (-x ).

n

, f : Sn Rn , f (-x ) = -f (x ). (2) f : Sn Rn x Sn , f (x ) = 0, .. Zf = {f -1 (0)} .




J. Matou Using the Borsuk-Ulam theorem, Springer-Verlag, sek, Berlin, 2003.




­




(1941)

X Y F : X 2Y . (Y ) 2Y , Y Rm , F : X (Y ). : X Rn , , F : X (X ) , . F x X , x F (x ).




(1945)

() ­ Bd , . L : V () {+1, -1, +2, -2, . . . , +d , -d } , . . L(-v ) = -L(v ) v Bd "" ("complementary edge"): .











­ Sd . , L : V () {+1, -1, +2, -2, . . . , +d , -d } , . . : L(-v ) = -L(v ). (complimentary edge).




(1950)
() S1 , . . . , Sn () u1 , . . . , un X = S1 в . . . в Sn . , Si ­ , ui si . . s1 , . . . , sn i si Si :
ui (s1 , . . . , si 1 , si , si 1 , sn ) ui (s1 , . . . , s - + i -1

, si , s

i +1

, sn ).




. .

T . . , . . . , L : V (T ) {+1, -1, +2, -2, +3, -3} , . . L(-v ) = -L(v ) v . , . a, b, c , |a| = 1, |b| = 2, |c | = 3, T (a, b, c ) (-a, -b, -c ) ­ .




. .
-1


2


-3


-2

3 1 2

1


-2


-1



3




-1


-3



3



-2

1

.:




. .

(a, b, c ) = (1, 2, 3), (1, -2, 3), (1, 2, -3) (1, -2, -3) , (a, b, c ) (-a, -b, -c ) ­ . SN(a, b, c ). : SN(1, 2, 3) = 3, SN(1, -2, 3) = 1, SN(1, 2, -3) = 3, SN(1, -2, -3) = 3.






p (1, -2)+p (-2, 3)+p (3, -1)+p (-1, 2)+p (2, -3)+p (-3, 1) 1(2) d (1, -2) + d (-2, 3) + d (3, -1) 1 (mod 2) + 7 (a, b) (-a, -b) () " " (1, -2), (-2, 3), (3, -1), (1, -2, -3) (-1, 2, 3). ()






McLennan, Tourky (2008) + . . area() = (x2 - x1 )(y3 - y1 ) - (y2 - y1 )(x3 - x1 ) 2






p (t ) := (x + (a - x )t , y + (b - y )t ), p (0) = (x , y ), p (1) = (a, b). area(, t ) - . S (t ) =
T

area(, t )

S (t ) =const= A1 A2 A3 =1. 1 = S (1) =
i

area(Spi )






J. A. De Loera, E. Peterson, and F. E. Su, A Polytopal Generalization of Sperner's Lemma, J. of Combin. Theory Ser. A, 100 (2002), 1-26. T P Rd n . L : V (T ) {1, 2, . . . , n} (n - d ) . . d ­.




MЁ obius band

.: MЁ us band. Diametrically opposite points of the inner boundary obi circle are to be identified. The outer circle is the boundary of the MЁ obius band.




Sperner's lemma for the MЁ obius band

1

2 1 2 3 2 1 3 2

1

1

3




Degree and Sperner's lemma

Let P be a convex polytope in Rd with vertices p1 , . . . , pn . Let X be a finite orientable d -dimensional simplicial complex. Let L : V (X ) {1, 2, . . . , n} be a labelling such that fL,P ( X ) P . Then there are at least (n - d )| deg(fL,P )| fully labelled d -simplices.




Degree and Sperner's lemma

Corollary Let X be a finite orientable d -dimensional simplicial complex. Let L : V (X ) {1, 2, . . . , d + 1} be any labelling. Then X contains at least | deg(fL,P )| fully labelled d -simplices. For Sperner's labelling deg(fL,P ) = 1. The theorem also implies Tucker's lemma. In this case P is a crosspolytope and deg(fL,P ) is odd.




1

2 3

3 2

1

2 3

1

2

2

3

3

.: deg(L, T ) = 3. There are three fully labelled triangles.




2 1 3 1 4 4 3 2 3 1 4

1

4 2 3 3

1

2

.: Octagon with two square holes. Here n = 4, deg(L, T )| = 4 and there are eight fully labelled triangles




2 -1 -2 -2 1 2 -1 -2

1 -2 -1

-2 1 -2

2

.: Since deg(L, T ) = 3, there are three complementary edges.




D 4 4 3 4 3 3 4 3 3 3

C

1

1

1

2

2

1 A

2

2

1

2 B

Sperner's labelling of 2 (4, 3). One edge is colored with (1, 3).




1 2 3 3 2 1

1 1

3 1

2 1

1 1

2 1

4
qqqqq qqqqq qqqqq

1

4

1

1

2

3

4

1

Since deg(L, Q ) = 2, there are two centrally labelled cells.




Sperner type lemma for quadrangulations

Let C d denote the d -dimensional cube. Theorem Let Q be a quadrangulation of an oriented d -dimensional manifold M . Suppose L : V (Q ) V (C d ) be a labelling such that fL ( Q ) C d . Then Q contains at least | deg(L, Q )| centrally labelled cells.