P tt ts r - - PowerPoint PPT Presentation

✶

P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❡①tr❡♠❛❧ ♣r♦❜❧❡♠s ✐♥ ❞✐s❝r❡t❡ ❣❡♦♠❡tr② ♦♥ ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡

◆✐❦♦❧❛✐ ❆. ❑✉❦❧✐♥ 30.08.2017

✷ m ✖ ❞✐♠❡♥s✐♦♥; m ≥ 2, q1, q2 =

m

• k=1

q(k)

1 q(k) 2 , q1, q2 ∈ Rm;

Sm−1 =

• q ∈ Rm

q, q − 1 = 0

• , em = (0, 0, . . . , 0, 1) ∈ Sm−1;

n ✖ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts; n ≥ 2, (Sm−1)n = Sm−1 × · · · × Sm−1; h: [−1, 1] → (−∞, ∞] ✖ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥; Wh(q1, q2, . . . , qn) =

• 1≤i<j≤n

h (qi, qj) : (Sm−1)n → (−∞, ∞].

✸

❚❤♦♠s♦♥ ♣r♦❜❧❡♠

P♦s❡❞ ❜② ❏.❏. ❚❤♦♠s♦♥ (1904) ❢♦r m = 3. φ(t) =

• (2 − 2t)−1/2,

t ∈ [−1, 1), ∞, t = 1. ωφ = min

• Wφ(q1, q2, . . . , qn) | q1, q2, . . . , qn ∈ S2

= ?

✹

❚❤♦♠s♦♥ ♣r♦❜❧❡♠

n = 2 n = 3 n = 4 n = 5 n = 6 n = 12 ❘.❊. ❙❝❤✇❛rt③ (2010) ❱.❆. ❨✉❞✐♥ (1993) ◆.◆. ❆♥❞r❡❡✈ (1996)

✺

❚❤♦♠s♦♥ ♣r♦❜❧❡♠

❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥

T ⊂ [−1, 1) ✖ ✜♥✐t❡ s❡t; hT ✖ ✉♥✐✈❛r✐❛t❡ ♣♦❧✐♥♦♠✐❛❧ ✇✐t❤:

• hT (t) = φ(t),

t ∈ T;

• h′

T (t) = φ′(t),

t ∈ T \ {−1} ;

• deg(hT ) = |T| + |T \ {−1}| − 1.

❈❧❛✐♠ ✭❱.❆. ❨✉❞✐♥ & ◆.◆. ❆♥❞r❡❡✈✮

hT (t) ≤ φ(t), t ∈ [−1, 1]; WhT ≤ Wφ ♦♥ (S2)n, ωhT ≤ ωφ; min

• WhT (q1, q2, . . . , qn) | q1, q2, . . . , qn ∈ S2

= ?

✻

❚❤♦♠s♦♥ ♣r♦❜❧❡♠

n = 2 n = 3 n = 4 T = {−1} T = {− 1

2}

T = {− 1

3}

deg(hT ) = 0 deg(hT ) = 1 deg(hT ) = 1 n = 5 n = 6 n = 12 T = {−1, − 1

2, 0}

T = {−1, 0} T = {−1, −

√ 5 5 , √ 5 5 }

deg(hT ) = 4 deg(hT ) = 2 deg(hT ) = 4

✼

P❛❝❦✐♥❣ ♣r♦❜❧❡♠

s ∈ [−1, 1); τ(m, s) = max {n | ∃ q1, q2, . . . , qn ∈ Sm−1 : qi, qj ≤ s, i = j}. s = 1 − 2

• R

1 + R 2 ; R =

• 2

1 − s − 1 −1 .

❑✐ss✐♥❣ ♥✉♠❜❡r ♣r♦❜❧❡♠

τ(m) = τ(m, 1/2).

✽

❑✐ss✐♥❣ ♥✉♠❜❡r ♣r♦❜❧❡♠

τ(2) = 6; ❑. ❙❝❤ütt❡ & ❇.▲. ✈❛♥ ❞❡r ❲❛❡r❞❡♥ (1953): τ(3) = 12; ❱.■. ▲❡✈❡♥s❤t❡✐♥; ❆.▼. ❖❞❧②③❦♦ & ◆.❏.❆. ❙❧♦❛♥❡ (1978): τ(8) = 240, τ(24) = 196560; ❖.❘. ▼✉s✐♥ (2003): τ(4) = 24; 40 ≤ τ(5) ≤ 44 (❍.❉. ▼✐tt❡❧♠❛♥ & ❋. ❱❛❧❧❡♥t✐♥, 2009, ❜❛s❡❞ ♦♥ ✇♦r❦ ♦❢ ❈. ❇❛❝❤♦❝ & ❋. ❱❛❧❧❡♥t✐♥, 2006).

