t r rs tr t - - PowerPoint PPT Presentation

❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✐♥ ❡❧❧✐♣t✐❝ t❤❡♦r②

❆♥t♦♥ ❙❛✈✐♥

P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮

✸✵✳✵✺✳✷✵✶✼

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶ ✴ ✺✹

■✳ ❘❡❧❛t✐✈❡ ❡❧❧✐♣t✐❝ t❤❡♦r②

❲❡ ❝♦♥s✐❞❡r ❡❧❧✐♣t✐❝ t❤❡♦r② ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ♣❛✐r (M, X)✱ ✇❤❡r❡

M ✐s ❛ s♠♦♦t❤ ♠❛♥✐❢♦❧❞❀ X ✐s ❛ s♠♦♦t❤ s✉❜♠❛♥✐❢♦❧❞ ✭♦❢ ❛r❜✐tr❛r② ❞✐♠❡♥s✐♦♥✮✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷ ✴ ✺✹

✶✳ ▼♦t✐✈❛t✐♦♥ ❢r♦♠ ♣❤②s✐❝s✳

▲❡❢t✿ ❛ ♥❡❡❞❧❡ ♣❛ss❡s t❤r♦✉❣❤ ❛ t❤✐♥ ✜❧♠ ✭❡❧❛st✐❝✐t② t❤❡♦r②✮✳ ❘✐❣❤t✿ ❛ ♥❡❡❞❧❡ ❤♦❧❞s ❛ r✐❣✐❞ ♣❧❛t❡ ✭♣❧❛st✐❝✐t② t❤❡♦r②✮✳ ▲❡t ✉s ❞❡s❝r✐❜❡ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ♠❛t❤❡♠❛t✐❝❛❧❧②✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸ ✴ ✺✹

■♥ R3 ✇❡ ❝♦♥s✐❞❡r ❛ ✷❉ r✐❣✐❞ ♣❧❛t❡ ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥t♦✉r Γ✳ ❚❤❡ ♣❧❛t❡ ✐s s✉♣♣♦rt❡❞ ❛t t❤❡ ♣♦✐♥t (x0, y0) ∈ R2✳ ❚❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣❧❛t❡ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ♣r♦❜❧❡♠        ∆2u ≡ 0 mod (x0, y0),

u|Γ = ϕ,

∂u ∂n

• Γ = ψ.

✭✶✮ ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥s ϕ, ψ ❛r❡ ❞❡✜♥❡❞ ♦♥ t❤❡ ❝♦♥t♦✉r Γ✱ n ✐s t❤❡ ♥♦r♠❛❧ ✈❡❝t♦r t♦ Γ✳ ❚❤❡ ❡q✉❛t✐♦♥ ❤♦❧❞s ❡①❝❡♣t t❤❡ ♣♦✐♥t (x0, y0)✱ ✐✳❡✳ t❤❡ ❞✐str✐❜✉t✐♦♥ ∆2u ✐s s✉♣♣♦rt❡❞ ❛t (x0, y0)✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ ❝♦♥t✐♥✉♦✉s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r♠

r2 ln r✱ ❤❡♥❝❡✱ ✇❡ ❝❛♥ ❡q✉✐♣ t❤❡ ❝♦♠♣❛r✐s♦♥ ✭✶✮ ✇✐t❤ ❜♦✉♥❞❛r②

❝♦♥❞✐t✐♦♥

u|(x0,y0) = u0

✭✷✮ ✇✐t❤ ❛ ❣✐✈❡♥ u0✳ Pr♦❜❧❡♠s ✇✐t❤ s✉❝❤ ❝♦♥❞✐t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❙♦❜♦❧❡✈ ♣r♦❜❧❡♠s✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹ ✴ ✺✹

✷✳ ❙t❛t❡♠❡♥t ♦❢ ❙♦❜♦❧❡✈ ♣r♦❜❧❡♠s

✭P❤❉ t❤❡s✐s✱ ✶✾✻✹✮

• M ✐s ❛ ❝❧♦s❡❞ s♠♦♦t❤ ♠❛♥✐❢♦❧❞ ✇✐t❤♦✉t ❜♦✉♥❞❛r②
• i : X ֒→ M ✐s ❛ s♠♦♦t❤ ❡♠❜❡❞❞✐♥❣ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ν✳

❙♦❜♦❧❡✈ ♣r♦❜❧❡♠

      

Du ≡ f mod X, u ∈ Hs(M), f ∈ Hs−m(M), i∗Bu = g, g ∈ Hs−b−ν/2(X).

✭✸✮

D ✐s ❛ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✭ψ❉❖✮ ♦❢ ♦r❞❡r m ♦♥ M✱ B

✐s ❛ ψ❉❖ ♦❢ ♦r❞❡r b ♦♥ M✳ ❝♦♠♣❛r✐s♦♥ ✐s ❝♦♥s✐❞❡r❡❞ ♠♦❞✉❧♦ s✉❜s♣❛❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥s s✉♣♣♦rt❡❞ ♦♥ X✳

i∗ : Hs(M) → Hs−ν/2(X)✱ i∗u(x) = u(i(x)) ✐s t❤❡ ❜♦✉♥❞❛r②

♦♣❡r❛t♦r✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥❞✐t✐♦♥s ❞❡♣❡♥❞s ♦♥ s✦

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺ ✴ ✺✹

✸✳ ❙♦❧✉t✐♦♥

▼♦r♣❤✐s♠s ✭♠❛tr✐① ♦♣❡r❛t♦rs✮

❙♦❜♦❧❡✈ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♠❛tr✐① ♦♣❡r❛t♦r

• D

i∗ i∗B

• ,

✇❤❡r❡ i∗ : Hs(X) −→ Hs−ν/2(M), s < 0 ✐s t❤❡ ❝♦❜♦✉♥❞❛r② ♦♣❡r❛t♦r ✭❞✉❛❧ t♦ i∗✮✳ ❆t✐②❛❤✱ ❉②♥✐♥✱ ❇♦✉t❡t ❞❡ ▼♦♥✈❡❧ ❈♦r♦❧❧❛r②✳ ❙♦❜♦❧❡✈ ♣r♦❜❧❡♠ (D, B) ✐s r❡❞✉❝❡❞ t♦ ♦♣❡r❛t♦r i∗BD−1i∗ ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ▲❡♠♠❛✳ ❚❤❡ ❧❛tt❡r ♦♣❡r❛t♦r ✐s ❛ ψ❉❖✳

❚❤❡♦r❡♠

✶✳ (D, B) ✐s ❡❧❧✐♣t✐❝ ⇔ D ✐s ❡❧❧✐♣t✐❝ ♦♥ ▼ ❛♥❞ i∗BD−1i∗ ✐s ❡❧❧✐♣t✐❝ ♦♥

X❀

✷✳ ■♥❞❡①✿ ind(D, B) = indD + indi∗BD−1i∗✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✻ ✴ ✺✹

✹✳ ❙❡✈❡r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥s

✶✳ ❙♦❜♦❧❡✈ ♣r♦❜❧❡♠s ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✳ ✷✳ ❚♦♣♦❧♦❣✐❝❛❧ ❛s♣❡❝ts ♦❢ tr❛❝❡s ◆♦✈✐❦♦✈ ❛♥❞ ❙t❡r♥✐♥ ✭✶✾✻✻✮ ✭❑✲t❤❡♦r②✮✳ ▲❡t D ❜❡ ❛♥ ♦♣❡r❛t♦r ♦♥ t❤❡ ❛♠❜✐❡♥t ♠❛♥✐❢♦❧❞ M✳ ■ts tr❛❝❡ ✐s t❤❡ ♦♣❡r❛t♦r ♦♥ t❤❡ s✉❜♠❛♥✐❢♦❧❞ X✿

i∗Di∗ : Hs(X) −→ Hs−m−ν(X).

✸✳ ❚r❛❝❡ ❛s ❛ ♠✐♥❝❡r✿ ◆❡✇ ❝❧❛ss❡s ♦❢ ♦♣❡r❛t♦rs✳ ❚r❛❝❡s ♦♥ s✉❜♠❛♥✐❢♦❧❞s ✇✐t❤ ❜♦✉♥❞❛r② ❣✐✈❡ ❱✐s❤✐❦✕❊s❦✐♥ ❡①t❡♥t✐♦♥ ❜② ③❡r♦✳ ❚r❛❝❡s ♦❢ s❤✐❢t ♦♣❡r❛t♦rs ❣✐✈❡ ❋♦✉r✐❡r✕▼❡❧❧✐♥ ♦♣❡r❛t♦rs✳✳✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✼ ✴ ✺✹

❘❡❢❡r❡♥❝❡s

❙t❡r♥✐♥✱ ❇✳ ❨✉✳ ❊❧❧✐♣t✐❝ ❛♥❞ ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❛ ❜♦✉♥❞❛r② ❝♦♥s✐st✐♥❣ ♦❢ ❝♦♠♣♦♥❡♥ts ♦❢ ❞✐✛❡r❡♥t ❞✐♠❡♥s✐♦♥✳ ❚r❛♥s✳▼♦s❝♦✇ ▼❛t❤✳ ❙♦❝✳✱ ✶✺✱ ✶✾✻✻✱ ✸✹✻✕✸✽✷✳ ◆♦✈✐❦♦✈✱ ❙✳ P✳❀ ❙t❡r♥✐♥✱ ❇✳ ❨✉✳ ❚r❛❝❡s ♦❢ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ♦♥ s✉❜♠❛♥✐❢♦❧❞s ❛♥❞ ❑✲t❤❡♦r②✳ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳ ✼✱ ✶✾✻✻✳ ◆♦✈✐❦♦✈✱ ❙✳ P✳❀ ❙t❡r♥✐♥✱ ❇✳ ❏✉✳ ❊❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ❛♥❞ s✉❜♠❛♥✐❢♦❧❞s✳ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳ ✼ ✶✾✻✻ ✶✺✵✽✕✶✺✶✷✳ ❙t❡r♥✐♥✱ ❇✳ ❨✉✳ ❊❧❧✐♣t✐❝ ✭❝♦✮❜♦✉♥❞❛r② ♠♦r♣❤✐s♠s✳ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳ ✽ ✶✾✻✼ ✹✶✕✹✺✳ ✭❘❡✈✐❡✇❡r✿ ▼✳ ❋✳ ❆t✐②❛❤✮ ❙t❡r♥✐♥✱ ❇✳ ❙✳ ▲✳ ❙♦❜♦❧❡✈ t②♣❡ ♣r♦❜❧❡♠s ✐♥ t❤❡ ❝❛s❡ ♦❢ s✉❜♠❛♥✐❢♦❧❞s ✇✐t❤ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s✐♥❣✉❧❛r✐t✐❡s✳ ✭❘✉ss✐❛♥✮ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❘✱ ✶✽✾✱ ✶✾✻✾✱ ✼✸✷✕✼✸✺✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✽ ✴ ✺✹

❙t❡r♥✐♥✱ ❇✳❨✉✳ ❊❧❧✐♣t✐❝ t❤❡♦r② ♦♥ ❝♦♠♣❛❝t ♠❛♥✐❢♦❧❞s ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✳ P✉❜❧✐s❤❡❞ ✐♥ ▼■❊▼✱ ▼♦s❝♦✇✱ ✶✾✼✹✳ ✶✵✽ ♣♣✳ ✭❘✉ss✐❛♥✮ ❙t❡r♥✐♥ ❇✳❨✉✳✱ ❙❤❛t❛❧♦✈ ❱✳❊✳✱ ✏❘❡❧❛t✐✈❡ ❡❧❧✐♣t✐❝ t❤❡♦r② ❛♥❞ t❤❡ ❙♦❜♦❧❡✈ ♣r♦❜❧❡♠✑✱ ❙❜✳ ▼❛t❤✳✱ ✶✽✼✿✶✶ ✭✶✾✾✻✮✱ ✶✻✾✶✕✶✼✷✵ ◆❛③❛✐❦✐♥s❦✐✐ ❱✳ ❊✳✱ ❙t❡r♥✐♥ ❇✳ ❨✉✳ ❘❡❧❛t✐✈❡ ❡❧❧✐♣t✐❝ t❤❡♦r② ✴✴ ❆s♣❡❝ts ♦❢ ❜♦✉♥❞❛r② ♣r♦❜❧❡♠s ✐♥ ❛♥❛❧②s✐s ❛♥❞ ❣❡♦♠❡tr②✱ ❖♣❡r❛t♦r ❚❤❡♦r②✿ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✖ ❇❛s❡❧✿ ❇✐r❦❤☎ ❛✉s❡r✱ ✷✵✵✹✳ ✖ ❱✳ ✶✺✶✱ ✖ P✳ ✹✾✺✕✺✻✵✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✾ ✴ ✺✹

■■✳ ❊❧❧✐♣t✐❝ t❤❡♦r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s

✶✳ ❆♥❛❧②t✐❝ ❛s♣❡❝ts✿ ❋r❡❞❤♦❧♠ ♣r♦♣❡rt②✱ ❛s②♠♣t♦t✐❝s ✭✐♥❝❧✉❞✐♥❣ r❡s✉r❣❡♥t ❛s②♠♣t♦t✐❝s✮ ❇♦♦❦s✿ ❇✳✲❲✳ ❙❝❤✉❧③❡✱ ❇✳❨✉✳ ❙t❡r♥✐♥✱ ❱✳❊✳ ❙❤❛t❛❧♦✈✱ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦♥ s✐♥❣✉❧❛r ♠❛♥✐❢♦❧❞s✳ ❙❡♠✐❝❧❛ss✐❝❛❧ t❤❡♦r② ❛♥❞ ♦♣❡r❛t♦r ❛❧❣❡❜r❛s✱ ❲✐❧❡②✲❱❈❍✱ ❲❡✐♥❤❡✐♠✱ ✶✾✾✽✱ ✸✼✻ ♣♣✳ ❱✳❊✳ ◆❛③❛✐❦✐♥s❦✐✐✱ ❇✳✲❲✳ ❙❝❤✉❧③❡✱ ❇✳❨✉✳ ❙t❡r♥✐♥✱ ◗✉❛♥t✐③❛t✐♦♥ ♠❡t❤♦❞s ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❚❛②❧♦r ❛♥❞ ❋r❛♥❝✐s✱ ▲♦♥❞♦♥✱ ✷✵✵✷✱ ✸✺✻ ♣♣✳ ✷✳ ❚♦♣♦❧♦❣✐❝❛❧ ❛s♣❡❝ts✿ s✉r❣❡r② t❡❝❤♥✐q✉❡s t♦ ❝♦♠♣✉t❡ ✐♥❞✐❝❡s✱ ❑✲❤♦♠♦❧♦❣② t❡❝❤♥✐q✉❡s t♦ ❝♦♠♣✉t❡ ❤♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✵ ✴ ✺✹

✶✳ ■♥❞❡① ❢♦r♠✉❧❛s ✭❙✉r❣❡r② ♠❡t❤♦❞s✮

M ✐s ❛ ♠❛♥✐❢♦❧❞ ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✳ D ✐s ❛♥ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r ♦♥ M✳

❉❡s✐r❡❞ ✐♥❞❡① ❢♦r♠✉❧❛✿ indD =

• ♦✈❡r str❛t❛

ind{❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ♦♥ str❛t❛}. ▼❡t❤♦❞✿ s✉r❣❡r② ✭❝✉t ❛♥❞ ♣❛st❡✮ ❊①❛♠♣❧❡ ⇒ ✐♥❞❡① ❢♦r♠✉❧❛

2indD = ind(♦♣❡r❛t♦r ♦♥ 2M) + ind(♦♣❡r❛t♦r ♦♥ s✉s♣❡♥s✐♦♥).

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✶ ✴ ✺✹

❘❡❢❡r❡♥❝❡s

❋♦r ♠♦r❡ ❞❡t❛✐❧s s❡❡✿ ◆❛③❛✐❦✐♥s❦✐✐✱ ❱✳❀ ❙❝❤✉❧③❡✱ ❇✳✲❲✳❀ ❙t❡r♥✐♥✱ ❇✳ ❚❤❡ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐♥ ✐♥❞❡① t❤❡♦r② ♦❢ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs✳ ❇✐r❦❤☎ ❛✉s❡r✴❙♣r✐♥❣❡r ❇❛s❡❧ ❆●✱ ❇❛s❡❧✱ ✷✵✶✹✳ ✈✐✐✐✰✶✶✼ ♣♣✳ ❙✉r❣❡r✐❡s ❛r❡ ❛❧s♦ ❛♣♣❧✐❡❞ t♦ ❝♦♠♣✉t❡ s♣❡❝tr❛❧ ✢♦✇s✱ ❡✳❣✳✱ s❡❡ ❑❛ts♥❡❧s♦♥✱ ▼✳■✳❀ ◆❛③❛✐❦✐♥s❦✐✐✱ ❱✳❊✳ ❚❤❡ ❆❤❛r♦♥♦✈✲❇♦❤♠ ❡✛❡❝t ❢♦r ♠❛ss❧❡ss ❉✐r❛❝ ❢❡r♠✐♦♥s ❛♥❞ t❤❡ s♣❡❝tr❛❧ ✢♦✇ ♦❢ ❉✐r❛❝✲t②♣❡ ♦♣❡r❛t♦rs ✇✐t❤ ❝❧❛ss✐❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤❡♦r❡t✳ ❛♥❞ ▼❛t❤✳ P❤②s✳ ✶✼✷ ✭✷✵✶✷✮✱ ♥♦✳ ✸✱ ✶✷✻✸✕✶✷✼✼✳ ✇❤❡r❡ t❤❡② ❛r❡ ❛♣♣❧✐❡❞ t♦ ❞❡s❝r✐❜❡ ❡❧❡❝tr♦♥ st❛t❡s ✐♥ ❣r❛♣❤❡♥❡✳ ❙♣❡❝tr❛❧ ✢♦✇ ✐s r❡❧❛t❡❞ t♦ t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥✲❤♦❧❡ ♣❛✐rs✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✷ ✴ ✺✹

✷✳ ❍♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ K✲❤♦♠♦❧♦❣②

M ✐s ❛ str❛t✐✜❡❞ ♠❛♥✐❢♦❧❞✳

❍♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠✿ ❍♦✇ ♠❛♥② ❞✐st✐♥❝t ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ✇❡ ❣❡t ♦♥ M❄ ◆❛t✉r❛❧ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ✭❆t✐②❛❤✕❙✐♥❣❡r✮✿ st❛❜❧❡ ❤♦♠♦t♦♣② ▲❡t D1✱ D2 ❜❡ t✇♦ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ♦♥ M ❛❝t✐♥❣ ♦♥ ✈❡❝t♦r ❜✉♥❞❧❡ s❡❝t✐♦♥s✳

❉❡✜♥✐t✐♦♥

D1 ❛♥❞ D2 ❛r❡

st❛❜❧② ❤♦♠♦t♦♣✐❝ ⇔ ❢♦r ✈❡❝t♦r ❜✉♥❞❧❡s E1 ❛♥❞ E2 ♦♥ M t❤❡r❡ ❡①✐sts ❛ ❤♦♠♦t♦♣②

D1 ⊕ 1E1 ∼ D2 ⊕ 1E2

❆ss✉♠♣t✐♦♥s✿ ❤♦♠♦t♦♣② t❤r♦✉❣❤ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs❀ ♦♣❡r❛t♦rs ♦❢ ♦r❞❡r ③❡r♦❀ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✸ ✴ ✺✹

❉❡✜♥✐t✐♦♥

Ell(M) := {❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ♦♥ M}/{st❛❜❧❡ ❤♦♠♦t♦♣②}

❘❡♠❛r❦✳ Ell(M) ✐s ❛ ❣r♦✉♣✳

Pr♦❜❧❡♠✿ ❝♦♠♣✉t❡ Ell✲❣r♦✉♣ ❢♦r ♦♣❡r❛t♦rs ♦♥ str❛t✐✜❡❞

♠❛♥✐❢♦❧❞s✳

❘❡♠❛r❦

■❢ M ✐s ❛ ❝❧♦s❡❞ s♠♦♦t❤ ♠❛♥✐❢♦❧❞✱ t❤❡♥

Ell(M) ≃ K 0

c (T ∗M)

✭✹✮ ✭❆t✐②❛❤✕❙✐♥❣❡r ❞✐✛❡r❡♥❝❡ ❝♦♥str✉❝t✐♦♥✮✳ ❇✉t ✭✹✮ ✐s ♥♦t ✈❛❧✐❞ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✦

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✹ ✴ ✺✹

❍♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭◆❛③❛✐❦✐♥s❦✐✐✱ ❙t❡r♥✐♥✱ ❛♥❞ ❙❛✈✐♥ ✷✵✵✻✮

❚❤❡r❡ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ Ell(M) ≃ K0(M) ❍❡r❡ K0 ✐s ❡✈❡♥ K✲❤♦♠♦❧♦❣② ❣r♦✉♣ ✭❣❡♥❡r❛❧✐③❡❞ ❤♦♠♦❧♦❣② t❤❡♦r② ❞✉❛❧ t♦ K✲t❤❡♦r②✮✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✺ ✴ ✺✹

❆♣♣❧✐❝❛t✐♦♥✳ ❖❜str✉❝t✐♦♥ t♦ ❋r❡❞❤♦❧♠ ♣r♦♣❡rt②

Pr♦❜❧❡♠✿ ▲❡t D ❜❡ ❛♥ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t M \ X✳ ❲❤❡♥ t❤❡r❡ ❡①✐sts D′ ❡❧❧✐♣t✐❝ ♦♥ M ✇✐t❤ t❤❡ s❛♠❡ ✐♥t❡r✐♦r s②♠❜♦❧ ♦♥ M \ X❄ ✭❝❢✳ t❤❡♦r② ♦❢ ❝❧❛ss✐❝❛❧ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✮

X ✖ s✐♥❣✉❧❛r✐t②

s❡t ♦❢ M ❙♦❧✉t✐♦♥ ✭❢♦r ♦♣❡r❛t♦rs ♠♦❞✉❧♦ st❛❜❧❡ ❤♦♠♦t♦♣②✮✿ ❈♦♥s✐❞❡r ❞✐❛❣r❛♠

Ell(M)

j ≃

• Ell(M \ X)

≃

• K0(M)

K0(M \ X)

∂

K1(X)

Ell(M \ X) ✖ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ♦♣❡r❛t♦rs ✇✐t❤ ❡❧❧✐♣t✐❝

✐♥t❡r✐♦r s②♠❜♦❧❀ ❧♦✇❡r r♦✇ ✖ ❡①❛❝t s❡q✉❡♥❝❡ ♦❢ ♣❛✐r X ⊂ M ✐♥ K✲❤♦♠♦❧♦❣②❀ ∂ ✖ ❜♦✉♥❞❛r② ♠❛♣ ♦❢ ❡①❛❝t s❡q✉❡♥❝❡✳ ❧✐❢t✐♥❣ [D′] ∈ Ell(M) ❡①✐sts ⇐⇒ ∂[D] = 0

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✻ ✴ ✺✹

▼❛♥✐❢♦❧❞s ✇✐t❤ ❝♦r♥❡rs

❉❡✜♥✐t✐♦♥

M ✐s ❛

♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦r♥❡rs ⇔

M ✐s ❧♦❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦

♠♦❞❡❧ ❝♦r♥❡r R

j + × Rn−j

♣♦✐♥t (0, 0) ∈ R

j + × Rn−j ✐s ♦❢ ❞❡♣t❤ j

✭j = 0 ❢♦r ❝❧♦s❡❞ ♠❛♥✐❢♦❧❞s✮ ❊①❛♠♣❧❡

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✼ ✴ ✺✹

❉✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs

❊①❛♠♣❧❡

❈♦♥s✐❞❡r ❝♦r♥❡r R

2 +

✇✐t❤ ❝♦♦r❞✐♥❛t❡s (x1, x2)✳ ❱❡❝t♦r ✜❡❧❞s t❛♥❣❡♥t t♦ ❢❛❝❡s ✭❣❡♥❡r❛t♦rs✮✿ x1 ∂ ∂x1 , x2 ∂ ∂x2 ⇓ ❉✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✿

D =

• ij

aij(x)

• x1

∂ ∂x1 i

x2

∂ ∂x2 j ✇❤❡r❡ ❝♦❡✣❝✐❡♥ts aij(x) ❛r❡ ❢✉♥❝t✐♦♥s ♦♥ R

2 +

✭s♠♦♦t❤ ✉♣ t♦ t❤❡ ❝♦r♥❡r✮

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✽ ✴ ✺✹

▲♦❣❛r✐t❤♠✐❝ ❝♦♦r❞✐♥❛t❡s

❈❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s✿ x1 = e−t1, x2 = e−t2

x1

∂ ∂x1 = − ∂ ∂t1 ,

x2

∂ ∂x2 = − ∂ ∂t2 . ■♥ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ♦✉r ♦♣❡r❛t♦rs ❛r❡ ❥✉st ♦♣❡r❛t♦rs ✐♥ R2

+

✇✐t❤ ❡①♣♦♥❡♥t✐❛❧❧② st❛❜✐❧✐③✐♥❣ ❝♦❡✣❝✐❡♥ts ❛t ✐♥✜♥✐t② ✭✐♥ t✮✳

❘❡♠❛r❦

▼❡tr✐❝ dt2

1 + dt2 2 =⇒ s♣❛❝❡s L 2(M), Hs(M)

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✶✾ ✴ ✺✹

❍♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭◆❛③❛✐❦✐♥s❦✐✐✱ ❙t❡r♥✐♥✱ ❛♥❞ ❙❛✈✐♥ ✷✵✵✻✮

❚❤❡r❡ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ Ell(M) ≃ K0(M#) ❍❡r❡

K0 ✐s ❡✈❡♥ K✲❤♦♠♦❧♦❣② ❣r♦✉♣ ✭❣❡♥❡r❛❧✐③❡❞ ❤♦♠♦❧♦❣② t❤❡♦r②

❞✉❛❧ t♦ K✲t❤❡♦r②✮❀

M# ✐s s♦♠❡ ❡①♣❧✐❝✐t❧② ❝♦♥str✉❝t❡❞ ❝♦♠♣❛❝t t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡

✭✏❞✉❛❧✑ ♦❢ M✮✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✵ ✴ ✺✹

❉✉❛❧ ♠❛♥✐❢♦❧❞ M#

✶✳ ❋♦r♠❛❧ ❞❡✜♥✐t✐♦♥✳ ▲❡t C(M#) ⊂ C(M0) ❜❡ t❤❡ s✉❜❛❧❣❡❜r❛ ♦❢ ❢✉♥❝t✐♦♥s f s✉❝❤ t❤❛t ✐♥ ❛♥② ❧♦❣❛r✐t❤♠✐❝ ❝♦♦r❞✐♥❛t❡s

t1 = − ln x1,

...,

tj = − ln xj, xj+1,

..., xn ♥❡❛r ❛ ❝♦r♥❡r ♦❢ ❝♦❞✐♠❡♥s✐♦♥ j t❤❡r❡ ❡①✐st ❧✐♠✐ts

f(t1, ..., tj, xj+1, ..., xn) −→ g(t1 : . . . : tj),

❛s |t| → ∞✱ ✇❤❡r❡ g(t1 : . . . : tj) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦❢ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s (t1 : . . . : tj)✳

❉❡✜♥✐t✐♦♥

M# ✐s t❤❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡

❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛ C(M#)✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✶ ✴ ✺✹

✷✳ ■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥✳

M# ✐s ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s

✶ ❚♦ ❡❛❝❤ ❝♦❞✐♠❡♥s✐♦♥ j ≥ 1 ❢❛❝❡ F ✇❡

❛ss♦❝✐❛t❡ ❛ ❝❧♦s❡❞ s✐♠♣❧❡① ∆F ♦❢ ❞✐♠❡♥s✐♦♥ j − 1✳

✷ ❚✇♦ s✐♠♣❧✐❝❡s ❛r❡ ❣❧✉❡❞ ♦♥❡ t♦ ❛♥♦t❤❡r

❛❝❝♦r❞✐♥❣ t♦ ❛❞❥❛❝❡♥❝② ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛❝❡s✳

✸ ❚❤❡ s✐♠♣❧✐❝❡s ❛r❡ ❣❧✉❡❞ t♦ t❤❡ ✐♥t❡r✐♦r

s♠♦♦t❤ ♣❛rt M0✳ ❆s ❛ r❡s✉❧t ✇❡ ♦❜t❛✐♥ ❛ ❝♦♠♣❛❝t s♣❛❝❡ M#✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✷ ✴ ✺✹

❊①❛♠♣❧❡s

✶✳ ❉✐s❦✳ ✷✳ ❙q✉❛r❡✳ ✸✳ ❚❡❛r✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✸ ✴ ✺✹

✹✳ ❈✉❜❡✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✹ ✴ ✺✹

❑❡② ♦❜s❡r✈❛t✐♦♥

❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r D ♦❢ ♦r❞❡r ③❡r♦ ♦♥ M ⇓ ❝♦♠♣❛❝t♥❡ss ♦❢ ❝♦♠♠✉t❛t♦r [D, f] ❢♦r ❛♥② f ∈ C(M#) ⇓

D ❞❡✜♥❡s ❡❧❡♠❡♥t ✐♥ K 0(C(M#))

✖ ❛♥❛❧②t✐❝ K✲❤♦♠♦❧♦❣② ♦❢ ❛❧❣❡❜r❛ C(M#)✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✺ ✴ ✺✹

❘❡❢❡r❡♥❝❡s

P❛♣❡rs ❜② ◆❛③❛✐❦✐♥s❦✐✐✱ ❱✳❊✳❀ ❙❛✈✐♥✱ ❆✳❨✉✳❀ ❙t❡r♥✐♥✱ ❇✳❨✉✳

✶ ❖♥ t❤❡ ❤♦♠♦t♦♣② ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ♦♥

str❛t✐✜❡❞ ♠❛♥✐❢♦❧❞s✳ ■③✈✳ ▼❛t❤✳ ✼✶ ✭✷✵✵✼✮✱ ♥♦✳ ✻✱ ✶✶✻✼✕✶✶✾✷

✷ ❚❤❡ ❆t✐②❛❤✲❇♦tt ✐♥❞❡① ♦♥ str❛t✐✜❡❞ ♠❛♥✐❢♦❧❞s✳ ❏✳ ▼❛t❤✳ ❙❝✐✳

✭◆✳❨✳✮ ✶✼✵ ✭✷✵✶✵✮✱ ♥♦✳ ✷✱ ✷✷✾✕✷✸✼

✸ ❊❧❧✐♣t✐❝ t❤❡♦r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝♦r♥❡rs✳ ■✳ ❉✉❛❧ ♠❛♥✐❢♦❧❞s

❛♥❞ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✱ ✶✽✸✕✷✵✻✱ ❚r❡♥❞s ▼❛t❤✳✱ ❇✐r❦❤☎ ❛✉s❡r✱ ❇❛s❡❧✱ ✷✵✵✽✳

✹ ❊❧❧✐♣t✐❝ t❤❡♦r② ♦♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❝♦r♥❡rs✳ ■■✳ ❍♦♠♦t♦♣②

❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❑✲❤♦♠♦❧♦❣②✳ ❚r❡♥❞s ▼❛t❤✳✱ ❇✐r❦❤☎ ❛✉s❡r✱ ❇❛s❡❧✱ ✷✵✵✽✳

✺ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❣❡♦♠❡tr② ❛♥❞ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❡❧❧✐♣t✐❝

♦♣❡r❛t♦rs✳ ❏✳ ▼❛t❤✳ ❙❝✐✳ ✭◆✳❨✳✮ ✶✻✹ ✭✷✵✶✵✮✱ ♥♦✳ ✹✱ ✻✵✸✕✻✸✻ ❲♦r❦s ❜② ❈❛rr✐❧❧♦ ❘♦✉s❡✱ ❉❡❜♦r❞✱ ❍❛s❦❡❧❧✱ ▲❡s❝✉r❡✱ ▼❡❧r♦s❡✱ ▼♦♥t❤✉❜❡rt✱ ◆✐st♦r✱ P✐❛③③❛✱ ❘♦❝❤♦♥ ✳✳✳ ▼② ❛♣♦❧♦❣✐❡s✱ ✐❢ ■ ♠✐ss❡❞ s♦♠❡♦♥❡✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✻ ✴ ✺✹

■■■✳ ❊❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❣r♦✉♣ ❛❝t✐♦♥s

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✼ ✴ ✺✹

❖♣❡r❛t♦rs ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❣r♦✉♣ ❛❝t✐♦♥s

M ✐s ❛ ❝❧♦s❡❞ s♠♦♦t❤ ♠❛♥✐❢♦❧❞❀ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s g : M → M✳

G✲♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs

D =

• g

DgTg : Hs(M) −→ Hs−d(M)

✭✺✮

✶

Dg ❛r❡ ✭♣s❡✉❞♦✮❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ M ♦❢ ♦r❞❡rs ≤ d❀

✷

Tg ✐s s❤✐❢t ♦♣❡r❛t♦r✿ Tgu(x) = u(g−1(x));

✸

Hs(M) ❛♥❞ Hs−d(M) ❛r❡ ❙♦❜♦❧❡✈ s♣❛❝❡s✳

Pr♦❜❧❡♠✿ ❝♦♥str✉❝t ❡❧❧✐♣t✐❝ t❤❡♦r② ❢♦r ♦♣❡r❛t♦rs ✭✺✮✳ ❈❛r❧❡♠❛♥✱ ❆♥t♦♥❡✈✐❝❤✕▲❡❜❡❞❡✈✱ ❈♦♥♥❡s✕▼♦s❝♦✈✐❝✐✱ ◆❛③❛✐❦✐♥s❦✐✐✕❙❛✈✐♥✕❙t❡r♥✐♥✱ P❡rr♦t✱ ❡t❝✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✽ ✴ ✺✹

❊①❛♠♣❧❡✳ ❉✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ◆❈ t♦r✉s ✭❈♦♥♥❡s✮

❋✐① ♥✉♠❜❡r θ ∈ (0, 1)✳

G = Z

❖♥ t❤❡ r❡❛❧ ❧✐♥❡ ✇✐t❤ ❝♦♦r❞✐♥❛t❡ x ❝♦♥s✐❞❡r ♦♣❡r❛t♦rs

D =

• α+β≤m

aαβxα

• −i d

dx

β , ✇❤❡r❡ aαβ ❛r❡ ▲❛✉r❡♥t ♣♦❧②♥♦♠✐❛❧s ✐♥ ♦♣❡r❛t♦rs (V1u)(x) = u(x + θ), (V2u)(x) = e2πixu(x).

V1 ❛♥❞ V2 ❞♦ ♥♦t ❝♦♠♠✉t❡✿ V1V2 = e2πiθV2V1✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✷✾ ✴ ✺✹

▼❛✐♥ ♣r♦❜❧❡♠s✿

✶ ✭❋✐♥✐t❡♥❡ss t❤❡♦r❡♠✮ ❞❡s❝r✐❜❡ t❤❡ ❡❧❧✐♣t✐❝✐t② ❝♦♥❞✐t✐♦♥ ❢♦r D✱

✇❤✐❝❤ ✐♠♣❧✐❡s ❋r❡❞❤♦❧♠ ♣r♦♣❡rt②❀

✷ ✭■♥❞❡① t❤❡♦r❡♠✮ ❝♦♠♣✉t❡ ✐♥❞❡① ♦❢ D✳ ❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✵ ✴ ✺✹

❲❡ ✇❛♥t t♦ ❞❡✜♥❡ t❤❡ s②♠❜♦❧ σ(D)(x, ξ) ❛t ♣♦✐♥t (x, ξ) ∈ T ∗M \ 0 ❚❤❡ s②♠❜♦❧ s❤♦✉❧❞ ✐♥✈♦❧✈❡ t❤❡ ❡♥t✐r❡ ♦r❜✐t {∂g−1(x, ξ)}g∈G✱ ✇❤❡r❡ ∂g : T ∗M −→ T ∗M st❛♥❞s ❢♦r t❤❡ ❝♦❞✐✛❡r❡♥t✐❛❧ ♦❢ g✳

❉❡✜♥✐t✐♦♥ ✭tr❛❥❡❝t♦r② s②♠❜♦❧✮

❚❤❡ s②♠❜♦❧ ♦❢ D =

h DhTh : Hs(M) → Hs−d(M) ✐s t❤❡ ♦♣❡r❛t♦r

σ(D)(x, ξ) : l2(G, µx,ξ,s) −→ l2(G, µx,ξ,s−d), σ(D)(x, ξ)u(g) =

• h σ(Dh)
• ∂g−1(x, ξ)
• Thu
• (g).

❤❡r❡ l2(G, µx,ξ,s) ✐s t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s u = {u(g)} ♦♥ t❤❡ tr❛❥❡❝t♦r②✱ µ ✐s ❛ ❝❡rt❛✐♥ ♠❡❛s✉r❡ ♦♥ G❀ s②♠❜♦❧ ♦❢ Th ✐s r✐❣❤t s❤✐❢t ♦♣❡r❛t♦r✿ Thu(g) = u(gh)❀ s②♠❜♦❧ ♦❢ Dh ✐s ♦♣❡r❛t♦r ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ σ(Dh)(∂g−1(x, ξ))✳ µx,ξ,s(g) =

• det ∂g−1

∂x

• ·
• t∂g−1

∂x −1 (ξ)

• 2s

.

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✶ ✴ ✺✹

❊❧❧✐♣t✐❝✐t②✳ ❋✐♥✐t❡♥❡ss t❤❡♦r❡♠

❉❡✜♥✐t✐♦♥

❖♣❡r❛t♦r D : Hs(M) → Hs−d(M) ✐s ❡❧❧✐♣t✐❝ ✐❢ ✐ts s②♠❜♦❧ σ(D)(x, ξ) : l2(Z, µx,ξ,s) → l2(Z, µx,ξ,s−d) ✐s ✐♥✈❡rt✐❜❧❡ ❢♦r ❛❧❧ (x, ξ) ∈ T ∗M \ 0✳

❚❤❡♦r❡♠ ✭❋✐♥✐t❡♥❡ss✮

❊❧❧✐♣t✐❝ ♦♣❡r❛t♦r D ✐s ❋r❡❞❤♦❧♠ ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s

D : Hs(M) → Hs−d(M).

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✷ ✴ ✺✹

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❉✐✣❝✉❧t②✿ t❤❡ s②♠❜♦❧ σ(D)(x, ξ) ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ✐♥ x, ξ ⇒ ❞✐✣❝✉❧t t♦ ❝♦♥str✉❝t D−1 ✉s✐♥❣ σ(D)(x, ξ)✳

❯s❡ C∗✲❛❧❣❡❜r❛s✿

✶✳ ❘❡❞✉❝❡ D t♦ ♦♣❡r❛t♦r

D0 =

• g

D0,g Tg : L 2(M) → L 2(M),

✇❤❡r❡

Tg ✐s ✉♥✐t❛r②✳

✷✳ ❚❤❡♥ D0 ❤❛s ♥❛t✉r❛❧❧② s②♠❜♦❧ ✐♥ ♠❛①✐♠❛❧ C∗✲❝r♦ss❡❞ ♣r♦❞✉❝t

C(S∗M) ⋊ G✳

✸✳ ❆❧♠♦st ✐♥✈❡rs❡ ♦♣❡r❛t♦r ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② q✉❛♥t✐③✐♥❣ t❤❡ ✐♥✈❡rs❡ s②♠❜♦❧ ✐♥ C(S∗M) ⋊ G✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✸ ✴ ✺✹

❊①❛♠♣❧❡ ✶✳ ■s♦♠❡tr✐❝ ❛❝t✐♦♥s

■❢ g ♣r❡s❡r✈❡s ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ M✱ t❤❡♥ t❤❡ ♠❡❛s✉r❡s µx,ξ,s ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ st❛♥❞❛r❞ ♠❡❛s✉r❡ ✇✐t❤ ❞❡♥s✐t② 1 ♦♥ G✳

❈♦r♦❧❧❛r②

❊❧❧✐♣t✐❝✐t② ❛♥❞ ✐♥❞❡① ♦❢ G✲♦♣❡r❛t♦r

D : Hs(M) −→ Hs−d(M)

❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ s✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✹ ✴ ✺✹

❊①❛♠♣❧❡ ✷✳ ❖♣❡r❛t♦rs ✇✐t❤ ❞✐❧❛t✐♦♥s

❈♦♥s✐❞❡r t❤❡ ♦♣❡r❛t♦r

D = 1 + αT : Hs(S2) −→ Hs(S2)