✾

P❛❝❦✐♥❣ ♣r♦❜❧❡♠

❚❤❡♦r❡♠

❋♦r n ≥ 3 t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t:

• τ(m, s) < n;
• t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ h s✉❝❤ t❤❛t

✶ h(t) ≤ 0, t ∈ [−1, s]; ✷ Wh > 0 ♦♥ (Sm−1)n.

❈♦♥❥❡❝t✉r❡

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐s s✉✐t❛❜❧❡ ❢♦r m = 3, s = 1/2, n = 13: h(t) = (t + 1)

• t + 3

5 2 t + 1 5 2 t − 1 2

• .

✶✵

P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

❋♦r ❣✐✈❡♥ m ≥ 2, n ≥ 2 ❛♥❞ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ h ✜♥❞ ωh = min

• Wh(q1, q2, . . . , qn) | q1, q2, . . . , qn ∈ Sm−1

; Wh(q1, q2, . . . , qn) =

• 1≤i<j≤n

h (qi, qj). ◆♦t❡ t❤❛t Wh ✐s ♠✉❧t✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ mn ✈❛r✐❛❜❧❡s ✇✐t❤ deg(Wh) = 2 deg(h). ❈♦♥❞✐t✐♦♥s q1, q2, . . . , qn ∈ Sm−1 ❛r❡ ♣♦❧②♥♦♠✐❛❧ t♦♦: qj, qj − 1 = 0, 1 ≤ j ≤ n. ❲❡ ❝❛♥ ✜① ♦♥❡ ♣♦✐♥t: ωh = min

• Wh(q1, q2, . . . , qn−1, em) | q1, q2, . . . , qn−1 ∈ Sm−1

.

✶✶

❙❡♠✐❞❡✜♥✐t❡ ♣r♦❣r❛♠♠✐♥❣ (❙❉P)

A0, A1, . . . , Ap ✖ ❣✐✈❡♥ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ♦❢ s✐③❡ l × l; c1, c2, . . . , cp ∈ R;      max X • A0; X • Ai = ci, 1 i p; X 0; X • Y =

l

• i=1

l

• j=1

XijYij.

✶✷

❙✉♠ ♦❢ sq✉❛r❡s

❈❧❛✐♠

▲❡t A ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ R✲❛❧❣❡❜r❛, f ∈ A. ❋♦r ❛ ❧✐♥❡❛r s✉❜s♣❛❝❡ L = span {b1, . . . , bl} ⊂ A t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t: (1) f =

r

• k=1

f2

k,

f1, . . . , fr ∈ L; (2) f =

l

• i=1

l

• j=1

Yijbibj ❢♦r s♦♠❡ ♠❛tr✐① Y 0 ♦❢ s✐③❡ l × l.

✶✸

❈♦♥✈❡① r❡❧❛①❛t✐♦♥

P♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ωh = min

• Wh(q1, q2, . . . , qn−1, em) | q1, q2, . . . , qn−1 ∈ Sm−1

❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❉P ♣r♦❜❧❡♠s (d ≥ deg(h)):         

• ωh(d) = sup γ,

γ ∈ R; Wh(q1, . . . , qn−1, em) − γ = Q0 +

n−1

• j=1

(Q+

j − Q− j ) · (qj, qj − 1);

Q±

j ✐s s✉♠ ♦❢ sq✉❛r❡s ♦❢ ❡❧❡♠❡♥ts ♦❢ A≤d, 0 ≤ j ≤ n − 1.

A = R[q(k)

j ]1≤j≤n−1, 1≤k≤m;

A≤d = {f ∈ A | deg(f) ≤ d}.

• ωh(d) ≤ ωh.

✶✹ ❲❡ ❝❛♥ ✉s❡ ●rö❜♥❡r ❜❛s❡s (❲. ●rö❜♥❡r, ❇. ❇✉❝❤❜❡r❣❡r, 1965):      ωh(d) = sup γ, γ ∈ R; Wh(q1, . . . , qn−1, em) − γ = Q0 (mod I); Q0 ✐s s✉♠ ♦❢ sq✉❛r❡s ♦❢ ❡❧❡♠❡♥ts ♦❢ (A / I)≤d; I =   

n−1

• j=1

fj · (qj, qj − 1)

• fj ∈ A

   ⊂ A, A / I ✐s q✉♦t✐❡♥t ❛❧❣❡❜r❛, (A / I)≤d ✐s ✐♠❛❣❡ ♦❢ A≤d ❜② q✉♦t✐❡♥t ♠❛♣.

• ωh(d) ≤ ωh(d) ≤ ωh.

❙❝❤♠ü❞❣❡♥ P♦s✐t✐✈st❡❧❧❡♥s❛t③ (❑. ❙❝❤♠ü❞❣❡♥, 1991)

ωh(d) ր ωh, d → ∞.