✭✻✮ ✇❤❡r❡ α ❛♥❞ s ❛r❡ s♦♠❡ ♥✉♠❜❡rs ❛♥❞ Tu(x) = u(qx)✳ ▲❡t q < 1✳

Pr♦♣♦s✐t✐♦♥

❖♣❡r❛t♦r ✭✻✮ ✐s ❡❧❧✐♣t✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ ✐♥❡q✉❛❧✐t✐❡s ✐s ✈❛❧✐❞ |α| < q|s−1| ♦r |α| > q−|s−1|. ✭✼✮

s |α| 1 1 |α| = qs−1 |α| = q1−s

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✺ ✴ ✺✹

■♥❞❡① t❤❡♦r❡♠

▼❡t❤♦❞✿ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ✉♥✐❢♦r♠✐③❛t✐♦♥

✶✳ ❘❡❞✉❝❡ ♦♣❡r❛t♦r D t♦ ❛♥ ❡❧❧✐♣t✐❝ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r D ♦♥ ❛ ❝❧♦s❡❞ ♠❛♥✐❢♦❧❞ ❛♥❞ s✉❝❤ t❤❛t indD = indD. ✷✳ ❯s❡ ❆t✐②❛❤✕❙✐♥❣❡r ❢♦r♠✉❧❛ t♦ ❝♦♠♣✉t❡ indD✳

❘❡♠❛r❦

✭s②♠❜♦❧✐❝ ✉♥✐❢♦r♠✐③❛t✐♦♥✮ ■t s✉✣❝❡s t♦ ❞❡✜♥❡ ♦♥❧② t❤❡ s②♠❜♦❧ σ(D)✱ s✐♥❝❡ t❤❡ ✐♥❞❡① ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s②♠❜♦❧✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✻ ✴ ✺✹

❯♥✐❢♦r♠✐③❛t✐♦♥ ✭♣r❡❧✐♠✐♥❛r✐❡s✮

♦♣❡r❛t♦r D ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ♦♣❡r❛t♦r ♦♥ t❤❡ q✉♦t✐❡♥t✲s♣❛❝❡ M/G ✭♦r❜✐t s♣❛❝❡✮❀ ✐❢ t❤❡ ❛❝t✐♦♥ ✐s ❢r❡❡ ❛♥❞ ♣r♦♣❡r✱ t❤❡♥ M/G ✐s ❛ s♠♦♦t❤ ♠❛♥✐❢♦❧❞ ❛♥❞ D ✐s ❛ ψ❉❖ ♦♥ ✐t ✇✐t❤ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ s②♠❜♦❧ ✭❡①♣❧❛♥❛t✐♦♥✿ ❞✐✛❡♦♠♦r♣❤✐s♠s g ♠♦✈❡ ♣♦✐♥ts ❛❧♦♥❣ ♦r❜✐ts✱ ❤❡♥❝❡✱ ❣✐✈❡ ✐❞❡♥t✐t② ♠❛♣♣✐♥❣ ♦❢ ♦r❜✐t s♣❛❝❡✮❀ ✉♥❢♦rt✉♥❛t❡❧②✱ ❛❝t✐♦♥ ♦❢ ✐♥✜♥✐t❡ ❣r♦✉♣ ♦♥ ❛ ❝♦♠♣❛❝t s♣❛❝❡ ✐s ♥❡✈❡r ♣r♦♣❡r ✭❛♥❞ ♦❢t❡♥ M/G ✐s ♥♦t ❡✈❡♥ ❍❛✉s❞♦r✛✦✮✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✼ ✴ ✺✹

❯♥✐❢♦r♠✐③❛t✐♦♥ ✭s❦❡t❝❤✮

❚❤❡r❡❢♦r❡✱ ✇❡ ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶ t❛❦❡ ❛ ❢r❡❡ ❛♥❞ ♣r♦♣❡r ❛❝t✐♦♥ ♦❢ G ♦♥ ❛✉①✐❧✐❛r② ♠❛♥✐❢♦❧❞ Y❀ ✷ t❛❦❡ ♣r♦❞✉❝t M × Y ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❞✐❛❣♦♥❛❧ ❛❝t✐♦♥

• g : M × Y

−→

M × Y

(x, y) −→ (g(x), g(y)) ✭✽✮ t❤✐s ✐s ❛ ❢r❡❡ ♣r♦♣❡r ❛❝t✐♦♥❀

✸ ❝♦♥str✉❝t ❛♥ ♦♣❡r❛t♦r ♦♥ M × Y ❛ss♦❝✐❛t❡❞ ✇✐t❤

❞✐✛❡♦♠♦r♣❤✐s♠ ✭✽✮✱ ✇❤♦s❡ ✐♥❞❡① ✐s ❡q✉❛❧ t♦ t❤❡ ✐♥❞❡① ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♦♣❡r❛t♦r❀

✹ r❡♣r❡s❡♥t ♦♣❡r❛t♦r ♦♥ M × Y ❛s ψ❉❖ ♦♥ ♦r❜✐t s♣❛❝❡

(M × Y)/G✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✽ ✴ ✺✹

❊①❛♠♣❧❡ ✶

M = S1

x✱ G = Z✱ Y = R✱ g(x) = x + α

M × R = S1

x × Rt✱

• g(x, t) = (x + α, t + 1)

(M × R)/Z = ❢✉♥❞❛♠✳❞♦♠❛✐♥ ✇✐t❤ ✐❞❡♥t✐✜❡❞ ❜♦✉♥❞❛r✐❡s

• = {t♦r✉s}

❆❝t✐♦♥ ♦♥ ❢✉♥❞❛♠❡♥t❛❧ ❞♦♠❛✐♥ ✐s tr✐✈✐❛❧✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✸✾ ✴ ✺✹

❆✉①✐❧✐❛r② s♣❛❝❡ Y ❛♥❞ s②♠❜♦❧ ♦♥ ✐t

▲❡t Y ❜❡ ❛ ❝♦♠♣❧❡t❡ s✐♠♣❧②✲❝♦♥♥❡❝t❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ♦❢ ♥♦♥♣♦s✐t✐✈❡ s❡❝t✐♦♥❛❧ ❝✉r✈❛t✉r❡ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ❢r❡❡ ♣r♦♣❡r ❝♦❝♦♠♣❛❝t ❛❝t✐♦♥ ♦❢ G✳ ▼✐s❤❝❤❡♥❦♦✬s ✐❞❡❛s✳

❉✉❛❧ ❉✐r❛❝ ❡❧❡♠❡♥t β ∈ KK G(C, C0(T∗Y))

β ✐s ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ ❡❧❧✐♣t✐❝ s②♠❜♦❧

a =

• a∗

+

a+

• : π∗Λ(Y) −→ π∗Λ(Y),

π : T ∗Y → Y,

a(η)u = (iη + ρdρ) ∧ u +

• (iη + ρdρ)∧

∗u, ✇❤❡r❡ ρ(y) = dist(y0, y) ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ✜①❡❞ ♣♦✐♥t y0 ∈ Y✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✵ ✴ ✺✹

❯♥✐❢♦r♠✐③❡❞ s②♠❜♦❧ σ(D) ♦♥ (M × Y)/G

❋✐❜❡r ❜✉♥❞❧❡ (M × Y)/G −→ Y/G ✇✐t❤ ✜❜❡r ▼ ⇒ ♦♥ T ∗(M × Y)/G ❝♦♦r❞✐♥❛t❡s (x, y, ξ, η)✳

❯♥✐❢♦r♠✐③❡❞ s②♠❜♦❧

σ(D) : π∗V −→ π∗V σ(D)(x, ξ, y, η) = (|ξ|2 + |η|2 + (hρ)2 + 1)−1/2        |ξ|σ(D)(x, ξ) −ha∗

+(y, η)

ha+(y, η)

|ξ|σ(D)∗(x, ξ)        . ✇❤❡r❡ V ✐s t❤❡ ❍✐❧❜❡rt ❜✉♥❞❧❡ V = (M × Λ(Y) ⊗ l2(G))/G ♦✈❡r (M × Y)/G✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✶ ✴ ✺✹

Pr♦♣♦s✐t✐♦♥

❚❤❡ s②♠❜♦❧ σ(D)(x, ξ, y, η) ❤❛s t❤❡ ♣r♦♣❡rt✐❡s✿ ✶✮ ✐t ✐s ❝♦♥t✐♥✉♦✉s ✐♥ ♥♦r♠❀ ✷✮ ✐t ❤❛s ❝♦♠♣❛❝t ✈❛r✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦✈❛r✐❛❜❧❡s ξ, η✱ ✐✳❡✳✱ t❤❡ ❞✐✛❡r❡♥❝❡ σ(D)(x, ξ, y, η) − σ(D)(x, ξ0, y, η0) ✭✾✮ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❢♦r ❛❧❧ x, ξ, y, η, ξ0, η0❀ ✸✮ ✐t ✐s ✐♥✈❡rt✐❜❧❡ ❢♦r ❛❧❧ x, ξ, y, η s✉❝❤ t❤❛t |ξ|2

y + |η|2 ≥ R2✱ ✇❤❡r❡

R ✐s ❧❛r❣❡ ❡♥♦✉❣❤❀ ♠♦r❡♦✈❡r✱ t❤❡ ♥♦r♠ ♦❢ t❤❡ ✐♥✈❡rs❡ ✐s

✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞✳

❚❤❡♦r❡♠ ✭❯♥✐❢♦r♠✐③❛t✐♦♥✮

▲❡t D ❜❡ ❛ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✇✐t❤ s②♠❜♦❧ σ(D)✳ ❚❤❡♥ indD = indD.

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✷ ✴ ✺✹

❊①❛♠♣❧❡✳ G = Z

▲❡t G = Z ❛♥❞ ❛✉①✐❧✐❛r② s♣❛❝❡ ❜❡ Y = R✳ ▲❡t D = DkT k ❜❡ ❛♥ ♦♣❡r❛t♦r ♦♥ M ❛♥❞

A = d dt + t : Hs(R) −→ Hs−1(R)

❜❡ ❛♥ ❛✉①✐❧✐❛r② ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r✳ indA = 1✳ ✭❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦r✮

❋♦r ✐s♦♠❡tr✐❝ ❛❝t✐♦♥✿

❚❤❡♥ D ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❡①t❡r♥❛❧ ♣r♦❞✉❝t ♦❢ D ❛♥❞ A

D#A =

Dk

T k A

−A ∗ ( Dk

T k)∗

• : Hs(M×R, C2) −→ Hs−1(M×R, C2),

✇❤❡r❡

Tu(x, t) = u(g−1(x), t − 1) ✐s t❤❡ s❤✐❢t ♦♣❡r❛t♦r ♦♥ M × R✳

❚❤✐s ✭♠♦r❡ ❞❡❧✐❝❛t❡✮ ♦♣❡r❛t♦r ✉♥✐❢♦r♠✐③❛t✐♦♥ ✇❛s ❞♦♥❡ ❜② ❙❝❤r♦❤❡✱ ❙t❡r♥✐♥✱ ❙✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✸ ✴ ✺✹

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢✿

❈♦♥s✐❞❡r t❤❡ ❞✐❛❣r❛♠✿

KK0(C, C0(T ∗M) ⋊ G)

×β

• ∂

KK0(C, K ⋊ G)

×β

• l∗

KK0(C, C) =

=

• KK0
• C,C0
• T ∗M×T ∗Y
• ⋊G
• ∂′
• ≃
• KK0
• C,
• K ⊗C0(

T ∗Y)

• ⋊G
• ≃
• K 0 (T ∗(M × Y)/G)

ind

K 0(T ∗(Y/G))

ind

Z

❍❡r❡ l : C ⋊ G → C ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠✳

• σ(D) ❣✐✈❡s ❝❧❛ss ✐♥ KK0(C, C0(T ∗M) ⋊ G)❀
• ♣❛t❤ t❤r♦✉❣❤ ✉♣♣❡r r✐❣❤t ❝♦r♥❡r ❣✐✈❡s indD❀
• ♣❛t❤ t❤r♦✉❣❤ ❧♦✇❡r ❧❡❢t ❝♦r♥❡r ❣✐✈❡s indD✳

❚♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠✱ ✐t s✉✣❝❡s t♦ s❤♦✇ t❤❛t t❤❡ ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✹ ✴ ✺✹

❘✐❣❤t r❡❝t❛♥❣❧❡✿ KK0(C, C ⋊ G)

×β

• l∗

KK0(C, C) = Z

=

• KK0 (C, C0(T ∗Y)) ⋊ G)

≃

• K 0(T ∗(Y/G))

ind

Z

❝♦♠♠✉t❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢

l∗ = l∗ ◦ (• ⊗ jG(γ)) : KK(C, C ⋊ G) −→ KK0(C, C) = Z.

✇❤❡r❡ γ ∈ KK G(C, C) ✐s t❤❡ γ✲❡❧❡♠❡♥t ❢♦r t❤❡ ❣r♦✉♣❀

jG : KK G(C, C) −→ KK(C ⋊ G, C ⋊ G)

✐s t❤❡ ❞❡s❝❡♥t ♠❛♣♣✐♥❣✳ ■❢ γ = 1✱ t❤❡♥ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✺ ✴ ✺✹

❙✉♠♠❛r②

✶ ❲❡ ❞❡❛❧ ✇✐t❤ tr✐♣❧❡s (D, M, G)❀ ✷ ❯♥✐❢♦r♠✐③❛t✐♦♥=r❡❞✉❝t✐♦♥ t♦ ❛ tr✐♣❧❡ ✇✐t❤ tr✐✈✐❛❧ ❣r♦✉♣

❛❝t✐♦♥✿ (D, M, G) −→ (σ(D), (M × Y)/G, Id). ❈♦♠♣✉t❡ ✐♥❞❡① ♦❢ D ❜② ❛♥ ❆t✐②❛❤✕❙✐♥❣❡r t②♣❡ ❢♦r♠✉❧❛ ♦♥ (M × Y)/G✳ ⇒ ✐♥❞❡① ❢♦r♠✉❧❛ ❢♦r D✳ ❲❤❛t ❛❜♦✉t ✐♥❞❡① ❢♦r♠✉❧❛ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ♠❛♥✐❢♦❧❞ M❄

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✻ ✴ ✺✹

■♥❞❡① ❢♦r♠✉❧❛ ✭✐s♦♠❡tr✐❝ ❝❛s❡✱ G = Z✮

indD =

• j

(j − 1)! (2πi)j(2j − 1)!

• S∗M

Td(T ∗

CM)(σ−1dσ)2j−1 00 ,

✭✶✵✮ ❍❡r❡✿ Td(T ∗

CM) ✐s ❛ g✲✐♥✈❛r✐❛♥t ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠ ♦♥ M t❤❛t

r❡♣r❡s❡♥ts t❤❡ ❚♦❞❞ ❝❧❛ss ♦❢ t❤❡ ❝♦♠♣❧❡①✐✜❡❞ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡❀ σ = σ(D) ✐s t❤❡ s②♠❜♦❧ ♦❢ ♦♣❡r❛t♦r❀ ❢♦r ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r a ❛❝t✐♥❣ ♦♥ l2(Z) ✇✐t❤ ❝♦♠♣♦♥❡♥ts (aij) ✇❡ s❡t (a)00 = a00✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✼ ✴ ✺✹

■♥❞❡① ❢♦r♠✉❧❛ ✭♥♦♥✐s♦♠❡tr✐❝ ❝❛s❡✱ G = Z✮

❲❡ ✉s❡ ♠❡t❤♦❞s ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❣❡♦♠❡tr②✳ ✶✳ ❙②♠❜♦❧ ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ ❝r♦ss❡❞ ♣r♦❞✉❝t✳ ❚❛❦❡ ♦♣❡r❛t♦r D =

k DkT k ❛♥❞ ❝♦♥s✐❞❡r ❡❧❡♠❡♥t ♦❢ t❤❡ ❝r♦ss❡❞

♣r♦❞✉❝t σ(D) =

• k

σ(Dk)wk ∈ C∞(S∗M) ⋊ Z, ✭❤❡r❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ✐s ❣❡♥❡r❛t❡❞ ❜② ❢✉♥❝t✐♦♥s f ❛♥❞ ❡❧❡♠❡♥t w✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥ wfw−1 = (∂g)∗f✮✳ ❯♥✐❢♦r♠✐③❛t✐♦♥ ❣✐✈❡s t❤❡ ♠❛♣♣✐♥❣

K1(C∞(S∗M) ⋊ Z)

−→

K 0(T ∗MZ)

σ(D) σ(D).

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✽ ✴ ✺✹

✷✳ ❚♦ ❝♦♥str✉❝t t♦♣♦❧♦❣✐❝❛❧ ✐♥❞❡①✱ ✇❡ ✉s❡ ❡q✉✐✈❛r✐❛♥t ❝❤❛r❛❝t❡r✐st✐❝ ❝❧❛ss❡s ✐♥ ❝②❝❧✐❝ ❝♦❤♦♠♦❧♦❣②✳ ❈♦♥s✐❞❡r t❤❡ ❞✐❛❣r❛♠✿

K1(C∞(S∗M) ⋊ Z)

• ch
• K 0(T ∗MZ)

ch

• HPodd(C∞(S∗M) ⋊ Z)
• ,Tdg(T∗

CM)

• P

P P P P P P P P P P P P P

Hev(T ∗MZ)

• ,Td(T∗

CM)

tttttttttt

C ✭✶✶✮ ✇❤❡r❡ Tdg(T ∗

CM) ∈ HPodd(C∞(S∗M) ⋊ Z) ✐s ❡q✉✐✈❛r✐❛♥t ❚♦❞❞ ❝❧❛ss

✐♥ ❝②❝❧✐❝ ❝♦❤♦♠♦❧♦❣②⇒

❚❤❡♦r❡♠ ✭✐♥❞❡① ❢♦r♠✉❧❛✮

❚❤❡ ❞✐❛❣r❛♠ ✭✶✶✮ ✐s ❝♦♠♠✉t❛t✐✈❡✳ ❍❡♥❝❡✱ indD = ch[σ(D)], Tdg(T ∗

CM)

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✹✾ ✴ ✺✹

❊q✉✐✈❛r✐❛♥t ❈❤❡r♥ ❝❤❛r❛❝t❡r

❉✐✛❡♦♠♦r♣❤✐s♠ g ❞❡✜♥❡s Z✲❛❝t✐♦♥ ♦♥ M✳ ❊q✉✐✈❛r✐❛♥t ❜✉♥❞❧❡ E ∈ VectZ(M) ❤❛s ❡q✉✐✈❛r✐❛♥t ❈❤❡r♥ ❝❤❛r❛❝t❡r

chZ(E) ∈ HP∗(C∞(M) ⋊ Z) chZ(E) = {chk

Z(E; a0, a1, ..., ak)} ✐s ❞❡✜♥❡❞ ❛s✿

chk

Z(E; a0, a1, ..., ak) =

= (−1)(n−k)/2 ((n + k)/2)!

• i0+i1+...+ik= n−k

2

• M

trE

• a0θi0∇(a1)θi1∇(a2) . . . ∇(ak)θik

✭s❡❡ ●♦r♦❦❤♦✈s❦②✱ ❝❢✳ ❏❛✛❡✲▲❡s♥✐❡✇s❦✐✲❖st❡r✇❛❧❞❡r ❢♦r♠✉❧❛✮✳ ❍❡r❡

dim M = n, k = n, n − 2, n − 4, . . . , ∇E ✐s ❛ ❝♦♥♥❡❝t✐♦♥ ✐♥ E ❛♥❞

θ = ∇2

E ✐s ✐ts ❝✉r✈❛t✉r❡ ❢♦r♠✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺✵ ✴ ✺✹

❊q✉✐✈❛r✐❛♥t ❚♦❞❞ ❝❧❛ss

❯s❡ ♦♣❡r❛t✐♦♥s ✐♥ K✲t❤❡♦r②✳ ❚❤❡ ❡q✉✐✈❛r✐❛♥t ❚♦❞❞ ❝❧❛ss TdZ(E) ∈ HP∗(C∞(M) ⋊ Z) ♦❢ ❛ ❝♦♠♣❧❡① Z✲❜✉♥❞❧❡ E ♦♥ ❛ s♠♦♦t❤ ♠❛♥✐❢♦❧❞ M ✐s ❡q✉❛❧ t♦ TdZ(E) = chZ(Φ(E)), ❤❡r❡ Φ ✐s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♦♣❡r❛t✐♦♥ ✐♥ K✲t❤❡♦r②✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❢✉♥❝t✐♦♥ ϕ(t) = t−1(1 + t) ln(1 + t)✳ ❖♣❡r❛t✐♦♥ Φ ✐s ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ●r♦t❤❡♥❞✐❡❝❦ ♦♣❡r❛t✐♦♥s✿ Φ(E)=1+E − n

2

+−2(E2 − 2nE + n2) + 7(E + Λ2E − nE + n(n − 1)/2)

12

= 3n2 − 19n + 24

24

+ (−3n + 13)

12 E − 1 6E ⊗ E + 7 12Λ2E

✇❤❡r❡ n = dimE✳ ❍❡r❡ ✇❡ ❛ss✉♠❡ t❤❛t dim M ≤ 5✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺✶ ✴ ✺✹

❙♣❡❝✐❛❧ ❝❛s❡

❙✉♣♣♦s❡ t❤❛t Td(T ∗

CMZ) = 1✳

❚❤❡♥ ✇❡ ❤❛✈❡ ❛ s✐♠♣❧❡r ✐♥❞❡① ❢♦r♠✉❧❛✿ indD = (n − 1)! (2πi)n(2n − 1)!

• S∗M

(σ−1dσ)2n−1 , ✭✐♥ t❤✐s ❝❛s❡ t❤❡ ❝②❝❧✐❝ ❝❧❛ss Tdg(T ∗

CM) ✐s ❥✉st t❤❡

tr❛♥s✈❡rs❡ ❢✉♥❞❛♠❡♥t❛❧ ❝❧❛ss✮✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺✷ ✴ ✺✹

❘❡❢❡r❡♥❝❡s

✶ ❙❛✈✐♥ ❆✳❨✉✳✱ ❙t❡r♥✐♥ ❇✳❨✉✳ ❯♥✐❢♦r♠✐③❛t✐♦♥ ♦❢ ♥♦♥❧♦❝❛❧ ❡❧❧✐♣t✐❝

♦♣❡r❛t♦rs ❛♥❞ KK✲t❤❡♦r②✳ ❘✉ss✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✷✵✶✸✱ ❱✳ ✷✵✱ ◆ ✸✱ ✸✹✺✕✸✺✾✳

✷ ❙❛✈✐♥ ❆✳❨✉✳✱ ❙t❡r♥✐♥ ❇✳❨✉✳ ■♥❞❡① ♦❢ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ❢♦r

❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ ♠❛♥✐❢♦❧❞s✳ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❝♦♠♠✉t❛t✐✈❡

• ❡♦♠❡tr②✱ ✷✵✶✹✱ ❱✳ ✽✱ ◆♦✳ ✸✱ ✻✾✺✲✼✸✹✱ ❛r❳✐✈✿✶✶✵✻✳✹✶✾✺✳

✸ ❙❛✈✐♥ ❆✳❨✉✳✱ ❙❝❤r♦❤❡ ❊✳✱ ❙t❡r♥✐♥ ❇✳❨✉✳ ❯♥✐❢♦r♠✐③❛t✐♦♥ ❛♥❞

❛♥ ✐♥❞❡① t❤❡♦r❡♠ ❢♦r ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ ❛ ♠❛♥✐❢♦❧❞✳ ❘✉ss✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤✳ P❤②s✐❝s✱ ✷✵✶✺✱ ✈✳ ✷✷✱ ◆✳ ✸✱ ❛r❳✐✈✿✶✶✶✶✳✶✺✷✺✳

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺✸ ✴ ✺✹

❚❤❛♥❦ ②♦✉ ✈❡r② ♠✉❝❤ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❆♥t♦♥ ❙❛✈✐♥ ✭ P❡♦♣❧❡s ❋r✐❡♥❞s❤✐♣ ❯♥✐✈❡rs✐t② ♦❢ ❘✉ss✐❛ ✭❘❯❉◆ ❯♥✐✈❡rs✐t②✮ ✮ ❖♥ t❤❡ ✇♦r❦ ♦❢ ❇♦r✐s ❙t❡r♥✐♥ ✸✵✳✵✺✳✷✵✶✼ ✺✹ ✴ ✺✹