✶✺

❈♦♠♣❧❡①✐t②

❚❤❡ ❝♦♠♣❧❡①✐t② (s✐③❡ ♦❢ ♠❛tr✐❝❡s) ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❞❡♣❡♥❞s ♦♥ dim (A≤d) = ((n − 1)m + d)! ((n − 1)m)! d! ≍ d(n−1)m, d → ∞; dim ((A / I)≤d) ≍ d(n−1)(m−1), d → ∞.

✶✻

❙②♠♠❡tr②

❚❤❡ ♣♦❧②♥♦♠✐❛❧ Wh(q1, q2, . . . , qn−1, em), Wh(q1, . . . , qn) =

• 1≤i<j≤n

h (qi, qj), ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r:

• ❢♦r σ ∈ S(n − 1) (❜✐❥❡❝t✐♦♥ ♦❢ t❤❡ s❡t {1, 2, . . . , n − 1}):

Wh(qσ(1), qσ(2), . . . , qσ(n−1), em) = Wh(q1, q2, . . . , qn−1, em);

• ❢♦r ρ ∈ O(m − 1) (♦rt❤♦❣♦♥❛❧ ♠❛tr✐①) ✇❡ ❤❛✈❡

Wh(ρq1, ρq2, . . . , ρqn−1, em) = Wh(q1, q2, . . . , qn−1, em). ❙♦ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❣r♦✉♣ G = S(n − 1) × O(m − 1).

✶✼

■rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s

■rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❣r♦✉♣s S(n − 1) ❛♥❞ O(m − 1) ❛r❡ ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡, s♦ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ G ❛r❡ ❥✉st t❡♥s♦r ♣r♦❞✉❝ts ♦❢ S(n − 1)✬s ❛♥❞ O(m − 1)✬s ♦♥❡s✳

• ■rr❡♣s ♦❢ S(n − 1) ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❙♣❡❝❤t ♠♦❞✉❧❡s

(❲. ❙♣❡❝❤t, 1935);

• ✐rr❡♣s ♦❢ O(2) ❛r❡ s✐♥❡s✲❝♦s✐♥❡s + ❞❡t✲r❡♣r❡s❡♥t❛t✐♦♥;
• ✐rr❡♣s ♦❢ O(3) ❛r❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s + ❞❡t✲r❡♣r❡s❡♥t❛t✐♦♥s;
• · · ·

✶✽

■rr❡❞✉❝✐❜❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥

❲❡ ❝❛♥ ♦❜t❛✐♥ ✐rr❡❞✉❝✐❜❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ A / I ❜② ✉s✐♥❣ r❡❛❧ ✈❡rs✐♦♥ ♦❢ P❡t❡r✲❲❡②❧ t❤❡♦r❡♠ (❋. P❡t❡r & ❍. ❲❡②❧, 1927) ❢♦r G: A / I =

∞

• ν

Nν

• λ

V λ

ν , V i ν ∼

= V j

ν .

▲❛r❣❡ ❞❡♥s❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡ ♠❛tr✐① ❜❡❝♦♠❡s ❜❧♦❝❦ ❞✐❛❣♦♥❛❧:

• ❡❛❝❤ ν ❝♦rr❡s♣♦♥❞s t♦ ♦♥❡ ❜❧♦❝❦;
• t❤❡ s✐③❡ ♦❢ ν✲t❤ ❜❧♦❝❦ ✐s Nν × Nν;
• t❤✐s ♣r♦❝❡ss ❝♦rr❡s♣♦♥❞s t♦ s❡❧❡❝t✐♦♥ ♦❢ s②♠♠❡tr②✲❛✇❛r❡ ❜❛s✐s

♦❢ A / I. ❆❢t❡r s②♠♠❡tr② r❡❞✉❝t✐♦♥ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♥♦ ❧♦♥❣❡r ❞❡♣❡♥❞s ♦♥ m ❛♥❞ n, ✐t ❞❡♣❡♥❞s ♦♥❧② ♦♥ d.

✶✾

❘❡s✉❧ts

❲✐t❤ ❝✉rr❡♥t ♠❛t❤❡♠❛t✐❝❛❧ s♦❢t✇❛r❡ ✐t ✐s ❛❜❧❡ t♦ s♦❧✈❡ ♣♦❧②♥♦♠✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❢♦r d ≤ 3, ✐.❡. t♦ s♦❧✈❡ ❚❤♦♠s♦♥ ♣r♦❜❧❡♠ ❢♦r n = 2, 3, 4, 6 ❛♥❞ t♦ ♦❜t❛✐♥ ❧♦✇❡r ❜♦✉♥❞s ❢♦r n = 5, 7.

✷✵

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥!