✿ ▼❆❚❍❊▼❆❚■◗❯❊❙

♥ ✳ |❛

. = C 0 0 (E, Λ p E * ) ✿ t❤❡ s❡t ♦❢ ❝♦♥t✐♥✉♦✉s p✲❞✐♠❡♥s✐♦♥❛• ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ♦♥ E ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ■❢ dx 1 , ..., dx n ✐s ❛ ❜❛s✐s ♦❢ E * ✱ ❛♥② ω ∈ Ω p 0 (E) ❝❛♥ ❜❡ ✇r✐tt❡♥ ✉♥❞❡r t❤❡ ❣❡♥❡r❛• ❢♦r♠ ✿ ∀x ∈ E, ω x = 1≤i 1 <...<ip≤n a i 1 ,...,ip (x)dx i 1 ∧ ... ∧ dx ip ✇❤❡r❡ a i 1 ,...,ip ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱ ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ✲ Ω p 0 (E) ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ✐♥✜♥✐t❡ ♥♦r♠ ω ∞ = sup
♠❡❛s✉r❡s✱ ❛•r❡❛❞② ✐♥tr♦❞✉❝❡❞ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❢♦r ✉♥•❛❜❡•❡❞ ♣♦✐♥t s❡ts ❬✹✶(✳ ■♥❞❡❡❞✱ ✐♥ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡ p = 0✱ t❤❡ ♣r❡✈✐♦✉s ❞❡✜♥✐t✐♦♥ ✐s ❡①❛❝t•② t❤❡ ♦♥❡ ♦❢ ✉s✉❛• ❞✐str✐❜✉t✐♦♥s ♦♥ E ✭❞✉❛• ♦❢ t❤❡ s♣❛❝❡ C 0 (E, R)✮✳ ❏✉st ❛s ❢♦r ❞✐str✐❜✉t✐♦♥s✱ s✐♠♣•❡st ❡①❛♠♣•❡s ♦❢ ❝✉rr❡♥ts ❛r❡ ❣✐✈❡♥ ❜② ❉✐r❛❝ ❝✉rr❡♥ts δ ξ x ✇✐t❤ x ∈ E, ξ ∈ Λ p E s✉❝❤ t❤❛t ❢♦r ❛♥② ❞✐✛❡r❡♥t✐❛• ❢♦r♠ ω ∈ Ω p 0 (E)✱ ✇❡ ❤❛✈❡ δ ξ x (ω) = ω x (ξ)✳ ◆♦✇✱ t❤❡ r❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ s❤❛♣❡s ❛♥❞ ❝✉rr❡♥ts •✐❡s ❢✉♥❞❛♠❡♥t❛••② ✐♥ t❤❡ ❢❛❝t t❤❛t ❡✈❡r② ❞✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ X ♦❢ E ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡•❡♠❡♥t ♦❢ Ω d 0 (E) ′ ✳ ■t ✐s ✐♥❞❡❡❞ ❛ ❝•❛ss✐❝❛• r❡s✉•t ❢r♦♠ ✐♥t❡❣r❛t✐♦♥ t❤❡♦r② ✭❬✶✻(✮ t❤❛t ❛♥② d✲❞✐♠❡♥s✐♦♥❛• ❞✐✛❡r❡♥t✐❛• ❢♦r♠ ❝❛♥ ❜❡ ✐♥t❡❣r❛t❡❞ ❛•♦♥❣ X ❛♥❞ t❤✉s ✿ T X : ω → X ω ✭✶✳✶✵✮ ❞❡✜♥❡s ❛♥ ❡•❡♠❡♥t ♦❢ Ω d 0 (E) ′ ✳ ■❢✱ ❢♦r ✐♥st❛♥❝❡✱ X ✐s ❣✐✈❡♥ t❤r♦✉❣❤ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ❣✐✈❡♥ ❜② ❛ ❝❡rt❛✐♥ s♠♦♦t❤ ✐♠♠❡rs✐♦♥ γ : U → E ✇✐t❤ U ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d ✱ t❤❡♥ ✇❡ ❤❛✈❡ ❡①♣•✐❝✐t•② ✿ T X (ω) =
♥❞ ❝♦♥t❛✐♥s ♠❛♥② ♦t❤❡r ✐♥t❡r❡st✐♥❣ ♦❜•❡❝ts✱ ❛♠♦♥❣ t❤❡♠ ❛r❡ t❤❡ ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ♦❢ E✱ ❢♦r ✇❤✐❝❤ ♦♥❡ ❤❛s ❛ ❢♦r♠✉•❛ s✐♠✐•❛r t♦ ❡q✳✭✶✳✶✵✮✳ ■♥❞❡❡❞✱ ❛s ❡①♣•❛✐♥❡❞ ♣r❡✈✐♦✉s•②✱ ❢♦r ❛♥② d✲ ❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t X✱ t❤❡r❡ ❡①✐sts ❢♦r H d ✲❛•♠♦st ❛•• x ∈ X ❛♥ ♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡ ❛♥❞ t❤✉s ❛ ✉♥✐t d✲✈❡❝t♦r τ (x) = τ 1 (x) ∧ ... ∧ τ d (x)✱ ✇❤❡r❡ τ 1 (x), ..., τ d (x) ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡✳ ❚❤❡♥✱ s✐♠✐•❛r•② t♦ t❤❡ s✉❜♠❛♥✐❢♦•❞ ❝❛s❡✱ t♦ X ❝♦rr❡s♣♦♥❞s t❤❡ d✲❝✉rr❡♥t ✿ T X (ω) . = X ω x (τ (x))dH d (x) . ✭✶✳✶✶✮ ❆❣❛✐♥✱ t❤✐s ❡①♣r❡ss✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♦r✐❡♥t❡❞ ♦rt❤♦♥♦r♠❛• ❜❛s✐s τ 1 , ..., τ d ❛t ❡❛❝❤ ♣♦✐♥t x ∈ X✳ ❆s ❡❛s✐•② ❝❤❡❝❦❡❞✱ ✐❢ t✇♦ r❡❝t✐✜❛❜•❡ s✉❜s❡ts X ❛♥❞ Y ❛r❡ s✉❝❤ t❤❛t T X = T Y ✱ t❤❡♥ X ❛♥❞ Y ❝♦✐♥❝✐❞❡ ✉♣ t♦ s❡t ♦❢ ③❡r♦ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡✳ ❚❤❡ tr❛♥s♣♦rt ♦❢ ❝✉rr❡♥ts ❜② ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ E ❝❛♥ ❜❡ ❡❛s✐•② ❡①♣r❡ss❡❞ t❤r♦✉❣❤ t❤❡ ♣✉••✲❜❛❝❦ ❛♥❞ ♣✉s❤✲❢♦r✇❛r❞ ♦♣❡r❛t✐♦♥s✳ ❚❤❡s❡ ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛•• T ∈ Ω d 0 (E) ′ ❛♥❞ φ ∈ ❉✐✛(E) ❜② t❤❡ r❡•❛t✐♦♥s ✿ (φ ♯ T ) (ω) = T φ ♯ ω ✭✶✳✶✷✮ ✷✽ ✇❤❡r❡ φ ♯ ω ✐s t❤❡ ♣✉••✲❜❛❝❦ ♦❢ ❛ ❞✐✛❡r❡♥t✐❛• ❢♦r♠ ❜② φ ❞❡✜♥❡❞ ❢♦r ❛•• x ∈ E ❛♥❞ ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E ✿ φ ♯ ω x (ξ) = ω φ(x) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ p )) ✭✶✳✶✸✮ d x φ ❜❡✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✉s❡ ❢♦r t❤❡ ❞✐✛❡r❡♥t✐❛• ♦❢ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ ❛t ♣♦✐♥t x✳ ❚❤❡ r❡❛s♦♥ ♦❢ s✉❝❤ ❞❡✜♥✐t✐♦♥s ✐s t❤❛t ✐t ❣✐✈❡s ❛ ❝♦♠♣❛t✐❜✐•✐t② ❜❡t✇❡❡♥ t❤❡ tr❛♥s♣♦rt ♦❢ s❤❛♣❡s ❛♥❞ t❤❡ ♦♥❡ ♦❢ ❝✉rr❡♥ts ✿ Pr♦♣❡rt② ✶✳✸✳✷✳ ❋♦r ❛♥② r❡❝t✐✜❛❜•❡ s✉❜s❡t X ♦❢ E ❛♥❞ ❞✐✛❡♦♠♦r♣❤✐s♠ φ ∈ ❉✐✛(E)✱ ✇❡ ❤❛✈❡ ✿ φ ♯ T X = T φ(X) .
(d + 1)✲✈♦•✉♠❡ ❞❡•✐♠✐t❡❞ ❜② t❤❡ t✇♦ s❤❛♣❡s✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤❡ ✢❛t ♥♦r♠ ❤❛s ♥♦ ❝•♦s❡❞ ❢♦r♠ ✐♥ ❣❡♥❡r❛• ✇❤✐❝❤ ♠❛❦❡s ✐t ✈✐rt✉❛••② ✐♠♣♦ss✐❜•❡ t♦ ✉s❡ ✐♥ ❛♥② ♥✉♠❡r✐❝❛• ♣✉r♣♦s❡✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✱ t❤❡ ✐❞❡❛ ♦❢ ✐♥tr♦❞✉❝✐♥❣ ♦♥❝❡ ❛❣❛✐♥ |❡♣r♦❞✉❝✐♥❣ ❑❡r♥❡• ❍✐•❜❡rt ❙♣❛❝❡s ✇❛s ♣r♦♣♦s❡❞ ✐♥ ❬✹✶(✳ ■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ •✐♥❡❛r ❢♦r♠s ♦♥ Ω d 0 (E)✱ ♦♥❡ ✐♥t❡♥❞s t♦ r❡str✐❝t t♦ ❛ ❍✐•❜❡rt s♣❛❝❡ W ❡♠❜❡❞❞❡❞ ✐♥t♦ Ω d 0 (E)✳ ❆ ❝♦♥✈❡♥✐❡♥t s❡tt✐♥❣ ❢♦r ❜✉✐•❞✐♥❣ s✉❝❤ ❍✐•❜❡rt s♣❛❝❡s ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ✐s t❤❡ ♦♥❡ ♦❢ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡•s ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✷✳✹✳ ❲❡ s❤❛•• ✉s❡ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ Λ d E * ✇✐t❤ Λ d E ❜❛s❡❞ ♦♥ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ❞❡✜♥❡❞ ♣r❡✈✐♦✉s•② ♦♥ Λ d E t♦ ❝♦♥s✐❞❡r ✐♥❞✐✛❡r❡♥t•② t❤♦s❡ t✇♦ s♣❛❝❡s✳ ❚❤❛t ❜❡✐♥❣ s❛✐❞✱ ✇✐t❤ t❤❡ t❡r♠✐♥♦•♦❣② ♦❢ s❡❝t✐♦♥ ✶✳✷✳✹✱ ✇❡ ❛r❡ •❡❛❞ t♦ ❝♦♥s✐❞❡r ✐♥ t❤✐s ❝❛s❡ Λ d E✲❦❡r♥❡•s ♦♥ E ❛♥❞ ✇❡ ❤❛✈❡ t❤❡ ❡①❛❝t s❛♠❡ ♣r♦♣❡rt② ❛s ✐♥ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✷ ✿ Pr♦♣♦s✐t✐♦♥ ✶✳✸✳✶✳ ❆ss✉♠❡ t❤❛t K ✐s ❛ Λ d E✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ E s✉❝❤ t❤❛t K ✐s ❝♦♥t✐♥✉♦✉s•② ❞✐✛❡r❡♥t✐❛❜•❡ ♦❢ ♦r❞❡r 2k ❛♥❞ s✉❝❤ t❤❛t ❛•• t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ K ✉♣ t♦ t❤❡ ♦r❞❡r k ❛r❡ ❜♦✉♥❞❡❞✳ ❆ss✉♠❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t ❢♦r ❛♥② x ∈ E✱ t❤❡ ❢✉♥❝t✐♦♥ K(x, .) ❛♥❞ ❛•• ✐ts ❞❡r✐✈❛t✐✈❡s ✉♣ t♦ ♦r❞❡r k ✈❛♥✐s❤ ❛t ✐♥✜♥✐t②✳ ❚❤❡♥✱ W ✱ t❤❡ |❑❍❙ ❛ss♦❝✐❛t❡❞ t♦ K✱ ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ Ω d 0,k (E)✳ ■❢ t❤❡ ♣r❡✈✐♦✉s ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❦❡r♥❡• ❛r❡ ❢✉•✜••❡❞ ❢♦r k = 0✱ W ✐s t❤❡♥ ❝♦♥t✐♥✲ ✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ t❤❡ s♣❛❝❡ ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s Ω d 0 (E)✳ ❲❡ t❤❡♥ ♦❜t❛✐♥ t❤❡ ❞✉❛• ❍✐•❜❡rt✐❛♥ str✉❝t✉r❡ ♦♥ W ′ t❤❛t ✐♥❞✉❝❡s ❛ ❞✐st❛♥❝❡ ♦♥ t❤❡ s♣❛❝❡ ♦❢ d✲❝✉rr❡♥ts ♣r♦✈✐❞❡❞ W ✐s ❞❡♥s❡ ✐♥ Ω d 0 (E)✳ ❖♥❡ ❝❛♥ ❛•s♦ s❤♦✇ ❛ ❢❡✇ str❛✐❣❤t❢♦r✇❛r❞ ❝♦♥tr♦• ♣r♦♣❡rt✐❡s ♦❢ s✉❝❤ ♠❡tr✐❝s✳ ■♥❞❡❡❞✱ t❤❡ ❝♦♥t✐♥✉♦✉s ❡♠❜❡❞❞✐♥❣ W ֒→ Ω d 0 (E) ❣✐✈❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦♥st❛♥t c W > 0 s✉❝❤ t❤❛t ❢♦r ❛•• ω ∈ W ✿ ω ∞ ≤ c W ω W ◆♦✇✱ ❢♦r ❛♥② T ∈ W ′ ✱ s✐♥❝❡ T W ′ = sup ω W =1 |T (ω)|✱ ✐t r❡s✉•ts ❡❛s✐•② t❤❛t ✿ ∀T ∈ W ′ , T W ′ ≤ c W M (T ) ✭✶✳✶✻✮ ❛♥❞ ✐♥ t❤❡ s♣❡❝✐❛• ❝❛s❡ ♦❢ ❝✉rr❡♥ts T X ✇❤❡r❡ X ✐s ❛ r❡❝t✐✜❛❜•❡ s✉❜s❡t✱ ✇❡ s❡❡ t❤❛t ✿ T X W ′ ≤ c W H d (X) ✭✶✳✶✼✮ ❆ s✐♠✐•❛r ❝♦♥tr♦• ❝❛♥ ❜❡ ❛•s♦ ♦❜t❛✐♥❡❞ ❜② t❤❡ ✢❛t ♥♦r♠ ✐❢ ♦♥❡ ♠❛❦❡s t❤❡ str♦♥❣❡r ❛ss✉♠♣t✐♦♥ t❤❛t K s❛t✐s✜❡s t❤❡ ❤②♣♦t❤❡s❡s ♦❢ ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✶ ❢♦r k = 1✱ ✐♠♣•②✐♥❣ t❤❛t W ֒→ Ω d 0,1 (E)✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t cW s✉❝❤ t❤❛t ω ∞ + dω ∞ ≤ cW ω W ❛♥❞ •✉st ❛s ❜❡❢♦r❡ ✿ ∀T ∈ W ′ , T W ′ ≤ cW F (T ) ✭✶✳✶✽✮ ▲❛st ❜✉t ♥♦t •❡❛st✱ t❤❡ ❢❛❝t t❤❛t W ✐s ❛ |❑❍❙ ✐♠♣•✐❡s ✐♥ ♣❛rt✐❝✉•❛r t❤❛t t❤❡ ❉✐r❛❝ ❝✉rr❡♥ts δ ξ x ❜❡•♦♥❣ t♦ W ′ ❛♥❞ t❤❡ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ♣r♦♣❡rt② ❣✐✈❡s ✐♥ ❛❞❞✐t✐♦♥ t❤❛t ❢♦r ❛•• x 1 , x 2 ∈ E ❛♥❞ ξ 1 , ξ 2 ∈ Λ d E ✿ δ ξ 1 x 1 , δ ξ 2 x 2 W ′ = K(x 1 , .)ξ 1 , K(x 2 , .)ξ 2 W = ξ 1 , K(x 1 , x 2 )ξ 2 L v ω . = d dt t=0 (φ t ) ♯ ω . ✭✶✳✷✵✮ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❜❛s✐❝❛••② ❝♦♥♥❡❝ts t❤✐s ♥♦t✐♦♥ ♦❢ ❞❡r✐✈❛t✐✈❡ ✇✐t❤ t❤❡ ♠♦r❡ ✉s✉❛• ❡①t❡r✐♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ d✳ ❋♦r t❤✐s✱ ♦♥❡ ♥❡❡❞s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥tr❛❝t✐♦♥ ♦♣❡r❛t♦r ♦♥ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✳ ❋♦r ❛ ✈❡❝t♦r ✜❡•❞ v ❛s ♣r❡✈✐♦✉s•②✱ t❤❡ ❝♦♥tr❛❝t✐♦♥ ♦♣❡r❛t✐♦♥ ı v : Ω d+1 0 (E) → Ω d 0 (E) ✐s ♦❜t❛✐♥❡❞ ❜② ✿ ∀ω ∈ Ω d+1 0 (E), (ı v ω) x (ξ) . = ω x (v(x) ∧ ξ) ✭✶✳✷✶✮ ❢♦r ❛♥② x ∈ E ❛♥❞ ξ ∈ Λ d E✳ ◆♦✇ t❤❡ r❡s✉•t ✐s t❤❡ ❢♦••♦✇✐♥❣ ✿ Pr♦♣♦s✐t✐♦♥ ✶✳✸✳✷✳ ❋♦r ❛•• ω ∈ Ω d 0,1 (E) ❛♥❞ s♠♦♦t❤ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ✈❡❝t♦r ✜❡•❞ v ♦❢ E✱ L v ω = ı v (dω) + d(ı v ω) . ❚❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❛♥ ❡•❡♠❡♥t❛r② ✇❛② ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ ❞✐r❡❝t•② (φ t ) ♯ ω ✉s✲ ✐♥❣ ❡q✳✭✶✳✶✸✮ ❛♥❞ r❡❛s♦♥ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥ d ❛s ❞♦♥❡ ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✼✶(✳ ❖t❤❡r ♣r♦♦❢s ❜❛s❡❞ ♦♥ ♠♦r❡ ❛❜str❛❝t ❝♦♥str✉❝t✐♦♥s ❝❛♥ ❜❡ ❛•s♦ ❢♦✉♥❞ ✐♥ ❬✺✷(✳ ❚❤✐s r❡s✉•t ♦♥ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ❣✐✈❡s ❛♥ ✐♥t❡r❡st✐♥❣ ❝♦r♦••❛r② ✐♥ t❡r♠s ♦❢ ❝✉rr❡♥ts ❛ss♦❝✐❛t❡❞ t♦ ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞s ✿ ✸✶ ❚❤❡♦r❡♠ ✶✳✸✳✷✳ ▲❡t X ❜❡ ❛ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ ✭♣♦ss✐❜•② ✇✐t❤ ❜♦✉♥❞❛r②✮✱ v ❛ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ s♠♦♦t❤ ✈❡❝t♦r ✜❡•❞ ❛♥❞ φ t ✐ts ✢♦✇✳ ❚❤❡♥ ❢♦r ❛♥② C 1 d✲ ❞✐✛❡r❡♥t✐❛• ω ♦♥ E ✿ d dt t=0 T φt(X) (ω) = X L v ω = X ı v (dω) + ∂X ı v ω .
Pr♦♦❢✳ ■t ✐s ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞ ❢r♦♠ ❈❛rt❛♥✬s ❢♦r♠✉•❛✳ ■♥❞❡❡❞✱ t❤❛♥❦s t♦ ♣r♦♣✲ ❡rt② ✶✳✸✳✷✱ ✇❡ ❤❛✈❡ ✿

T φt(X) (ω) = φ ♯ t T (ω) = T X ((φ t ) ♯ ω) = X (φ t ) ♯ ω . ❙✐♥❝❡ X ✐s ❝♦♠♣❛❝t✱ ♦♥❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛• ❛♥❞ ❣❡t ✿ d dt t=0 T φt(X) (ω) = X L v ω = X ı v (dω) + X d(ı v ω) .
❋r♦♠ ❙t♦❦❡s✬ t❤❡♦r❡♠✱ t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ t❤❡ s✉♠ ❡q✉❛•s ✿

X d(ı v ω) = ∂X ı v ω ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢✳
|♦✉❣❤•② s♣❡❛❦✐♥❣✱ t❤✐s r❡s✉•t ❡①♣r❡ss❡s ❤♦✇ t❤❡ ❝✉rr❡♥t ❛ss♦❝✐❛t❡❞ t♦ ❛ s❤❛♣❡ X ✈❛r✐❡s ✇❤❡♥ X ✐s ❞❡❢♦r♠❡❞ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ✈❡❝t♦r ✜❡•❞ v✳ ■t ❝❛♥ ❜❡ t❤✉s ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❡r♠s T X (ω) ✇✐t❤ r❡s♣❡❝t t♦ X✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❣r❛❞✐❡♥ts ♦❢ t❤❡ ♣r❡✈✐♦✉s•② ❞❡✜♥❡❞ |❑❍❙ ♠❡tr✐❝s ❛r❡ ❛•s♦ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡♦r❡♠ ✶✳✸✳✷ ❀ ✐❢ t❤❡ ❦❡r♥❡• ✐s s✉❝❤ t❤❛t W ֒→ Ω d 0,1 (E)✱ s✐♥❝❡ ❢♦r ❛♥② ❝✉rr❡♥t T ∈ W ′ ✱ T X , T W ′ = T X (K W T ) ✭✇❡ r❡♠✐♥❞ t❤❛t K W ✐s t❤❡ ♥❛t✉r❛• ✐s♦♠❡tr② ❜❡t✇❡❡♥ W ′ ❛♥❞ W ✮✱ ✇❡ ❤❛✈❡ t❤❡♥ T X , T W ′ = T X (ω) ✇❤❡r❡ ω = K W T ∈ W ✐s ❛ C 1 d✲❞✐✛❡r❡♥t✐❛• ❢♦r♠ t❤❛♥❦s t♦ t❤❡ ❡♠❜❡❞❞✐♥❣ ♦❢ W ✳ ❚❤✉s t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♣♣•✐❡s t♦ ❡①♣r❡ss ✈❛r✐❛t✐♦♥s ♦❢ ♥♦r♠s ❛♥❞ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❝✉rr❡♥ts✳ ❚❤❡ ✐♥t❡r❡st ♦❢ s✉❝❤ ❢♦r♠✉•❛s ✐s t❤❛t t❤❡② ❣✐✈❡ q✉❛•✐t❛t✐✈❡ ♣r♦♣❡rt✐❡s ❛❜♦✉t t❤❡ ❣r❛✲ ❞✐❡♥ts ♦❢ t❤❡ ♠❡tr✐❝s ✇❡ ❛r❡ ✉s✐♥❣✱ ✇❤✐❝❤ ❞♦ ♥♦t ❛♣♣❡❛r ♦❜✈✐♦✉s ❛t ❛•• ✇❤❡♥ ✇r✐t✐♥❣ t❤❡ ❞✐s❝r❡t❡ ❡①♣r❡ss✐♦♥s ♦❢ ❣r❛❞✐❡♥ts ✉s❡❞ ✐♥ t❤❡ ♥✉♠❡r✐❝❛• ♣❛rt✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ✜rst s❡❡ t❤❡ ♣r❡s❡

♥❝❡ ♦❢ ❛ ❜♦✉♥❞❛r② t❡r♠ s✐❣♥✐❢②✐♥❣ ❛ ♣❛rt✐❝✉•❛r ❜❡❤❛✈✐♦r ❢♦r ♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r② ✇❤❡r❡❛s✱ ✐♥ t❤❡ ✐♥s✐❞❡ ♦❢ X✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✜rst ✐♥t❡❣r❛• ✐♥ t❤❡ ❢♦r♠✉•❛ ❞❡♣❡♥❞s ✐♥ ❢❛❝t ♦♥•② ✐♥ t❤❡ ❝♦♠♣♦♥❡♥t ♦❢ v ♥♦r♠❛• t♦ X ❛t ❡❛❝❤ ♣♦✐♥t ♦❢ X✱ s❤♦✇✐♥❣ ✐♥❝✐❞❡♥t❛••② t❤❛t t❤❡ ❣r❛❞✐❡♥t ✐s ♦rt❤♦❣♦♥❛• t♦ t❤❡ s❤❛♣❡ ❛t ❛•• ♣♦✐♥ts ♦❢ X✱ •❡t ❛•♦♥❡ ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ❲❡ r❡❢❡r t♦ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛♣
♦t ❝♦♠♣❛r❛❜•❡ ❢r♦♠ ♦♥❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ t♦ ❛♥♦t❤❡r ✭❡①❝❡♣t ✐❢ t❤❡ s✉♣♣♦rts ❛r❡ ✐❞❡♥t✐❝❛•✮✳ ❉✐✛❡♦♠♦r♣❤✐s♠s ♦❢ E ❝❛♥ ❛❝t ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❜② s✐♠♣•❡ tr❛♥s♣♦rt ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt✳ ■❢ φ ∈ ❉✐✛(E)✱ φ.(X, f ) ✐s t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ♦❢ s✉♣♣♦rt φ(X) ❛♥❞ s✐❣♥❛• f • φ -1 ✱ ✇❤✐❝❤ ✇❡ ✇r✐t❡ ✿ φ.(X, f ) = (φ(X), f • φ -1 ) ❚❤✐s ❣❡♥❡r❛•✐③❡s t❤❡ tr❛♥s♣♦rt ♦❢ ✐♠❛❣❡s ❞❡s❝r✐❜❡❞ ✐♥ ❡①❛♠♣•❡ ✶✳✷✳✷✳ ■♥ t❤❛t s✐t✉✲ ❛t✐♦♥✱ t❤❡ |✐❡♠❛♥♥✐❛♥ s❡tt✐♥❣ ♦❢ ▲❉❉▼▼ ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✷ ❛♣♣•✐❡s ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞•②✳ ◆❡✈❡rt❤❡•❡ss✱ t❤✐s ♠♦❞❡• ✐s •✐♠✐t❡❞ t♦ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ❝❤❛♥❣❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❛♥❞ ❝❛♥♥♦t ❛❝❝♦✉♥t ❢♦r s♦♠❡ r❡s✐❞✉❛• s✐❣♥❛• ✈❛r✐❛t✐♦♥s t❤❛t ❛r❡ ♦❢t❡♥ ♥❡❝❡ss❛r② t♦ ❝♦♥s✐❞❡r ✐♥ s❡✈❡r❛• ❛♣♣•✐❝❛t✐♦♥s✳ ❋♦r ❝•❛ss✐❝❛• ✐♠❛❣❡s✱ s✉❝❤ ❣❡♥✲ ❡r❛•✐③❡❞ tr❛♥s❢♦r♠❛t✐♦♥s ❝♦♥s✐st✐♥❣ ♦❢ ❜♦t❤ ❛ ❞❡❢♦r♠❛t✐♦♥ ❛♥❞ r❡s✐❞✉❛• ❞✐s♣•❛❝❡♠❡♥ts ✸✸ ✇❛s ✐♥tr♦❞✉❝❡❞ t❤r♦✉❣❤ t❤❡ ❝♦♥❝❡♣t ♦❢ ♠❡t❛♠♦r♣❤♦s✐s ✐♥ ❬✼✽(✳ ■t ❝❛♥ ❜❡ ❣❡♥❡r❛•✐③❡❞ t♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✐♥ ❛ ♥❛t✉r❛• ✇❛②✳ ❋♦r ❛ ❢s❤❛♣❡ (X, f )✱ ✇❡ ❝♦♥s✐❞❡r ❞❡❢♦r♠❛t✐♦♥s (φ, ζ) ✇❤❡r❡ φ ∈ ❉✐✛(E) ❛♥❞ ζ ∈ L 2 (X) t❤❛t ❛❝ts ♦♥ (X, f ) ❜② t❤❡ ❢♦r♠✉•❛ ✿ (φ, ζ).(X, f ) = (φ(X), (f + ζ) • φ -1 ) .
✭✶✳✷✷✮ ❆s ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♠❛❣❡s✱ t❤✐s ❛❝t✐♦♥ ✐s ♥♦t tr❛♥s✐t✐✈❡ ♦♥ t❤❡ s❡t ♦❢ ❛•• ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛♥❞ ♦♥❡ ❤❛s t♦ r❡str✐❝t t♦ s♣❡❝✐✜❝ ♦r❜✐ts✳ ❲❡ ✐♥tr♦❞✉❝❡✱ ❛s ✐♥ ✶✳✷✳✸✱ ❛ ❣r♦✉♣

G V ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ E✱ V ❜❡✐♥❣ ❛ ❍✐•❜❡rt s♣❛❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (E, E)✳ ❋♦r ❛ ✜①❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t X 0 ♦❢ ❞✐♠❡♥s✐♦♥ d✱ ✇❡ ❝❛•• X 0 . = G V .X 0 t❤❡ ♦r❜✐t ♦❢ ❛•• t❤❡ φ(X 0 ) ❢♦r φ ∈ G V ❛♥❞ ❞❡✜♥❡ t❤❡ ❢♦••♦✇✐♥❣ s✉❜s❡t ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✿ F = {(X, f ) | X ∈ X 0 , f ∈ L 2 (X)} . ❚❤❡♥ ✐t✬s str❛✐❣❤t❢♦r✇❛r❞ t❤❛t ❢♦r ❛•• (X, f ) ❛♥❞ (X ′ , f ′ ) ✐♥ F✱ t❤❡r❡ ❡①✐sts (φ, h) ∈ G V × L 2 (X) s✉❝❤ t❤❛t (X ′ , f ′ ) = (φ, h).(X, f )✳ ◆♦✇✱ ❛s ❢♦r ♠❡t❛♠♦r♣❤♦s❡s✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡s❡ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛s r❡s✉•t✐♥❣ ❢r♦♠ ✐♥t❡❣r❛t✐♦♥ ♦❢ ✐♥✜♥✐t❡s✐♠❛• ✈❛r✐❛t✐♦♥s✳ ▲❡t (X, f ) ❜❡ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✐♥ F✳ ❋♦r t❤❡ ❣❡♦♠❡tr✐❝ ♣❛rt✱ ❞✐✛❡♦♠♦r♣❤✐s♠s φ ❛r❡ ❛❣❛✐♥ ❣❡♥❡r❛t❡❞ ❛s t❤❡ ✢♦✇ φ v t ♦❢ ❛ t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞ v ∈ L 2 V ✇❤✐•❡ t❤❡ r❡s✐❞✉❛• s✐❣♥❛•s ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❛s ζ h t = t 0 h s ds✱ h s ∈ L 2 (X) ❜❡✐♥❣ t❤❡ ✐♥st❛♥t❛♥❡♦✉s s♣❡❡❞ ❛t t✐♠❡ s✳ ❲❡ ❡q✉✐♣ t❤❡s❡ ❞❡❢♦r♠❛t✐♦♥s ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ ❡♥❡r❣② ♠❡❛s✉r❡ ✿ E X (v, h) . = γ V 2 1 0 |v t | 2 V dt + γ f 2 1 0 φ v t (X) |h t • (φ v t ) -1 | 2 (x)dH d (x) ✭✶✳✷✸✮ ✇❤❡r❡ γ V ❛♥❞ γ f ❛r❡ •✉st ♣♦♥
φ v t (X) |h t • (φ v t ) -1 | 2 (x)dH d (x) = X |h t | 2 (x)|d x φ v t .ξ(x)|dH d (x)dt ✇❤❡r❡ ξ(x) = ξ 1 (x) ∧ ... ∧ ξ d (x) ✐s t❤❡ ✉♥✐t d✲✈❡❝t♦r r❡♣r❡s❡♥t✐♥❣ t❤❡ t❛♥❣❡♥t s♣❛❝❡ t♦ X ❛t x ❛♥❞ |d x φ v t .ξ(x)| ✐s t❤❡ ♥♦r♠ ♦❢ t❤❡ d✲✈❡❝t♦r d x φ v t (ξ 1 ) ∧ ... ∧ d x φ v t (ξ d )✳ ❚❤✉s t❤❡ ❡♥❡r❣② ❡q✉❛•s ❛•s♦ ✿ E X (v, h) = γ V 2 1 0 |v t | 2 V dt + γ f 2 1 0 X |h t | 2 (x)|d x φ v t .ξ(x)|dH d (x)dt . ✭✶✳✷✹✮ ■t ✐s ❛•s♦ ❛ ❝•❛ss✐❝❛• ❝♦♥tr♦• r❡s✉•t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ C : R + → R + s✉❝❤ t❤❛t✱ ❢♦r ❛•• x ∈ E✱ t ∈ [0, 1] ❛♥❞ ξ = ξ 1 ∧ ... ∧ ξ d ∈ Λ d E✱ ✇❡ ❤❛✈❡ |d x φ v t (ξ 1 ) ∧ ... ∧ d x φ v t (ξ d )| ≤ C 1 0 v t 1,∞ dt |ξ 1 ∧ ... ∧ ξ d | . ❲❡ r❡❢❡r t♦ ❬✽✻( ❢♦r ✐♥st❛♥❝❡ ♦r t♦ ✷✳✷✳✸ ✐♥ ❝❤❛♣t❡r ✷✳ ❙✐♥❝❡ v 1,∞ ≤ c V v V ✱ ✇❡ ❡✈❡♥t✉❛••② ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ C s✉❝❤ t❤❛t ❢♦r ❛•• (v, h)✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❜♦✉♥❞ ♦♥ t❤❡ ❡♥❡r❣② ✿ E X (v, h) ≤ γ V 2 1 0 v t 2 V dt + γ f 2 .C 1 0 v t 2 V dt 1 0 h t 2 L 2 (X) dt .

✭✶✳✷✺✮ ✸✹

❆ |✐❡♠❛♥♥✐❛♥ ❢r❛♠❡✇♦r❦ ❢♦r ❢s❤❛♣❡s ♠❡t❛♠♦r♣❤♦s❡s ❲❡ ❝❛♥ ♥♦✇ ❡①t❡♥❞ t❤❡ ♣r❡✈✐♦✉s ✐❞❡❛ ♦❢ ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡s t♦ t❤✐s s✐t✉❛t✐♦♥ ♦❢ ❢s❤❛♣❡s✬ ♠❡t❛♠♦r♣❤♦s❡s✳ ❲❡ r❡str✐❝t✱ ❛s ♣r❡✈✐♦✉s

•②✱ t♦ ❛ s❡t F ❢♦r ✇❤✐❝❤ X 0 = G.X 0 ❛♥❞ X 0 ✐s ❛•s♦ ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♠♣❛❝t✳ ■t ✐♠♣•✐❡s t❤❛t ❢♦r ❛•• X ∈ X 0 ✱ X ✐s ❝♦♠♣❛❝t✳ ❲❡ t❤❡♥ ❞❡✜♥❡✱ ❢♦r ❛•• (X, f ) ❛♥❞ (X ′ , f ′ ) ✐♥ F ✿ d F ((X, f ), (X ′ , f ′ )) . = inf (v,h) E X (v, h) 1 2 | (φ v 1 , ζ h 1 ).(X, f ) = (X ′ , f ′ ) . ✭✶✳✷✻✮ ❚❤❡♦r❡♠ ✶✳✹✳✶✳ ❚❤❡ ❢✉♥❝t✐♦♥ d F ❣✐✈❡♥ ❜② ❡q✳✭✶✳✷✻✮ ❞❡✜♥❡s ❛ ❞✐st❛♥❝❡ ♦♥ F ✐❡✳ ✐s s②♠♠❡tr✐❝✱ s❛t✐s✜❡s t❤❡ tr✐❛♥❣•❡ ✐♥❡q✉❛•✐t② ❛♥❞ d F ((X, f ), (X ′ , f ′ )) = 0 ✐❢ ❛♥❞ ♦♥•② ✐❢ X = X ′ ❛♥❞ f = f ′ ∈ L 2 (X)✳ Pr♦♦❢✳ ❆ ✜rst ♣♦✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ ✐s t❤❛t t❤❡ ❞✐st❛♥❝❡ d F ✐s ✜♥✐t❡ ❜❡t✇❡❡♥ ❛♥② ♣❛✐r ♦❢ ❢s❤❛♣❡s✳ ■♥❞❡❡❞✱ ❜② ❞❡✜♥✐t✐♦♥ t❤❡r❡ ❡①✐sts ❜❡t✇❡❡♥ ❛♥② X ❛♥❞ X ′ ∈ X 0 ❛ ♣❛t❤ t → φ v t .X ✇✐t❤ v ∈ L 2 ([0, 1], V )✳ ◆♦✇✱ ✐❢ h t = (f ′ • φ v 1 -f )✱ ✇❡ ❣❡t (v, h) ∈ L 2 ([0, 1], V × L 2 (X)) ❛♥❞ f ′ = (f + ζ v 1 ) • (φ v 1 ) -1 s♦ t❤❛t d F ((X, f ), (X ′ , f ′ )) ≤ E X (v, h) 1/2 < ∞ . ❚❤❡ s②♠♠❡tr② ❝♦♠❡s ❢r♦♠ ❛ ✉s✉❛• ♣r♦❝❡ss ♦❢ t✐♠❡ r❡✈❡rs❛•✳ ▲❡t (X, f ), (X ′ , f ′ ) ∈ F ❛♥❞ (v, h) ∈ L 2 ([0, 1], V × L 2 (X)) s✉❝❤ t❤❛t X ′ = φ v 1 .X, f ′ = (f + ζ h 1 ) • (φ v 1 ) -1 . ✭✶✳✷✼✮ ■❢ ✇❡ ❞❡✜♥❡ ṽt . = -v 1-t ❛♥❞ ht . = -h 1-t • (φ v 1 ) -1 ❢♦r ❛♥② t ∈ [0, 1]✱ t❤❡♥ ❢♦r X t . = φ v t .X ✭s♦ t❤❛t X 0 = X ❛♥❞ X 1 = X ′ ✮ ✇❡ ❤❛✈❡ φ v t .X 0 = X t = φ ṽ 1-t .X 1 ❛♥❞ (ṽ, h) ∈ L 2 V × L 2 (X ′ ) ✇✐t❤ E X (v, h) = E X ′ (ṽ, h) . ✭✶✳✷✽✮ ❙✐♥❝❡ ♦♥❡ ❡❛s✐•② ❝❤❡❝❦s t❤❛t ζ h 1 = -ζ h 1 • (φ v 1 ) -1 s♦ t❤❛t ✐❢ f ′ = (f + ζ h 1 ) • (φ v 1 ) -1 ✇❡ ❤❛✈❡ f = (f ′ + ζ h 1 ) • (φ ṽ 1 ) -1 ✇✐t❤ X = φ ṽ 1 .X ′ ❚❤✐s ❣✐✈❡s ✐♠♠❡❞✐❛t❡•② t❤❡ s②♠♠❡tr②✳ ❆s ❢♦r t❤❡ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t②✱ ✐t ❢♦••♦✇s ❢r♦♠ ❛ ✉s✉❛• ♣r♦❝❡ss ♦❢ ♣❛t❤ ❝♦♥❝❛t❡♥❛t✐♦♥✳ ▲❡t (X, f )✱ (X, f ′ ) ❛♥❞ (X ′′ , f ′′ ) ❜❡ t❤r❡❡ ❢s❤❛♣❡s ✐♥ F s✉❝❤ t❤❛t d F ((X, f ), (X ′ , f ′ )) > 0 ❛♥❞ d F ((X ′ , f ′ ), (X ′′ , f ′′ )) > 0✳ ❖♥❡ ❡❛s✐•② ❝❤❡❝❦s t❤❛t ❢♦r (v, h) ∈ L 2 ([0, 1], V × L 2 (X)) ❛♥❞ (v ′ , h ′ ) ∈ L 2 ([0, 1], V × L 2 (X ′ )) ✇✐t❤ (X ′ , f ′ ) = (φ v 1 .X, (f + ζ h 1 ) • (φ v 1 ) -1 ) ❛♥❞ (X ′′ , f ′′ ) = (φ v ′ 1 .X ′ , (f ′ + ζ h ′ 1 ) • (φ v ′ 1 ) -1 ) t❤❡♥ ❞❡♥♦t✐♥❣ ❢♦r α, β > 1 s✉❝❤ t❤❛t 1/α + 1/β = 1 ❛♥❞ s ∈ [0, 1] ❈❛t α ((v ′ , h ′ ), (v, h)) s . = β(v ′ β(s-1/α) , h ′ β(s-1/α) • φ)1 s≥1/α + α(v αs , h αs )1 0≤s<1/α ✭✶✳✷✾✮ ✇✐t❤ φ = φ v 1 ✱ ✇❡ ❤❛✈❡ ❢♦r (ṽ, h) . = ❈❛t α ((v ′ , h ′ ), (v, h)) t❤❛t (ṽ, h) ∈ L 2 ([0, 1], V × L 2 (X)) ❛♥❞ (X ′′ , f ′′ ) = (φ ṽ 1 .X, (f + ζ h 1 ) • (φ ṽ 1 ) -1 ) ✭✶✳✸✵✮ ✸✺ s♦ t❤❛t d F ((X, f ), (X ′′ , f ′′ )) ≤ E X (ṽ, h) 1/2 . ✭✶✳✸✶✮ ❍♦✇❡✈❡r✱ ❢♦r α * = (E X (v, h) 1/2 + E X ′ (v ′ , h ′ ) 1/2 )/E X (v, h) 1/2 ✱ ✇❡ ❝❤❡❝❦ ❡❛s✐•② t❤❛t E X (ṽ, h) 1/2 = E X (v, h) 1/2 + E(v ′ , h ′ ) 1/2 ✭✶✳✸✷✮ ❛♥❞ t❤❡ tr✐❛♥❣•❡ ✐♥❡q✉❛•✐t② ❢♦••♦✇s ✐♠♠❡❞✐❛t❡•②✳ ◆♦✇✱ ✐❢ d F ((X, f ), (X ′ , f ′ )) = 0 t❤❡♥ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ φ n s✉❝❤ t❤❛t φ n .X = φ n (X) = X ′ ❛♥❞ φ n → ■❞ ♦♥ X✳ ■♥ ♣❛rt✐❝✉•❛r✱ X ′ ✐s ❞❡♥s❡ ✐♥ X✳ ❆s X ′ ✐s ❝♦♠♣❛❝t✱ X ⊂ X ′ ✳ ❇② s②♠♠❡tr② ✇❡ ❣❡t X = X ′ ✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ζ n ∈ L 2 (X) s✉❝❤ t❤❛t ζ → 0 ✐♥ L 2 (X) ❛♥❞ f ′ = (f + ζ n ) • φ -1 n ✇✐t❤ φ n (X) = X✳ ❲❡ ❣❡t X (f ′ -f ) 2 ≤ 2 X (f • φ -1 n -f ) 2 + 2 X (ζ n • φ -1 n ) 2 → 0 ❛♥❞ t❤❡ r❡s✉•t ✐s ♣r♦✈❡❞✳ ❚❤❡ ❞✐st❛♥❝❡ d F ❡①t❡♥❞s t♦ ❢s❤❛♣❡s t❤❡ ♠❡t❛♠♦r♣❤♦s✐s ♠❡tr✐❝s ♦♥ ✐♠❛❣❡s ❡①❛♠✐♥❡❞ ✐♥ ❬✼✽(✳ ❆ ❢✉♥❞❛♠❡♥t❛• q✉❡st✐♦♥ ✐s t❤❡♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐♥✐♠✐③✐♥❣ ❣❡♦❞❡s✐❝s ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✐♥ t❤❡ ♦r❜✐t F✳ ❚❤✐s ✐s ❛♥s✇❡r❡❞ ❜② t❤❡ r❡s✉•t ❜❡•♦✇ ✿ ❚❤❡♦r❡♠ ✶✳✹✳✷✳ ❋♦r ❛♥② (X, f )✱ ❛♥❞ (X ′ , f ′ ) ✐♥ F✱ t❤❡r❡ ❡①✐sts (v, h) ∈ L 2 ([0, 1], V × L 2 (X)) s✉❝❤ t❤❛t E X (v, h) 1/2 = d F ((X, f ), (X ′ , f ′ ))✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ (X t , f t ) . = (φ v t .X, (f + t 0 h s ds) • (φ v t ) -1 )
✭✶✳✸✸✮ t❤❡ ♣❛t❤ t → (X t , f t ) ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ♠✐♥✐♠✐③✐♥❣ ❣❡♦❞❡s✐❝ ❜❡t✇❡❡♥ (X, f ) ❛♥❞ (X ′ , f ′ )✳

Pr♦♦❢✳ ■t ✐s s✉✣❝✐❡♥t t♦ s❤♦✇ t❤❛t (v, h) → E X (v, h) ✐s •♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s ❢♦r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s♣❛❝❡ L 2 ([0, 1], V × L 2 (X))✳ ■♥❞❡❡❞✱ ✐❢ t❤✐s ✐s t❤❡ ❝❛s❡✱ t❤❡♥ ❢r♦

♠ ❛♥② ♠✐♥✐♠✐③✐♥❣ s❡q✉❡♥❝❡ (v n , h n ) s✉❝❤ t❤❛t E X (v n , h n ) → d F ((X, f ), (X ′ , f ′ )) 2 ✇❡ ❞❡❞✉❝❡ t❤❛t v n ✐s ❜♦✉♥❞❡❞ ♦♥ L 2 ([0, 1], V ) ❛♥❞ ✿ 1 0 X |h n,t (x)| 2 dH d (x)dt ≤ 1 0 (sup x∈X |d x φ vn t (ξ(x))| -1 ) X (|h n,t (x)| 2 )|d x φ vn t .ξ(x)|dH d (x) ◆♦✇✱ sup x∈X |d x φ vn t .ξ(x)| -1 = 1/ inf x∈X |d x φ vn t .ξ(x)| ❛♥❞✱ ✐♥ ❛ s✐♠✐•❛r ✇❛② ❛s ♠❡♥✲ t✐♦♥❡❞ ❡❛r•✐❡r✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ C : R + → R * + ✐♥❞❡♣❡♥❞❡♥t ♦❢ v s✉❝❤ t❤❛t ❢♦r ❛•• x ∈ E✱ |d x φ v t .ξ(x)| ≥ C 1 0 v t 2 V dt . ■t r❡s✉•ts t❤❛t t❤❡r❡ ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ C s✉❝❤ t❤❛t ✿ 1 0 X |h n,t (x)| 2 dH d (x)dt ≤ C 1 0 v n,t 2 V dt X (|h n,t (x)| 2 )|d x φ vn t .ξ(x)|dH d (x) ≤ C 1 0 v n,t 2 V dt E X (v n , h n ) ✸✻ ❛♥❞ ✇❡ ❣❡t t❤❛t t❤❡ s❡q✉❡♥❝❡ (h n ) ✐s ❜♦✉♥❞❡❞ ✐♥ L 2 ([0, 1], L 2 (X))✳ ❍❡♥❝❡✱ ❜② ✇❡❛❦ ❝♦♠♣❛❝t♥❡ss ♦❢ str♦♥❣ ❜❛••s ✐♥ L 2 ([0, 1], V × L 2 (X))✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t✱ ✉♣ t♦ t❤❡ ❡①tr❛❝t✐♦♥ ♦❢ ❛ s✉❜✲s❡q✉❡♥❝❡✱ (v n , h n ) ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦✇❛r❞s (v ∞ , h ∞ ) ∈ L 2 ([0, 1], V × L 2 (X)) ❛♥❞ ❜② •♦✇❡r s❡♠✐✲❝♦♥t✐♥✉✐t② ♦❢ E X ✱ ✇❡ ❞❡❞✉❝❡ t❤❛t E X (v ∞ , h ∞ ) ≤ d F ((X, f ), (X ′ , f ′ )) 2 . ❲❡ ♦♥•② ♥❡❡❞ t♦ ❝❤❡❝❦ t❤❛t (X ′ , f ′ ) = (φ v∞ 1 (X), (f + ζ h∞ 1 ) • (φ v∞ 1 ) -1 )✳ ❚❤✐s •❛st r❡s✉•ts ❢♦••♦✇ ❢r♦♠ t❤❡ ❢❛❝t ✐❢ (v n , h n ) ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦ (v ∞ , h ∞ ) t❤❡♥ φ vn 1 → φ v∞ 1 ✉♥✐❢♦r♠•② ♦♥ ❛♥② ❝♦♠♣❛❝t s❡ts ✭✇❤✐❝❤ ✐s ❛ ✇❡••✲❦♥♦✇♥ r❡s✉•t ♣r♦✈❡❞ ✐♥ ❬✽✻( t❤❡♦r❡♠ ✶✷✳✶✷ ❢♦r ✐♥st❛♥❝❡✮ ❛♥❞ (f + ζ hn 1 ) • (φ vn 1 ) -1 ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦ (f + ζ h∞ 1 ) • (φ v∞ 1 ) -1 ✳ ❲❡ ♥♦✇ ❝♦♠❡ t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ •♦✇❡r s❡♠✐✲❝♦♥t✐♥✉✐t② ♦❢ E X ✳ ❈♦♥s✐❞❡r✐♥❣ ❛❣❛✐♥ ❛ s❡q✉❡♥❝❡ (v n , h n ) t❤❛t ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦✇❛r❞ (v ∞ , h ∞ ) ✐♥ L 2 ([0, 1], V × L 2 (X))✱ ✇❡ ✜rst ❦♥♦✇ ❢r♦♠ ❛ ❝•❛ss✐❝❛• r❡s✉•t ♦♥ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t❤❛t 1 0 v ∞,t 2 V dt ≤ lim inf n→∞ 1 0 |v n,t | 2 V dt✳ ❇❡s✐❞❡s✱ ✇❡ ❤❛✈❡ 1 0 X |h ∞,t (x)| 2 |d x φ v∞ t .ξ(x)|dH d (x) = lim n→∞ 1 0 h n,t (x)h ∞,t (x)|d x φ v∞ t .ξ(x)|dH d (x) ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ (h n )✳ ◆♦✇✱ s✐♥❝❡ d x φ vn t → d x φ v∞ t ✉♥✐❢♦r♠•② ♦♥ t ∈ [0, 1] ❛♥❞ x ∈ X✱ ✇❡ ❣❡t 1 0 X |h ∞,t (x)| 2 |d x φ v∞ t .ξ(x)|dH d (x) = lim n→∞ 1 0 X h n,t (x)h ∞,t (
♥❛• s❤❛♣❡s✳ ■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ r❡s✐❞✉❛• s✐❣♥❛•s ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐♥t❡❣r❛t✐♥❣ ✈❡•♦❝✐t✐❡s ζt = h t ❛♥❞ ❝♦♠♣✉t❡ ❛♥ ❡♥❡r❣② ❛•♦♥❣ t✐♠❡✲❞❡♣❡♥❞❡♥t s♣❛❝❡s L 2 (X t )✱ ✇❡ ❝❛♥ ❢r❡❡③❡ t❤❡ ♠❡tr✐❝ t♦ ✐ts ✐♥✐t✐❛• s✉♣♣♦rt X✱ ♥❛♠❡•② r❡♣•❛❝❡ t❤❡ r❡s✐❞✉❛• ♣❛rt ♦❢ t❤❡ ❡♥❡r❣② ❜② ✿ 1 0 X | ζt (x)| 2 dH d (x)dt . ❈♦♥tr❛r✐•② t♦ ❡q✳✭✶✳✷✹✮✱ t❤✐s ❡♥❡r❣② ❞♦❡s ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s✉♣♣♦rt ♦❢ t❤❡ s✐❣♥❛• ζ t t❤r♦✉❣❤ t❤❡ •♦❝❛• ❝❤❛♥❣❡ ♦❢ ✈♦•✉♠❡ |d x φ v t .ξ(x)|✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ✐s t❤❛t ❢♦r ❛ ✜①❡❞ ❡♥❞✲♣♦✐♥t ❝♦♥❞✐t✐♦♥ ζ = ζ 1 ✱ t❤❡ ♦♣t✐♠❛• ♣❛t❤ ζ t ❢♦r t❤✐s ❝♦st ✐s ❛•✇❛②s ζ t = tζ ❛♥❞ ✐t r❡s✉•ts t❤❛t t❤❡ ❡♥❡r❣② ❝❛♥ ❜❡ s✐♠♣•② ✇r✐tt❡♥ ✐♥ t❤❛t ❝❛s❡ ✿ ẼX (v, ζ) = γ V 2 1 0 v t 2 V dt + γ f 2 ζ 2 L 2 (X) . ✭✶✳✸✺✮ ❚❤❡♥✱ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ✿ dF ((X, f ), (X ′ , f ′ )) . = inf (v,ζ)∈L 2 V ×L 2 (X) ẼX (v, ζ) 1 2 | (φ v 1 , ζ).(X, f ) = (X ′ , f ′ ) . ✭✶✳✸✻✮ ❯♥•✐❦❡ t❤❡ ♣r❡✈✐♦✉s d F ❢✉♥❝t✐♦♥✱ dF ✐s ♥♦ ♠♦r❡ ❛ ❞✐st❛♥❝❡ ♦♥ F ✭✐t ❞♦❡s♥✬t s❛t✐s❢② s②♠♠❡tr② ❛♥❞ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t②✮✳ (❡t ✐t st✐•• ❣✐✈❡s ❛ ❢♦r♠✉•❛t✐♦♥ ❢♦r t❤❡ ❡①❛❝t ♠❛t❝❤✐♥❣ ♣r♦❜•❡♠ ❜❡t✇❡❡♥ t✇♦ ❢s❤❛♣❡s (X, f ) ❛♥❞ (X ′ , f ′ ) ❜❡•♦♥❣✐♥❣ t♦ ❛ ❝♦♠♠♦♥ ♦r❜✐t F✱ ✇❤✐❝❤ ❛♠♦✉♥ts ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛t✐♦♥ ✿ inf (v,ζ)∈L 2 V ×L 2 (X)
• ❈❤❛♣t❡r ✸ ✿ ❛❢t❡r ❡①♣♦s✐♥❣ t❤❡ ♣r✐♥❝✐♣❛• ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s ❝♦♠✐♥❣ ❛•♦♥❣ ✇✐t❤ ❝✉rr❡♥ts✱ ✇❡ t✉r♥ t♦ t❤❡ ❛•t❡r♥❛t✐✈❡ ❝♦♥❝❡♣t ♦❢ ✈❛r✐❢♦•❞s ✐♥tr♦❞✉❝❡❞ ♦r✐❣✐♥❛••② ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r②✳ ❲❡ ❡①♣•❛✐♥ ❤♦✇ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s ❝❛♥ ❡♠❜❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ♦❢ E ✇✐t❤♦✉t t❤❡ ♥❡❡❞ ♦❢ ❞❡✜♥✐♥❣ ❛♥ ♦r✐❡♥t❛t✐♦♥✳ ❚❤❡ r❡st ♦❢ t❤❡ ❝❤❛♣t❡r ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❦❡r♥❡• ♥♦r♠s ♦♥ ✈❛r✐❢♦•❞s ❛♥❞ st✉❞✐❡s t❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ ❝♦♠♣❛r❡❞ t♦ ❝✉rr❡♥ts✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ s❤♦✇ t❤❛t✱ ✉♥❞❡r s✐♠♣•❡ ❛ss✉♠♣t✐♦♥s✱ ♥♦r♠s ♦❢ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛r❡ ❜♦✉♥❞❡❞ ❜❡•♦✇ ❜② t❤❡ ❛❝t✉❛• ✈♦•✉♠❡ ♦❢ t❤❡ s❡t ✭t❤❡♦r❡♠ ✸✳✹✳✶ ❛♥❞ ❝♦r♦••❛r② ✸✳✹✳✶✮✱ ✇❤✐❝❤ ♣r❡✈❡♥ts s❤❛♣❡ ❝❛♥❝❡••❛t✐♦♥ ♣❤❡♥♦♠❡♥❛ ♦❝❝✉rr✐♥❣ ✇✐t❤ ❝✉rr❡♥ts✳ ❲❡ ❛•s♦ ❞❡r✐✈❡ ❛ ✈❛r✐❛t✐♦♥ ❢♦r♠✉•❛ ❢♦r ✈❛r✐❢♦•❞ ♥♦r♠s ✭t❤❡♦r❡♠ ✸✳✹✳✷✮✳ ❚❤❡ ❡♥❞ ♦❢ ❝❤❛♣t❡r ✸ ❛❞❛♣ts t❤❡ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠ t♦ t❤❡ ❝❛s❡ ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s r❡♣r❡s❡♥t❡❞ ❛s ✈❛r✐❢♦•❞s ❛♥❞ s❤♦✇✱ ♦♥ ❛ ❢❡✇ s✐♠♣•❡ ❡①❛♠♣•❡s✱ t❤❡ ♣♦t❡♥t✐❛• ✐♥t❡r❡st ♦❢ t❤✐s ❢r❛♠❡✇♦r❦ ❝♦♠♣❛r❡❞ t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ♦❢ ❝✉rr❡♥ts✳ ■t ❝♦♥❝•✉❞❡s ❜② ♠❡r❣✐♥❣ t❤✐s ❛♣♣r♦❛❝❤ ✇✐t❤ t❤❡ ♦♥❡ ♦❢ ❝❤❛♣t❡r ✷ r❡s✉•t✐♥❣ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞ s❡tt✐♥❣ t♦ tr❡❛t ♥♦♥✲♦r✐❡♥t❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ • ❈❤❛♣t❡r ✹ ✿ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❡✈❡♥t✉❛••② ❝♦♠❡ t♦ t❤❡ r❡❛• ❤❡❛rt ♦❢ t❤❡ ♣r♦❜•❡♠✲ ❛t✐❝ ♦❢ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦♥ ♣♦♣✉•❛t✐♦♥s✳ ❚❤❡ ♦❜•❡❝t✐✈❡ ✐s t♦ ♣r♦♣♦s❡ ❛ ❝♦❤❡r❡♥t ❛♥❞ r♦❜✉st ❢r❛♠❡✇♦r❦ t♦ ♠♦❞❡• st❛t✐st✐❝❛••② t❤❡ ✈❛r✐❛❜✐•✐t② ❛♠♦♥❣ ❛ ❣r♦✉♣ ♦❢ ❢✉♥❝t✐♦♥❛• ✹✵ s❤❛♣❡s ❜♦t❤ ✐♥ t❤❡✐r ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛•✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✇❡ r❡•② ♦♥ t❤❡ t♦♦•s ❞❡✈❡•♦♣❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs ❛♥❞ st❛rt ❜② ❡①t❡♥❞✐♥❣ t❤❡ s✐♠♣•❡r s✐t✉❛t✐♦♥ ♦❢ r❡❣✐str❛t✐♦♥ ♦❢ ♦♥•② t✇♦ s✉❜•❡❝ts t♦ ❣r♦✉♣✲❛♥❛•②s✐s✳ ❚❤❡ ❢r❛♠❡✇♦r❦ t❤❛t ✐s ❞❡✈❡•♦♣❡❞ ❡st✐♠❛t❡s ✐♥ ❛ s✐♠✉•t❛♥❡♦✉s ♠✐♥✐♠✐③❛t✐♦♥ ❛ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ r❡♣r❡s❡♥t✐♥❣ s♦♠❡ ❦✐♥❞ ♦❢ ♣r♦t♦t②♣❡ ❢s❤❛♣❡ ❢♦r t❤❡ ❣r♦✉♣✱ t♦❣❡t❤❡r ✇✐t❤ ❞❡❢♦r♠❛t✐♦♥s ❛♥❞ r❡s✐❞✉❛• s✐❣♥❛•s ❢♦r ❡❛❝❤ s✉❜•❡❝t✳ ■♥ ❛ s❡❝♦♥❞ st❡♣✱ ✇❡ ❛❞❞r❡ss t❤❡ ♣r♦❜•❡♠ ♦❢ •❡❛r♥✐♥❣ ♣r✐♥❝✐✲ ♣❛• ♠♦❞❡s ♦❢ ✈❛r✐❛t✐♦♥s ❢♦r t❤❡s❡ s❡ts ♦❢ ❞❡❢♦r♠❛t✐♦♥s ❛♥❞ r❡s✐❞✉❛•s ❛♥❞ ♣❡r❢♦r♠✐♥❣ ❝•❛ss✐✜❝❛t✐♦♥ ❢♦r ♠✉•t✐✲❝•❛ss ❞❛t❛s❡ts ❜❛s❡❞ ♦♥ t❤❡s❡✳ ❚❤❡r❡❜②✱ ✇❡ ❛❞❛♣t s♦♠❡ st❛✲ t✐st✐❝❛• •❡❛r♥✐♥❣ ♠❡t❤♦❞s s✉❝❤ ❛s P❈❆ ❛♥❞ ▲❉❆ t♦ t❤✐s s✐t✉❛t✐♦♥✳ ❙♦♠❡ r❡s✉•ts ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❜♦t❤ ❣❡♦♠❡tr✐❝ ❛♥❞ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❛t❛s❡ts ❢r♦♠ ❜✐♦•♦❣②✱ ❛s ✇❡•• ❛s ❛ ❢❡✇ ❝•❛ss✐✜❝❛t✐♦♥ r❡s✉•ts✱ ❛r❡ ❛•s♦ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ✹✶ ✹✷ ❈❤❛♣t❡r ✷ ❆♥ ❡①t❡♥❞❡❞ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ ❛♥❛•②s✐s ♦❢ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❛t❛ ❈♦♥t❡♥ts ✷✳✶ ❋✉♥❝t✐♦♥❛• s❤❛♣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✶✳✶ ❙t❛t❡ ♦❢ t❤❡ ❛rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✶✳✷ ❋✉♥❝t✐♦♥❛• s❤❛♣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✶✳✸ |❡♣r❡s❡♥t❛t✐♦♥ ❜② ❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✷ ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✿ ♠❛t❤❡♠❛t✐❝❛• ❝♦♥s✐❞❡r❛t✐♦♥s ✳ ✳ ✳ ✺✶ ✷✳✷✳✶ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✺✶ ✷✳✷✳✷ |❡♣r♦❞✉❝✐♥❣ ❑❡r♥❡• ❍✐•❜❡rt ❙♣❛❝❡ str✉❝t✉r❡s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✷✳✸ Pr♦♣❡rt✐❡s ♦❢ ❦❡r♥❡• ♥♦r♠s ♦♥ ❢❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✷✳✸ ❆•❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✷✳✸✳✶ ❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ▼❛t❝❤✐♥❣ P✉rs✉✐t ♦♥ ❢❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✸✳✷ |❡❣✐str❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✷✳✸✳✸ ◆✉♠❡r✐❝❛• ❝♦♥s✐❞❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✷✳✶✳✸ |❡♣r❡s❡♥t❛t✐♦♥ ❜② ❝✉rr❡♥ts ❙t❛rt✐♥❣ ❢r♦♠ t❤❡ ✐❞❡❛ ♦❢ ❝✉rr❡♥ts ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ ✇❡ ❡✈❛•✉❛t❡ t✇♦ ❞✐✛❡r❡♥t ✇❛②s ♦❢ ❡♠❜❡❞❞✐♥❣ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✭❛t •❡❛st ✐♥ s♦♠❡ s♣❡❝✐❛• ❝❛s❡s✮ ✐♥ s♣❛❝❡s ♦❢ ❝✉rr❡♥ts✳ ❈♦•♦r❡❞ ❝✉rr❡♥ts ❋✐rst ❛tt❡♠♣ts t♦ ✐♥❝•✉❞❡ s✐❣♥❛•s s✉♣♣♦rt❡❞ ❣❡♦♠❡tr✐❝❛••② t❤r♦✉❣❤ t❤❡ ❝✉rr❡♥t r❡♣✲ r❡s❡♥t❛t✐♦♥ ✇❡r❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✷✽( ✇✐t❤ t❤❡ ✐❞❡❛ ♦❢ ❝♦•♦r❡❞ ❝✉rr❡♥ts✳ ❚❤✐s r❡•✐❡s ❜❛s✐❝❛••② ♦♥ t❤❡ ❢❛❝t ❛•r❡❛❞② ♠❡♥t✐♦♥❡❞ t❤❛t t❤❡ s❡t ♦❢ d✲❝✉rr❡♥ts ✐s ❛•r❡❛❞② ❛ ❜✐❣ s♣❛❝❡ t❤❛t ❝♦♥t❛✐♥s ❛ ✇✐❞❡r ✈❛r✐❡t② ♦❢ ♦❜•❡❝ts t❤❛♥ d✲❞✐♠❡♥s✐♦♥❛• s✉❜♠❛♥✐❢♦•❞s✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡✐❣❤t❡❞ s✉❜♠❛♥✐❢♦•❞s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❝✉rr❡♥ts ✿ ✐❢ f ✐s ❛ r❡❛•✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥ ♦♥ X t❤❡♥ ✇❡ ❝❛♥ ❛ss♦❝✐❛t❡ t♦ (X, f ) ❛ d✲❝✉rr❡♥t ✐♥ E ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ s✐❣♥❛• ❛s ❛ ✇❡✐❣❤t ♦♥ X ✿ T (X,f ) (ω) = X f ω ❇② t❤❡ ♣r❡✈✐♦✉s s❡tt✐♥❣✱ ✇❡ ❡♠❜❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ♦♥ E ✐ts❡•❢✳ ❆•t❤♦✉❣❤ t❤✐s s❡❡♠s t♦ ❜❡ t❤❡ ♠♦st str❛✐❣❤t❢♦r✇❛r❞ ✇❛② t♦ ❛♣♣•② ❝✉rr❡♥ts t♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ✐t q✉✐❝❦•② ❛♣♣❡❛rs t❤❛t s✉❝❤ ❛ r❡♣r❡s❡♥t❛t✐♦♥ s✉✛❡rs ❢r♦♠ s❡✈❡r❛• ✐♠♣♦rt❛♥t ❞r❛✇❜❛❝❦s✳ ❋✐rst ✐s t❤❡ ✐♠♣♦ss✐❜✐•✐t② t♦ ❣❡♥❡r❛•✐③❡ ❝♦•♦r❡❞ ❝✉rr❡♥ts ❢♦r s✐❣♥❛•s t❤❛t ❛r❡ ♥♦t s✐♠♣•② r❡❛•✲✈❛•✉❡❞✱ ♣❛rt✐❝✉•❛r•② ✐❢ t❤❡ s✐❣♥❛• s♣❛❝❡ ✐s ♥♦t ❛ ✈❡❝t♦r s♣❛❝❡✳ ❚❤❡ s❡❝♦♥❞ ♣♦✐♥t ❛r✐s❡s ✇❤❡♥ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥ ✐s ❞✐s❝r❡t✐③❡❞ ✐♥t♦ ❉✐r❛❝ ❝✉rr❡♥ts✱ ✇❤✐❝❤ •❡❛❞s t♦ ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❢♦r♠ k=1..N f (x k )δ ξ k x k ✳ ❲❡ ♥♦t✐❝❡ ❛♥ ❛♠❜✐❣✉✐t② ❛♣♣❡❛r✐♥❣ ❜❡t✇❡❡♥ t❤❡ s✐❣♥❛• ❛♥❞ t❤❡ ✈♦•✉♠❡ ❡•❡♠❡♥t ξ s✐♥❝❡ ❢♦r ❛♥② r = 0✱ f (x k )δ ξ k x k = rf (x k )δ ξ k /r x k ❙❡♣❛r❛t✐♥❣ ❣❡♦♠❡tr② ❢r♦♠ s✐❣♥❛• ✐♥ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ❛♣♣❡❛rs ❛s ❛ ❢✉♥❞❛♠❡♥t❛• ❞✐✣❝✉•t②✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❡♥❡r❣② ♦❢ ❉✐r❛❝ t❡r♠s ❛r❡ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ✈❛•✉❡ ♦❢ t❤❡ s✐❣♥❛• ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t ✇❤✐❝❤ ✐♥❞✉❝❡s ❛♥ ❛s②♠♠❡tr② ❜❡t✇❡❡♥ •♦✇ ❛♥❞ ❤✐❣❤✲ ✈❛•✉❡❞ s✐❣♥❛•s✳ ■♥ t❤✐s s❡tt✐♥❣✱ ❛r❡❛s ❤❛✈✐♥❣ ✈❡r② s♠❛•• s✐❣♥❛•s ❜❡❝♦♠❡ ♥❡❣•✐❣✐❜•❡ ✐♥ t❡r♠s ♦❢ ❝✉rr❡♥t✱ ✇❤✐❝❤ ♥♦t •✉st✐✜❡❞ ✐♥ ❣❡♥❡r❛• ❛♥❞ ❝❛♥ ❞r❛st✐❝❛••② ❛✛❡❝t t❤❡ ♠❛t❝❤✐♥❣ ♦❢ ❝♦•♦r❡❞ ❝✉rr❡♥ts✳ ❲❡ s❤♦✇ ❛ s✐♠♣•❡ ✐••✉str❛t✐♦♥ ♦❢ t❤✐s ✐ss✉❡ ✇❤❡♥ ♠❛t❝❤✐♥❣ t✇♦ ❝♦•♦r❡❞ ❡••✐♣s♦✐❞s ✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤ ✐♥ ✜❣✉r❡ ✷✳✹✳ ❋✐♥❛••② ❛♥ ❛❞❞✐t✐♦♥❛• •✐♠✐t❛t✐♦♥ ✐♥ ✉s✐♥❣ ❝♦•♦r❡❞ ❝✉rr❡♥ts ✐s t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ✐s ♥♦ ✢❡①✐❜✐•✐t② t♦ tr❡❛t t❤❡ s✐❣♥❛•s ❛t ❞✐✛❡r❡♥t s❝❛•❡ •❡✈❡•s t❤❛♥ ❣❡♦♠❡tr②✱ ♠❛❦✐♥❣ t❤✐s ❛♣♣r♦❛❝❤ ❤✐❣❤•② s❡♥s✐t✐✈❡ t♦ ♥♦✐s❡✳ ❈✉rr❡♥ts ✐♥ ♣r♦❞✉❝t s♣❛❝❡ ❆♥♦t❤❡r ♣♦ss✐❜•❡ ❛♥❞ ✐♥t❡r❡st✐♥❣ ✇❛② t♦ r❡♣r❡s❡♥t ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❜② ❛ ❝✉rr❡♥t ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❣r❛♣❤ G (X,f ) ❛s ❛ s❤❛♣❡ ✐♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ E × M ✱ ❡①t❡♥❞✐♥❣ t❤❡ ✐❞❡❛ ♦❢ s❡❡✐♥❣ ❛ ✷❉ ✐♠❛❣❡ ❛s ❛ ✸❉ s✉r❢❛❝❡✳ ◆♦✇✱ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ❛s ❛ d✲❝✉rr❡♥t ✐♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ E × M ✳ ❚❤✐s ✐s ♥♦t str❛✐❣❤t❢♦r✇❛r❞ t❤♦✉❣❤✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ❣❡♥❡r❛• s✐❣♥❛• ♠❛♥✐❢♦•❞ M ✐s ❢♦r ✐♥st❛♥❝❡ ♠♦r❡ ❞✐✣❝✉•t t♦ ❤❛♥❞•❡ ✐♥ t❤✐s s❡tt✐♥❣ s✐♥❝❡ ✐t r❡q✉✐r❡s t♦ ❝♦♥s✐❞❡r ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ❞❡✜♥❡❞ ♦♥ M ✿ ❢♦r s✐♠♣•✐❝✐t②✱ ✇❡ ✇✐•• ❛ss✉♠❡ ❢♦r t❤❡ r❡st ♦❢ t❤✐s s❡❝t✐♦♥ t❤❛t M ✐s ❛ ✈❡❝t♦r s♣❛❝❡✳ ❆ s❡❝♦♥❞ ♣♦✐♥t ♠❡♥t✐♦♥❡❞ ❡❛r•✐❡r ✐s t❤❛t ✇❡ ♥❡❡❞ t♦ ✐♠♣♦s❡ ♠♦r❡ r❡❣✉•❛r✐t② ♦♥ t❤❡ s✐❣♥❛• f ✱ ❢♦r ✐♥st❛♥❝❡ ♣✐❡❝❡✇✐s❡ •♦❝❛••② ▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s s♦ t❤❛t G (X,f ) ✐s r❡❝t✐✜❛❜•❡ ❛♥❞ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❝✉rr❡♥t ✐♥ Ω d 0 (E × M )✳ ❚❤❡s❡ ❞✐✣❝✉•t✐❡s ❛♣❛rt✱ t❤❡r❡ st✐•• ❛r❡ s♦♠❡ ✐♠♣♦rt❛♥t ❡•❡♠❡♥ts t♦ ♣♦✐♥t ♦✉t✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ✐♥❝r❡❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥❛•✐t② ✹✽ ❋✐❣✉r❡ ✷✳✹✿ ❆♥ ❡①❛♠♣•❡ ♦❢ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❡••✐♣s♦✐❞s✳ ❖♥ t❤❡ •❡❢t✱ t❤❡ ♠❛t❝❤✐♥❣ ✐s ❞♦♥❡ ❜② ❝♦♠♣❛r✐♥❣ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ t❤❡ ❝♦•♦r❡❞ ❝✉rr❡♥ts✬ r❡♣r❡✲ s❡♥t❛t✐♦♥✳ ❱❛•✉❡s ♦❢ t❤❡ s✐❣♥❛•s ❛r❡ t✇♦ ❞✐✛✉s❡❞ st❛✐♥s ❜♦t❤ ♦♥ t❤❡ s♦✉r❝❡ ❡••✐♣s♦✐❞ ✭✐♥s✐❞❡ s✉r❢❛❝❡✮ ❛♥❞ t❤❡ t❛r❣❡t ♦♥❡ ✭❡①t❡r✐♦r s❤❛❞❡❞ s✉r❢❛❝❡✮✳ ❲❡ ❞✐s♣•❛② ✐♥ ❜•✉❡ tr❛✲ •❡❝t♦r✐❡s ♦❢ t❤❡ ♣♦✐♥ts✳ ❚❤❡ ♣♦✐♥ts ❝♦♠♣♦✉♥❞✐♥❣ t♦ ③❡r♦✲✈❛•✉❡❞ ❛r❡❛ ♦❢ t❤❡ s✐❣♥❛• ✐♥ t❤❡ s♦✉r❝❡ s❤❛♣❡ ❛r❡ ♥♦t ♠❛t❝❤❡❞ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ✐♥ t❤❡ t❛r❣❡t s✉r❢❛❝❡✳ ❖♥ t❤❡ r✐❣❤t✱ ✇❡ s❤♦✇ ✇❤❛t s❤♦✉•❞ ❜❡ t❤❡ ❡①♣❡❝t❡❞ r❡s✉•t✳ ■t ✐s ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♣r❡s❡♥t❡❞ ✐♥ ✷✳✸✳✷✳ ♦❢ t❤❡ ❛♣♣r♦❛❝❤✱ ❜❡❝❛✉s❡✱ ✇❤✐•❡ ✇❡ ❛r❡ st✐•• ❝♦♥s✐❞❡r✐♥❣ s❤❛♣❡s ♦❢ ❞✐♠❡♥s✐♦♥ d✱ t❤❡ ❝♦✲ ❞✐♠❡♥s✐♦♥ ✐s ❤✐❣❤❡r ✿ t❤❡ s♣❛❝❡ ♦❢ d✲✈❡❝t♦rs ❝❤❛r❛❝t❡r✐③✐♥❣ •♦❝❛• ❣❡♦♠❡tr② Λ d (E ×M ) ✐s ♥♦✇ ♦❢ ❞✐♠❡♥s✐♦♥ n+dim(M ) d ✱ ✇✐t❤ s✐❣♥✐✜❝❛♥t ❝♦♥s❡q✉❡♥❝❡s ❢r♦♠ ❛ ❝♦♠♣✉t❛t✐♦♥❛• ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❋r♦♠ ❛ ♠♦r❡ t❤❡♦r❡t✐❝❛• ❛♥❣•❡✱ ✇❡ s❡❡ t❤❛t✱ ✐♥ s✉❝❤ ❛♥ ❛♣♣r♦❛❝❤✱ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ❛♥❞ s✐❣♥❛• ♣•❛② ❛ s②♠♠❡tr✐❝ r♦•❡✳ ■♥ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✱ t❤❡ ♠♦❞❡•❡❞ t♦♣♦•♦❣② ✐s ♥♦ ♠♦r❡ t❤❡ ♦♥❡ ♦❢ t❤❡ ♦r✐❣✐♥❛• s❤❛♣❡ ❜❡❝❛✉s❡ ✇❡ ❛•s♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ✈❛r✐❛t✐♦♥s ✇✐t❤✐♥ t❤❡ s✐❣♥❛• s♣❛❝❡✳ ❲❤❡t❤❡r t❤✐s ✐s ❛ str❡♥❣t❤ ♦r ❛ ✇❡❛❦♥❡ss ✐s ♥♦t ♦❜✈✐♦✉s ❛ ♣r✐♦r✐ ❛♥❞ ✇♦✉•❞ ❞❡♣❡♥❞ ♦♥ t❤❡ ❦✐♥❞ ♦❢ ❛♣♣•✐❝❛t✐♦♥s✳ ❲❤❛t ✇❡ ❝❛♥ st❛t❡ ✐s t❤❛t t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♥♦t r♦❜✉st t♦ t♦♣♦•♦❣✐❝❛• ❝❤❛♥❣❡s ♦❢ t❤❡ s❤❛♣❡ ✿ ✐♥ ♣r❛❝t✐❝❡✱ t❤❡ ❝♦♥♥❡❝t✐✈✐t② ❜❡t✇❡❡♥ ❛•• ♣♦✐♥ts ❜❡❝♦♠❡s ❝r✉❝✐❛•✱ ✇❤✐❝❤ ✇❡ ✐••✉str❛t❡ ✐♥ ✜❣✉r❡ ✷✳✺✳ ❋✐❣✉r❡ ✷✳✻ s❤♦✇s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❛t ✐❢ t✇♦ ❝✉r✈❡s ❝❛rr②✐♥❣ s✐❣♥❛•s ❛r❡ ♠❛t❝❤❡❞ ✉♥❞❡r t❤✐s ♠♦❞❡•✳ ■t s❤♦✇s t❤❛t ✐t ❝❛♥ ❛❝t✉❛••② •❡❛❞ t♦ ✈❡r② ✉♥♥❛t✉r❛• r❡❣✐str❛t✐♦♥ ✐♥ ❝❛s❡s ✇❤❡r❡ s♦♠❡ ❞✐s❝♦♥♥❡❝t✐♦♥s ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ s✐❣♥❛•s ❛♣♣❡❛r✳ ■♥ t❤❡ ✜❡•❞ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠②✱ ✇❤❡♥ ♣r♦❝❡ss✐♥❣ ❞❛t❛ s✉❝❤ ❛s ✜❜❡r ❜✉♥❞•❡s✱ ✇❤❡r❡ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ♣♦✐♥ts ♦❢ t❤❡ ✜❜❡rs ❛r❡ ♥♦t ❛•✇❛②s r❡•✐❛❜•❡✱ ✇❡ ❛r❣✉❡ t❤❛t t❤✐s ✇♦✉•❞ ❜❡ ❛ ❝•❡❛r ❞r❛✇❜❛❝❦✳ ❚♦ s✉♠♠❛r✐③❡ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ t✇♦ ❞✐r❡❝t ✇❛②s t♦ s❡❡ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❛s ❛ ❝✉rr❡♥t✳ ❚❤❡ ❝♦•♦r❡❞ ❝✉rr❡♥t s❡tt✐♥❣✱ ❛•t❤♦✉❣❤ ❜❡✐♥❣ ✈❡r② ❝•♦s❡ t♦ t❤❡ ♠♦❞❡••✐♥❣ ♦❢ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s✱ ✐s ♥♦t ❛❝❝❡♣t❛❜•❡ ♠❛✐♥•② ❜❡❝❛✉s❡ ✐t ♠✐①❡s ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛• ✐♥ ❛♥ ✐♥❝♦♥s✐st❡♥t ✇❛②✳ ❆s ❢♦r t❤❡ s❡❝♦♥❞ ✐❞❡❛ ♦❢ ✐♠♠❡rs✐♥❣ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✐♥ ❛ ♣r♦❞✉❝t s♣❛❝❡✱ ✇❡ ❤❛✈❡ ❡①♣•❛✐♥❡❞ ✐ts •✐♠✐ts ❜♦t❤ ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐✣❝✉•t② ✐♥ ♣r❛❝t✐❝❛• ✐♠♣•❡♠❡♥t❛t✐♦♥✱ t❤❡ ✐♥❝r❡❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥ ❛♥❞ ♦❢ t❤❡ •❛❝❦ ♦❢ r♦❜✉st♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ t♦♣♦•♦❣② ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt✳ ❙♦♠❡❤♦✇✱ t❤❡ s♣❛❝❡ ♦❢ ❝♦•♦r❡❞ ❝✉rr❡♥ts ✐s ♥♦t r✐❝❤ ❡♥♦✉❣❤ t♦ ❡♥❝♦♠♣❛ss ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇❤❡r❡❛s ❝✉rr❡♥ts ✐♥ t❤❡ ♣r♦❞✉❝t ✐s ♣r♦❜❛❜•② t♦♦ ❜✐❣✳ ❚❤❡s❡ ♦❜s❡r✈❛t✐♦♥s ❝♦♥st✐t✉t❡ ♦✉r ♠♦t✐✈❛t✐♦♥ t♦ r❡❞❡✜♥❡ ❛ ♣r♦♣❡r ❝•❛ss ♦❢ ♠❛t❤❡♠❛t✐❝❛• ♦❜•❡❝ts t❤❛t ✇♦✉•❞ ♣r❡s❡r✈❡ t❤❡ ✉s❡❢✉•♥❡ss ♦❢ ❝✉rr❡♥ts ✇❤✐•❡ ♦✈❡r❝♦♠✐♥❣ t❤❡ ♣r❡✈✐♦✉s ❞r❛✇❜❛❝❦s✳ ✹✾ ❋✐❣✉r❡ ✷✳✺✿ Pr♦❞✉❝t ❝✉rr❡♥ts ❛♥❞ t♦♣♦•♦❣②✳ ❖♥ t❤❡ •❡❢t✱ ✇❡ s❤♦✇ ❛ ❞✐s❝♦♥♥❡❝t❡❞ ✷❉ ❝✉r✈❡ ✇✐t❤ s✐❣♥❛• ✈❛•✉❡s 0 ✐♥ ❜•✉❡ ❛♥❞ 1 ✐♥ r❡❞ ❛s ✇❡•• ❛s t❤❡ ❝♦♥♥❡❝t❡❞ ❝✉r✈❡ ✐♥ ❞❛s❤❡❞ •✐♥❡✳ ❖♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❛r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝✉r✈❡s ✐♥ t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛• ❣❡♦♠❡tr② × s✐❣♥❛• s♣❛❝❡✳ ❲❤❛t ✇❡ ✇❛♥t t♦ ❡♠♣❤❛s✐③❡ ❤❡r❡ ✐s t❤❡ ❢❛❝t t❤❛t ♥♦ |❑❍❙ ♥♦r♠ ♦♥ ♣r♦❞✉❝t ❝✉rr❡♥ts ✇♦✉•❞ ♣r♦✈✐❞❡ ❛ ❝♦♥t✐♥✉✐t② ♦❢ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♥♥❡❝t✐✈✐t② ✿ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❝✉r✈❡s ✐s t❤❡ ♠❛❣❡♥t❛ ❞❛s❤❡❞ ♣❛rt ✇❤✐❝❤ r❡♣r❡s❡♥ts ❛ ♣✉r❡ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ s✐❣♥❛• ❞♦♠❛✐♥✳ ❋✐❣✉r❡ ✷✳✻✿ ▲❉❉▼▼ ♠❛t❝❤✐♥❣ ♦❢ t✇♦ ♣•❛♥❛r ❝✉r✈❡s ✇✐t❤ ❞✐s❝♦♥t✐♥✉♦✉s s✐❣♥❛•s ❛♥❞ t♦♣♦•♦❣✐❝❛• ❞✐s❝♦♥♥❡❝t✐♦♥s r❡♣r❡s❡♥t❡❞ ❛s ❝✉rr❡♥ts ✐♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ R 2 ×R✳ ❊❛❝❤ ❝✉r✈❡ ❤❛s t✇♦ ♣♦✐♥ts ♦❢ ❢✉♥❝t✐♦♥❛• ❞✐s❝♦♥t✐♥✉✐t②✱ ♦♥❡ ♦❢ t❤❡♠ ❜❡✐♥❣ ❛•s♦ ❛ ❞✐s❝♦♥✲ ♥❡❝t✐♦♥ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✭❜ ❛♥❞ ❜✬✮✳ ❚❤❡ r❡s✉•t✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ✐s ♠✉❝❤ ♣❡rt✉r❜❡❞ ❜② t❤❡ ❞✐s❝♦♥♥❡❝t✐♦♥s s✐♥❝❡ t❤❡ ❛•❣♦r✐t❤♠ ✐♥t❡♥❞s t♦ ♠❛t❝❤ ❝♦♥♥❡❝t❡❞ ♣❛rt ♦❢ t❤❡ s♦✉r❝❡ s❤❛♣❡ ♦♥ ❛ ❝♦♥♥❡❝t❡❞ ♣❛rt ♦❢ t❤❡ t❛r❣❡t s❤❛♣❡✳ ✺✵ ✷✳✷ ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✿ ♠❛t❤❡♠❛t✐❝❛• ❝♦♥s✐❞❡r❛t✐♦♥s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦♣♦s❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♥♦t✐♦♥ ♦❢ ❝✉rr❡♥ts t♦ r❡♣r❡s❡♥t ❢✉♥❝✲ t✐♦♥❛• s❤❛♣❡s✳ ❚❤❡ ♥❡✇ ♠❛t❤❡♠❛t✐❝❛• ♦❜•❡❝ts ✇❡ ✐♥tr♦❞✉❝❡✱ ♥❛♠❡❞ ❢✉♥❝t✐♦♥❛• ❝✉r✲ r❡♥ts✱ ❛r❡ ♥♦t ✉s✉❛• ❝✉rr❡♥ts str✐❝t•② s♣❡❛❦✐♥❣✱ ❛s ♦♣♣♦s❡❞ t♦ t❤❡ ♠❡t❤♦❞s ♣r❡s❡♥t❡❞ ✐♥ ✷✳✶✳✸✳ ❚❤❡② ✇♦✉•❞ r❛t❤❡r ❞❡r✐✈❡ ❢r♦♠ t❤❡ ✈❡r② ❣❡♥❡r❛• ❝♦♥❝❡♣t ♦❢ ❞♦✉❜•❡ ❝✉rr❡♥t ✐♥tr♦❞✉❝❡❞ ♦r✐❣✐♥❛••② ❜② ❉❡ |❤❛♠ ✐♥ ❬✼✵(✳ ✷✳✷✳✶ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❝♦♥s✐❞❡r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s (X, f )✱ ✇✐t❤ X ❛ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ t❤❡ n✲❞✐♠❡♥s✐♦♥❛• ❊✉❝•✐❞❡❛♥ s♣❛❝❡ E ❛♥❞ f ❛ ♠❡❛s✉r❛❜•❡ ♠❛♣♣✐♥❣ ❢r♦♠ X t♦ t❤❡ s✐❣♥❛• ♠❛♥✐❢♦•❞ M ✳ ■♥ t❤❡ t❤❡♦r❡t✐❝❛• ❢r❛♠❡✇♦r❦ t❤❛t ✇❡ ♣r❡s❡♥t ❤❡r❡✱ M ❝❛♥ ❜❡ ❛♥② |✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦•❞✳ ❋✉♥❝t✐♦♥❛• p✲❢♦r♠s ❛♥❞ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❲❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛❣❛✐♥ ❛s t❤❡ ❞✉❛• ♦❢ ❛ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢♦r♠s ✿ ❉❡✜♥✐t✐♦♥ ✷✳✷✳✶✳ ❲❡ ❝❛•• ❛ ❢✉♥❝t✐♦♥❛• p✲❢♦r♠ ♦♥ (E, M ) ❛♥ ❡•❡♠❡♥t ♦❢ t❤❡ s♣❛❝❡ C 0 (E × M, Λ p E * ) ✇❤✐❝❤ ✇✐•• ❜❡ ❞❡♥♦t❡❞ ❜② Ω p 0 (E, M ) ❤❡r❡❛❢t❡r✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ✉♥✐✲ ❢♦r♠ ♥♦r♠ ♦♥ Ω p 0 (E, M ) ❞❡✜♥❡❞ ❜② ✿ ω ∞ = sup (x,m)∈E×M ω (x,m) ✳ ❆ ❢✉♥❝t✐♦♥❛• p✲❝✉rr❡♥t ✭♦r ❢❝✉rr❡♥t ✐♥ s❤♦rt✮ ✐s ❞❡✜♥❡❞ ❛s ❛ ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ ♦♥ Ω p 0 (E, M ) ❢♦r t❤❡ ✉♥✐❢♦r♠ ♥♦r♠✳ ❚❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• p✲❝✉rr❡♥ts ✇✐•• ❜❡ t❤❡r❡❢♦r❡ ❞❡♥♦t❡❞ Ω p 0 (E, M ) ′ ✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✱ ♦♥❡ s❡❡s t❤❛t ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛r❡ ❛✉❣♠❡♥t✐♥❣ ✉s✉❛• ❝✉rr❡♥ts ✇✐t❤ ♣♦s✐t✐♦♥s ✐♥ t❤❡ s✐❣♥❛• ❞♦♠❛✐♥ M ✳ ❆ ❢✉♥❝t✐♦♥❛• p✲❢♦r♠ ✐s s♦♠❡❤♦✇ ❛ ✜❡•❞ ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ♦♥ t❤❡ s♣❛❝❡ E ❜✉t s♣r❡❛❞ ♦♥ t❤❡ ♣r♦❞✉❝t E × M ✳ ■♥❞❡❡❞ ❛ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥❛• p✲❢♦r♠ ❝❛♥ ❜❡ ✇r✐tt❡♥✱ ✐♥ ❛ ❝♦♦r❞✐♥❛t❡ s②st❡♠ (x 1 , ..., x n ) ♦❢ E✱ ω = 1≤i 1 <..<ip≤n a i 1 ,...,ip (x, m)dx i 1 ∧ ... ∧ dx ip ✭✷✳✻✮ ◆♦t❡✱ ❤♦✇❡✈❡r✱ t❤❡ ✐♠♣♦rt❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts Ω p 0 (E, M ) ′ ❛♥❞ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ✐♥ E × M ✱ Ω p 0 (E × M ) ′ ❞✐s❝✉ss❡❞ ♣r❡✈✐♦✉s•②✳ ❲✐t❤ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ t❤❡ •♦❝❛• ❣❡♦♠❡tr② ✐s st✐•• t❤❡ ♦♥❡ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s❤❛♣❡ r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡•❡♠❡♥t ♦❢ Λ p E ❛s ♦♣♣♦s❡❞ t♦ t❤❡ ♣r♦❞✉❝t ❝✉rr❡♥t s❡tt✐♥❣ t❤❛t ♠♦❞❡•s t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ •✐❢t❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✐♥ E × M ✱ r❡q✉✐r✐♥❣ t❤❡ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛• s♣❛❝❡ ♦❢ p✲✈❡❝t♦rs Λ p (E × M )✳ ❚❤✉s t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐s ❜✐❣❣❡r t❤❛♥ t❤❡ ♦♥❡ ♦❢ ❝✉rr❡♥ts ♦♥ E ❜✉t s♠❛••❡r t❤❛♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ♦♥ E ×M ✳ ◆♦✇✱ •✉st ❛s ♦♥❡ ❝❛♥ ❡st❛❜•✐s❤ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ s❤❛♣❡s ❛♥❞ ❝✉rr❡♥ts✱ t♦ ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✇❡ ♥♦✇ ❛ss♦❝✐❛t❡ ❛ ❢❝✉rr❡♥t✳ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✶✳ ▲❡t (X, f ) ❜❡ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡✱ ✇✐t❤ X ❛♥ ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ ❞✐♠❡♥s✐♦♥ d ❛♥❞ ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ f ❛ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ ❢r♦♠ X t♦ M ✳ ❋♦r ❛•• ω ∈ Ω d 0 (E, M )✱ x → ω (x,f (x)) ❝❛♥ ❜❡ ✐♥t❡❣r❛t❡❞ ❛•♦♥❣ X✳ ❲❡ s❡t ✿ C (X,f ) (ω) . = X ω (x,f (x)) . ✭✷✳✼✮ ❚❤❡♥ C (X,f ) ∈ Ω d 0 (E, M ) ′ ❛♥❞ t❤❡r❡❢♦r❡ (X, f ) → C (X,f ) ❛ss♦❝✐❛t❡s✱ t♦ ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡✱ ❛ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t✳ ✺✶ ❆s ❢♦r ✉s✉❛• ❝✉rr❡♥ts✱ ❛♥ ♦r✐❡♥t❛t✐♦♥ ♦♥ X ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ♣r❡✈✐♦✉s ✐♥t❡❣r❛• t♦ ❤❛✈❡ ❛♥② ♠❡❛♥✐♥❣✳ ❲❡ r❡♠✐♥❞ t❤❛t X ❜❡✐♥❣ r❡❝t✐✜❛❜•❡✱ t❤❡r❡ ❡①✐sts ❛♥ ♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• t❛♥❣❡♥t s♣❛❝❡ ❛t ❛•♠♦st ❡✈❡r② ♣♦✐♥t x ∈ X ❛♥❞✱ ✐❢ ✇❡ ❞❡♥♦t❡ τ (x) t❤❡ ✉♥✐t d✲✈❡❝t♦r ✐♥ Λ d E ❛ss♦❝✐❛t❡❞ t♦ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❛t x✱ t❤❡♥ t❤❡ ✐♥t❡❣r❛• ✐♥ ❡q✳✭✷✳✼✮ ✐s ✿ C (X,f ) (ω) . = X ω (x,f (x)) (τ (x))dH d (x) . ✭✷✳✽✮ ■♥ t❤❡ ♠♦r❡ r❡str✐❝t❡❞ ❝❛s❡ ✇❤❡r❡ X ✐s ❛ s✉❜♠❛♥✐❢♦•❞✱ t❤❡♥ t❤❡ ♣r❡✈✐♦✉s ❝❛♥ ❛•s♦ ❜❡ ✇r✐tt❡♥ t❤r♦✉❣❤ •♦❝❛• ♣❛r❛♠❡tr✐③❛t✐♦♥ ✇✐t❤ ❛ ❣✐✈❡♥ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✉♥✐t② ♦❢ X✳ ■❢ γ : U → E ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ X ✇✐t❤ U ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d ✱ t❤❡♥ C (X,f ) (ω) = U ω (γ(u),f •γ(u)) ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d du 1 ...du d . ❛♥❞ ✐t✬s ❛♥ ❡❛s② ❝❤❡❝❦ t❤❛t ❛ r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ❞♦❡s♥✬t ❝❤❛♥❣❡ t❤❡ ✐♥t❡❣r❛•✳ ❖❢ ❝♦✉rs❡✱ ❛s ❢♦r r❡❣✉•❛r ❝✉rr❡♥ts✱ t❤❡ ♣r❡✈✐♦✉s ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛♥❞ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐s ♥♦t s✉r•❡❝t✐✈❡✳ ❋♦r ✐♥st❛♥❝❡✱ ❛ s✉♠ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♦❢ t❤❡ ❢♦r♠ C (X,f ) ❞♦ ♥♦t ❣❡♥❡r❛••② ❞❡r✐✈❡ ❢r♦♠ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡✳ ■♥ t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t ❢r❛♠❡✇♦r❦✱ ❉✐r❛❝ ♠❛ss❡s ❛r❡ ♥❛t✉r❛••② ❣❡♥❡r❛•✐③❡❞ ❜② ❡•❡♠❡♥t❛r② ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♦r ❉✐r❛❝ ❢❝✉rr❡♥ts δ ξ (x,m) ❢♦r x ∈ X, m ∈ M ❛♥❞ ξ ∈ Λ p E s✉❝❤ t❤❛t δ ξ (x,m) (ω) . = ω (x,m) (ξ)✳ ■♥ t❤❡ s❛♠❡ ✇❛② ❛s ❡①♣•❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ♦♥❡ ❝❛♥ ❣✐✈❡ ❛ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛ss♦❝✐❛t❡❞ t♦ (X, f ) ✇❤❡♥ ❛ ♠❡s❤ ✐s ❞❡✜♥❡❞ ♦♥ X✳ C (X,f ) ✐s t❤❡♥ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ s✉♠ ♦❢ ❉✐r❛❝ ❢❝✉rr❡♥ts ✿ C (X,f ) ≈ k=1..N δ ξ k (x k ,m k ) ✭✷✳✾✮ ■♥ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡ ♦❢ ❛ tr✐❛♥❣✉•❛t❡❞ s✉r❢❛❝❡✱ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢❝✉r✲ r❡♥t ❝❛♥ ❜❡ s✐♠♣•② ♦❜t❛✐♥❡❞ ❛s ❡①♣•❛✐♥❡❞ ❢♦r ❝•❛ss✐❝❛• ❝✉rr❡♥ts ❛♥❞ ❜② ❛❞❞✐♥❣ t❤❡ ✬✐♥t❡r♣♦•❛t❡❞ ✈❛•✉❡✬ ♦❢ s✐❣♥❛• ❛t ❡❛❝❤ ❝❡♥t❡r ♣♦✐♥t ♦❢ tr✐❛♥❣•❡s ✭♦r ❢♦r ❛ ❣❡♥❡r❛• s✐❣♥❛• ♠❛♥✐❢♦•❞✱ t❤❡ ❋r❡❝❤❡t ♠❡❛♥ ✐♥ M ✮✱ ❛s ✐♥ t❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✷✳✼✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✱ ✇❡ tr② t♦ ❣✐✈❡ ❛ ♠♦r❡ ♣r❡❝✐s❡ s❡♥s❡ t♦ ❡q✳✭✷✳✾✮✳ ❲❡ s❤❛•• ❛ss✉♠❡ ❢♦r s✐♠✲ ♣•✐❝✐t② t❤❛t X ✐s ❛ C 1 ✲s✉❜♠❛♥✐❢♦•❞ ♦❢ E ✇❤✐❝❤ ❛❞♠✐ts ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ F : Ω → E ✇✐t❤ Ω ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d ❛♥❞ F ❛♥ ❡♠❜❡❞❞✐♥❣✳ ◆❛♠❡•②✱ ❛ss✉♠❡ t❤❛t ✇❡ ❛r❡ ❣✐✈❡♥ ❛ d✲❞✐♠❡♥s✐♦♥❛• ♠❡s❤ ♦♥ ❛ C 1 ✲s✉❜♠❛♥✐❢♦•❞ X ✇❤♦s❡ ❢❛❝❡s ✇✐•• ❜❡ ❞❡♥♦t❡❞ ❜② (∆ i ) i=1,..,n ❛♥❞ t❤❛t t❤❡ s✐❣♥❛• ✳ ❋r♦♠ t❤❡ ❞✐s❝r❡t❡ r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❤❛✈❡ ❛ ✈❡r② s✐♠♣•❡ ✐♥t❡r♣r❡t❛t✐♦♥✱ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ❛tt❛❝❤✲ ✐♥❣ ✈❛•✉❡s ♦❢ t❤❡ s✐❣♥❛• f t♦ t❤❡ ✉s✉❛• r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ X ❛s ❛ d✲❝✉rr❡♥t✳ ■♥ ❢❛❝t✱ ♦♥❡ ❝♦✉•❞ ❛•t❡r♥❛t✐✈❡•② ❞❡✜♥❡ ❢✉♥❝t✐♦♥❛• ❢♦r♠s ❛s ❛ t❡♥s♦r ♣r♦❞✉❝t s♣❛❝❡ ❜② ♠❛❦✐♥❣ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ Ω d 0 (E, M ) = Ω d 0 (E) ⊗ C 0 (M, R) ❛♥❞ t❤✉s ❝♦♥s✐❞❡r ❢❝✉rr❡♥ts ❛s t❡♥s♦r ♣r♦❞✉❝ts ♦❢ d✲❝✉rr❡♥ts ♦♥ E ❛♥❞ 0✲❝✉rr❡♥ts ✭✐✳❡ ♠❡❛s✉r❡s✮ ♦♥ M ✱ ✇❤✐❝❤ ✐s ❝•♦s❡r t♦ t❤❡ ♥♦t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ ❬✼✵(✳ ❉✐✛❡♦♠♦r♣❤✐❝ tr❛♥s♣♦rt ♦❢ ❢❝✉rr❡♥ts ❲❤❛t ❛❜♦✉t ❞✐✛❡♦♠♦r♣❤✐❝ tr❛♥s♣♦rt ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❄ ❚❤✐s q✉❡st✐♦♥ ❝❛♥♥♦t ❜❡ ❛❞❞r❡ss❡❞ ❛s s✐♠♣•② ❛s ✐♥ t❤❡ ❝•❛ss✐❝❛• ❝✉rr❡♥t s❡tt✐♥❣ ✐♥ ✐ts ❢✉•• ❣❡♥❡r❛•✐t②✱ ✇❤✐❝❤ ❡ss❡♥t✐❛••② r❡s✉•ts ❢r♦♠ t❤❡ ❞✐s❝✉ss✐♦♥ ✐♥ ✷✳✶✳✷✳ ❆s ✇❡ ❡①♣•❛✐♥❡❞✱ t❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ♣♦ss✐❜✐•✐t✐❡s ❜♦t❤ ❢♦r t❤❡ ❣r♦✉♣ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ♦♥❡ ❝❛♥ ❝❤♦♦s❡ ❛♥❞ t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡s❡ ❞❡❢♦r♠❛t✐♦♥s ♦♥ t❤❡ ❝♦♥s✐❞❡r❡❞ ❝•❛ss ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❚♦ r❡♠❛✐♥ ❣❡♥❡r❛•✱ s✉♣♣♦s❡ t❤❛t ❛ ❝❡rt❛✐♥ ❝•❛ss ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ❣r♦✉♣ G ⊂ ❉✐✛(E × ✺✷ ❋✐❣✉r❡ ✷✳✼✿ ❉✐r❛❝ ❢❝✉rr❡♥ts ❝♦♠♣✉t❡❞ ❢r♦♠ ❛ tr✐❛♥❣✉•❛t❡❞ ❢✉♥❝t✐♦♥❛• s✉r❢❛❝❡✳ M ) ❛♥❞ ❛ •❡❢t ❣r♦✉♣ ❛❝t✐♦♥ ❛r❡ ✜①❡❞✱ ✇❡ ✇✐•• ♥♦t❡ φ.(X, f ) t❤❡ ❛❝t✐♦♥ ♦❢ φ ∈ G ♦♥ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f )✳ ❚❤❡♥✱ ❉❡✜♥✐t✐♦♥ ✷✳✷✳✷✳ ❲❡ ❝❛•• ❛ ❞❡❢♦r♠❛t✐♦♥ ♠♦❞❡• ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛ •❡❢t ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ G ♦♥ Ω d 0 (E, M ) ′ ✇❤✐❝❤ ✐s s✉❝❤ t❤❛t ❢♦r ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ❛♥❞ ❛♥② ❞✐✛❡♦♠♦r♣❤✐s♠ φ✱ ✐❢ φ * st❛♥❞s ❢♦r t❤❡ ❛❝t✐♦♥ ♦♥ ❢❝✉rr❡♥ts✱ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt② ❤♦•❞s ✿ [φ * C (X,f ) ](ω) = C φ.(X,f ) (ω) ✭✷✳✶✵✮ ❢♦r ❛•• ω ∈ Ω d 0 (E, M )✳ ❊q✳✭✷✳✶✵✮ ✐s ❜❛s✐❝❛••② ❛ ❝♦♠♠✉t❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡♠❜❡❞❞✐♥❣ ♦❢ ❢s❤❛♣❡s ✐♥t♦ ❢❝✉rr❡♥ts✬ s♣❛❝❡ ✇✐t❤ t❤❡ ❞❡❢♦r♠❛t✐♦♥ tr❛♥s♣♦rt✳ ❲✐t❤ ✉s✉❛• ❝✉rr❡♥ts✱ t❤✐s ❛•✇❛②s ❤♦•❞s ✇✐t❤ t❤❡ ♣✉••✲❜❛❝❦ ❛♥❞ ♣✉s❤✲❢♦r✇❛r❞ ❛❝t✐♦♥s✳ ❍❡r❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❛❞❛♣t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛❝t✐♦♥ ♦♥ ❢❝✉rr❡♥ts t♦ ❜❡ ❝♦♠♣❛t✐❜•❡ ✇✐t❤ ❛ ❣✐✈❡♥ ❛❝t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✱ t❤✐s ✐s ✉s✉❛••② ♥♦t ❞✐✣❝✉•t✳ ■♥ t❤❡ ❝❛s❡ ♦❢ r❡❛• ✈❛•✉❡❞ s✐❣♥❛• ❞❛t❛ ❞❡❢♦r♠❡❞ ❛s ✐♥ ❡q✳✭✷✳✶✮✱ t❤❡ ❛❝t✐♦♥ ♦❢ φ ∈ ❉✐✛(E) ♦♥ ❛ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t C ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✐♥ ❛ ✈❡r② s✐♠✐•❛r ✇❛② ✿            φ * C(ω) . = C(φ * ω), ∀ω ∈ Ω d 0 (E, M ) ✇❤❡r❡ ❢♦r ❛•• ∀x ∈ E, m ∈ M, ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E (φ * ω) (x,m) (ξ) . = ω (φ(x),m) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ d )) . ✭✷✳✶✶✮ ■t ❝❛♥ ❜❡ ❡❛s✐•② ❝❤❡❝❦❡❞ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s t❤❛t ❢♦r ❛•• ❢✉♥❝t✐♦♥❛• s❤❛♣❡s (X, f )✱ ✇❡ ❤❛✈❡ φ * C (X,f ) = C (φ(X),f •φ -1 ) ❛s ✇❡ ❡①♣❡❝t❡❞ ✉♥❞❡r t❤✐s ♠♦❞❡•✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❣r♦✉♣ ♦❢ ❞❡❢♦r♠❛t✐♦♥s ❉✐✛(E) × M(E, R) ❛❝t✐♥❣ ❛s ✐♥ ❡q✳✭✷✳✷✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ♠♦❞❡• ♦♥ ❢❝✉rr❡♥ts ❜② s❡tt✐♥❣ ✿            (φ, h) * C(ω) . = C((φ, h) * ω), ∀ω ∈ Ω d 0 (E, M ) ✇❤❡r❡ ❢♦r ❛•• ∀x ∈ E, m ∈ M, ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E ((φ, h) * ω) (x,m) (ξ) . = ω (φ(x),m+h(x)) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ d )) . ✭✷✳✶✷✮ ❚❤❡s❡ ❛r❡ ❡ss❡♥t✐❛••② t❤❡ t✇♦ s✐t✉❛t✐♦♥s t❤❛t s❤❛•• ❜❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ♥✉♠❡r✐❝❛• ❛♣♣•✐✲ ❝❛t✐♦♥s ✭s❡❝t✐♦♥ ✷✳✸✳✷✮✳ (❡t t❤❡ s❛♠❡ t❤✐♥❣ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❢♦r t❤❡ ❣❡♥❡r❛• ❛❝t✐♦♥ ♦♥ ✺✸ ❢s❤❛♣❡s ❣✐✈❡♥ ❜② ❡q✳✭✷✳✺✮ ✇❤✐❝❤ tr❛♥s•❛t❡s t♦ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❜② t❤❡ r❡•❛t✐♦♥s ✿            ψ * C(ω) . = C(ψ * ω), ∀ω ∈ Ω d 0 (E, M ) ✇❤❡r❡ ❢♦r ❛•• ∀x ∈ E, m ∈ M, ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E (ψ * ω) (x,m) (ξ) . = ω ψ(x,m) (d x ψ g (ξ 1 ) ∧ ... ∧ d x ψ g (ξ d )) . ✭✷✳✶✸✮ |❡•❛t✐♦♥s❤✐♣ ✇✐t❤ ♦t❤❡r ❝✉rr❡♥ts ❲❡ ♠❡♥t✐♦♥❡❞ ❡❛r•✐❡r t❤❛t t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐s ✬❜✐❣❣❡r✬ t❤❛♥ t❤❡ ♦♥❡ ♦❢ ✉s✉❛• ❝✉rr❡♥ts✱ ❛s ✇❡ s❡❡ ❡❛s✐•② ✐❢ ♦♥❡ t❛❦❡s t❤❡ s✐❣♥❛• s♣❛❝❡ ❛s ❛ s✐♥❣•❡t♦♥ ✭M = {m}✮ ❛♥❞ ✐❞❡♥t✐✜❡s E × M ✇✐t❤ E✳ ❆ ♠♦r❡ r✐❣♦r♦✉s ✇❛② t♦ s❛② s♦ ✐s t❤❛t ✇❡ ❤❛✈❡ ❛ s✉r•❡❝t✐✈❡ ❛♣♣•✐❝❛t✐♦♥ s : Ω d 0 (E, M ) → Ω d 0 (E) ✇❤✐❝❤ ✇❡ ♦❜t❛✐♥ ❜② ✜①✐♥❣ m ∈ M ❛♥❞ ❛ss♦❝✐❛t✐♥❣ t♦ ω ∈ Ω d 0 (E, M )✱ s(ω) = ω (.,m) (.) ∈ Ω d 0 (E). ❚❤❡ ❞✉❛• ❛♣♣•✐❝❛t✐♦♥ j : C ∈ Ω d 0 (E) ′ -→ [C : ω → C(s(ω))] ∈ Ω d 0 (E, M ) ′ ✐s ❛ ♦♥❡ t♦ ♦♥❡ •✐♥❡❛r ♠❛♣✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ❝♦✉•❞ ❡①♣❡❝t✱ ✐t✬s ❛ ✬s♠❛••❡r✬ s♣❛❝❡ t❤❛♥ t❤❡ ♦♥❡ ♦❢ ❝✉rr❡♥ts ✐♥ t❤❡ ♣r♦❞✉❝t E × M ✳ ■♥❞❡❡❞✱ ❛ss✉♠✐♥❣✱ t♦ s✐♠♣•✐❢②✱ t❤❛t M ✐s ❛ ✈❡❝t♦r s♣❛❝❡✱ ✇❡ ❝❛♥ ❡♠❜❡❞ Λ d E ✐♥t♦ Λ d (E × M ) ❜② s❡tt✐♥❣ ❢♦r ξ 1 ∧ ... ∧ ξ d ∈ Λ d E, l(ξ 1 ∧ ... ∧ ξ d ) = (ξ 1 , 0) ∧ ... ∧ (ξ d , 0) ∈ Λ d (E × M )✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❛♣♣•✐❝❛t✐♦♥ ✿ p : Ω d 0 (E × M ) -→ Ω d 0 (E, M ) ω -→ p(ω)/ p(ω) (x,m) (ξ) = ω (x,m) ( l(ξ)) ❖♥ t❤❡ ♦t❤❡r ✇❛②✱ t❤❡r❡ ✐s ❛ s✉r•❡❝t✐✈❡ ❛♣♣•✐❝❛t✐♦♥ l ❢r♦♠ Λ d (E × M ) t♦ Λ d E ✇❤✐❝❤ ✐s s✉❝❤ t❤❛t ❢♦r ❛•• (ξ 1 , η 1 ), ..., (ξ d , η d ) ∈ E × M, l((ξ 1 , η 1 ) ∧ ... ∧ (ξ d , η d )) = ξ 1 ∧ ... ∧ ξ d ✱ ✇❤✐❝❤ ✐♥❞✉❝❡s t❤❡ ❛♣♣•✐❝❛t✐♦♥ ✿ p : Ω d 0 (E, M ) -→ Ω d 0 (E × M ) ω -→ p(ω)/ p(ω) (x,m) (ξ) = ω (x,m) (l(ξ)) ❛♥❞ ✇❡ ❡❛s✐•② s❡❡ t❤❛t p•p ✐s t❤❡ ✐❞❡♥t✐t② ♦♥ Ω d 0 (E, M )✳ ■t r❡s✉•ts t❤❛t p ✐s ❛ s✉r•❡❝t✐✈❡ ❛♣♣•✐❝❛t✐♦♥✳ ❲❡ ♦❜t❛✐♥ t❤❡r❡❢♦r❡ ❛ ❞✉❛• ❛♣♣•✐❝❛t✐♦♥ ✿ i : Ω d 0 (E, M ) ′ -→ Ω d 0 (E × M ) ′ C -→ [i(C) : ω → C(p(ω))] t❤❛t ✐s ♦♥❡ t♦ ♦♥❡✳ ■♥ ❝♦♥❝•✉s✐♦♥✱ ✇❡ •✉st ♣r♦✈❡❞ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❞✐❛❣r❛♠ ✿ Ω d 0 (E) ′ Ω d 0 (E, M ) ′ Ω d 0 (E × M ) ′ j i ✺✹ ✷✳✷✳✷ |❡♣r♦❞✉❝✐♥❣ ❑❡r♥❡• ❍✐•❜❡rt ❙♣❛❝❡ str✉❝t✉r❡s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♥♦✇ ❛❞❞r❡ss t❤❡ ❢✉♥❞❛♠❡♥t❛• q✉❡st✐♦♥ ♦❢ ❝♦♠♣❛r✐♥❣ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts t❤r♦✉❣❤ ❛♣♣r♦♣r✐❛t❡ ♠❡tr✐❝s✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✇❡ ❛❞❛♣t t❤❡ ✐❞❡❛s ♦❢ |❑❍❙ t❤❛t ✇❡r❡ ♣r❡s❡♥t❡❞ ❜r✐❡✢② ❢♦r ❝✉rr❡♥ts ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ❝❤❛♣t❡r✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ♣r♦♣♦s❡ ❛ ❣❡♥❡r✐❝ ✇❛② t♦ ❜✉✐•❞ ❦❡r♥❡•s t❤❛t ❡♠❜❡❞s ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥t♦ ❍✐•❜❡rt s♣❛❝❡s ❛♥❞ ♣r♦✈✐❞❡ ❝•♦s❡❞ ❢♦r♠ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❞♦t ♣r♦❞✉❝t ❜❡t✇❡❡♥ ❉✐r❛❝s✳ ❑❡r♥❡•s✬ ❝♦♥str✉❝t✐♦♥ ❆s ✇❡ ❤❛✈❡ s❡❡♥ ❢♦r ❝✉rr❡♥ts✱ t❤❡ t❤❡♦r② ♦❢ |❑❍❙ ❞❡✜♥❡s ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ❜❡t✇❡❡♥ ❝✉rr❡♥ts t❤r♦✉❣❤ ❛ ❝❡rt❛✐♥ ❦❡r♥❡• ❢✉♥❝t✐♦♥ s❛t✐s❢②✐♥❣ s♦♠❡ r❡❣✉•❛r✐t② ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ s❤❛•• t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ✐❞❡❛ t❤❛t ❢✉♥❝t✐♦♥❛• d✲❝✉rr❡♥ts ❝❛♥ ❛•s♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ d✲❝✉rr❡♥ts ♦♥ E ❛♥❞ 0✲❝✉rr❡♥ts ♦♥ M ✳ ❲❡ r❡♠✐♥❞ t❤❛t ❢♦r ❛ s❡t A ❛♥❞ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡ V ✱ ✇❡ s❛② t❤❛t K ✐s ❛ V ✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ A ✐❢ K : A × A → L(V )✱ K(a ′ , a) = K(a, a ′ ) * ❛♥❞ K s❛t✐s✜❡s t❤❡ ♣♦s✐t✐✈✐t② ♣r♦♣❡rt② ✿ ❢♦r ❛•• a 1 , ..., a N ∈ A ❛♥❞ v 1 , .., v N ∈ V ✱ N i,j=1 K(a i , a j )v i , v j ≥ 0✳ ▲❡t✬s ✜rst ♠❡♥t✐♦♥ t❤❡ ❢♦••♦✇✐♥❣ ✈❡r② ❣❡♥❡r❛• r❡s✉•t ❢r♦♠ ❦❡r♥❡• t❤❡♦r② ✿ ▲❡♠♠❛ ✷✳✷✳✶✳ ▲❡t A ❛♥❞ B ❜❡ t✇♦ s❡ts ❛♥❞ K A ❛ V A ✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ A✱ K B ❛ V B ✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ B ✇❤❡r❡ V A ❛♥❞ V B ❛r❡ t✇♦ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡s✳ ❚❤❡♥✱ t❤❡ t❡♥s♦r ♣r♦❞✉❝t K A ⊗ K B ❞❡✜♥❡❞ ❜② ✿ K A ⊗ K B (a, b), (a ′ , b ′ ) u A ⊗ u B , v A ⊗ v B = K A (a, a ′ )u A , v A . K B (b, b ′ )u B , v B ✐s ❛ V A ⊗ V B ✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ A × B✳ Pr♦♦❢✳ ▲❡t N ∈ N * ✱ a 1 , .., a N ∈ A ❛♥❞ b 1 , ..., b N ∈ B✱ u 1 , .., u N ∈ V A ❛♥❞ v 1 , .., u N ∈ V B ✳ ❲❡ ❤❛✈❡ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡• ✿ N i,j=1 K A ⊗ K B ((a i , b i ), (a j , b j )) u i ⊗ v i , u j ⊗ v j = N i,j=1 K A (a i , a j )u i , u j . K B (b i , b j )v i , v j = T r(M A M B ) ✇❤❡r❡ M A = ( K A (a i , a j )u i , u j ) i,j ❛♥❞ M B = ( K B (b i , b j )v i , v j ) i,j ❛r❡ ❜♦t❤ ♣♦s✐t✐✈❡ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ✭❜❡❝❛✉s❡ K A ❛♥❞ K B ❛r❡ ♣♦s✐t✐✈❡ ❦❡r♥❡•s✮✳ ■t ✐s t❤❡♥ ❛ ✇❡••✲ ❦♥♦✇♥ ❢❛❝t t❤❛t T r(M A M B ) ≥ 0 ❛♥❞ t❤✉s K A ⊗ K B ✐s ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡•✳ ◆♦t❡ t❤❛t ✐❢ K B ✐s ❛ r❡❛• ❦❡r♥❡• ✭✐✳❡ V B = R✮ t❤❡♥ K A ⊗K B ✐s ❛ V A ⊗R ≈ V A ✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ A × B✳ ◆♦✇✱ ✇❡ ❛♣♣•② •❡♠♠❛ ✷✳✷✳✶ t♦ t❤❡ ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✷✳ ▲❡t K g : E×E → L(Λ d E) ❜❡ ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ t❤❡ ❣❡♦♠❡tr✐❝❛• s♣❛❝❡ E ❛♥❞ k f : M ×M → R ❛ r❡❛• ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ t❤❡ s✐❣♥❛• s♣❛❝❡ M ✳ ❲❡ ❛ss✉♠❡ t❤❛t ❜♦t❤ ❦❡r♥❡•s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ❜♦✉♥❞❡❞ ❛♥❞ ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ❚❤❡♥ K g ⊗ k f ❞❡✜♥❡s ❛ Λ d E✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ E × M ✇❤♦s❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡♣r♦❞✉❝✐♥❣ ❍✐•❜❡rt s♣❛❝❡ W ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ Ω d 0 (E, M )✳ Pr♦♦❢✳ ❋r♦♠ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ ❜♦t❤ ❦❡r♥❡•s✱ ✇❡ ❦♥♦✇ t❤❛t t♦ K g ❛♥❞ k f ❝♦rr❡s♣♦♥❞ t✇♦ |❑❍❙ W g ❛♥❞ W f t❤❛t ❛r❡ r❡s♣❡❝t✐✈❡•② ❡♠❜❡❞❞❡❞ ✐♥t♦ Ω p 0 (E) ❛♥❞ C 0 (M, R) ✭❝❢ s❡❝t✐♦♥ ✶✳✷✳✹ ✐♥ ❝❤❛♣t❡r ✶✮✳ ❇② ❛♣♣•②✐♥❣ •❡♠♠❛ ✷✳✷✳✶ ✇✐t❤ A = E✱ B = M ✱ ✺✺ V A = Λ d E ❛♥❞ V B = R✱ ✇❡ ❣❡t t❤❛t K . = K g ⊗ k f ❞❡✜♥❡s ❛ Λ d E✲♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ E × M ✳ ▼♦r❡♦✈❡r✱ s✐♥❝❡ k f ✐s r❡❛•✲✈❛•✉❡❞✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ❢♦r K ✿ K ((x 1 , m 1 ), (x 2 , m 2 )) = k f (m 1 , m 2 ).K g (x 1 , x 2 ) ✭✷✳✶✹✮ ❚❤❡ ❦❡r♥❡• K ❝♦rr❡s♣♦♥❞s t♦ ❛ ✉♥✐q✉❡ |❑❍❙ W t❤❛t ✐s t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r s♣❛❝❡ s♣❛♥♥❡❞ ❜② ❛•• t❤❡ ❢✉♥❝t✐♦♥s {k f (., m).K g (., x)ξ} ❢♦r x ∈ E, m ∈ M, ξ ∈ Λ d E✳ ❙✐♥❝❡ ❢✉♥❝t✐♦♥s k f (., m) ❛♥❞ K g (., x) ❛r❡ ❜♦t❤ ❝♦♥t✐♥✉♦✉s ❛♥❞ ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t② ❢r♦♠✱ t❤✐s ❤♦•❞s ❢♦r k f (., m).K g (., x)ξ ❛s ✇❡••✱ s♦ t❤❛t W ✐s ✐♥❞❡❡❞ ❡♠❜❡❞❞❡❞ ✐♥t♦ Ω d 0 (E, M )✳ ■t ♦♥•② r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t t❤❡ ❡♠❜❡❞❞✐♥❣ ✐s ❝♦♥t✐♥✉♦✉s✱ ✇❤✐❝❤ r❡❞✉❝❡s t♦ ❞♦♠✐♥❛t❡ t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ❜② . W ✳ ▲❡t ω ∈ W ✳ ❋♦r ❛•• (x, m) ∈ E × M ❛♥❞ ξ ∈ Λ d E s✉❝❤ t❤❛t |ξ| = 1✱ ✇❡ ❤❛✈❡ |ω (x,m) (ξ)| = |δ ξ (x,m) (ω)| . ✭✷✳✶✺✮ ❙✐♥❝❡ W ✐s ❛ |❑❍❙✱ ❛•• δ ξ (x,m) ❛r❡ ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠s ♦♥ W ✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ |✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ♣r♦✈✐❞❡s ❛♥ ✐s♦♠❡tr② K W : W ′ → W ✳ ❚❤❡♥ ✿ δ ξ 1 (x 1 ,m 1 ) , δ ξ 2 (x 2 ,m 2 ) W ′ = K W (δ ξ 1 (x 1 ,m 1 ) ), K W (δ ξ 2 (x 2 ,m 2 ) ) W = k f (., m 1 )K g (., x 1 )ξ 1 , k f (., m 2 )K g (., x 2 )ξ 2 W = k f (m 1 , m 2 ). K g (x 1 , x 2 )ξ 1 , ξ 2 ✭✷✳✶✻✮ ◆♦✇✱ ❜❛❝❦ t♦ ❡q✉❛t✐♦♥ ✭✷✳✶✺✮✱ ✇❡ ❤❛✈❡ ✿ |ω (x,m) (ξ)| ≤ δ ξ x,m W ′ ω W ≤ k f (m, m). K g (x, x)ξ, ξ ω W ❙✐♥❝❡ ✇❡ ❛ss✉♠❡ t❤❛t m → k f (m, m) ❛♥❞ x → K g (x, x) ❛r❡ ❜♦✉♥❞❡❞ ✇❡ ❞❡❞✉❝❡ t❤❛t k f (m, m). K g (x, x)ξ, ξ ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ x✱ m ❛♥❞ ξ ✇✐t❤ |ξ| = 1✳ ❍❡♥❝❡✱ ❜② t❛❦✐♥❣ t❤❡ s✉♣r❡♠✉♠ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥✱ ✇❡ ✜♥❛••② ❣❡t ω ∞ ≤ C ω W ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ❡♠❜❡❞❞✐♥❣ ✐s ❝♦♥t✐♥✉♦✉s✳ ❇② ❞✉❛•✐t②✱ ✇❡ ❤❛✈❡ ❛♥ ❛♣♣•✐❝❛t✐♦♥ ı * : Ω d 0 (E, M ) ′ → W ′ t❤❛t s❡♥❞s ❛♥② ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t ✐♥t♦ W ′ ✭✇❤✐❝❤ ✐s ❛•s♦ ❛ ❍✐•❜❡rt s♣❛❝❡✮ ❛♥❞ ✇❡ ❝❛♥ t❤✉s ❝♦♠♣❛r❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ t❤❡ ♥♦r♠ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❦❡r♥❡•✳ ❍♦✇❡✈❡r✱ ✐t✬s ✐♠♣♦rt❛♥t t♦ ♣♦✐♥t ♦✉t t❤❛t ı * ✐s ♥♦t ♥❡❝❡ss❛r✐•② ❛♥ ❡♠❜❡❞❞✐♥❣✳ ■❢ ❢♦r ✐♥st❛♥❝❡ M = S 1 ❛♥❞ ∀θ 1 , θ 2 ∈ S 1 , k f (θ 1 , θ 2 ) = 1✱ ✇❡ s❡❡ t❤❛t ❢♦r x ∈ E, ξ ∈ Λ d E ❛♥❞ θ 1 , θ 2 ∈ S 1 ✱ ✇❡ ❤❛✈❡ ı * (δ ξ (x,θ 1 ) ) = ı * (δ ξ (x,θ 2 ) ) ❛♥❞ ı * ✐s ♥♦t ✐♥•❡❝t✐✈❡✳ ❚❤❡ ❢❛❝t t❤❛t ı * ✐s ❛♥ ❡♠❜❡❞❞✐♥❣ ✐s ❝❛••❡❞ t❤❡ C 0 ✲✉♥✐✈❡rs❛•✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ❦❡r♥❡• ❛♥❞✱ ❛s ♣r♦✈❡❞ ✐♥ ❬✷✵(✱ ✐t ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ ♣r♦♣❡rt② ♦❢ W ❜❡✐♥❣ ❞❡♥s❡ ✐♥ Ω d 0 (E, M )✳ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✐♥ ♣❛rt✐❝✉•❛r ✐❢ ❜♦t❤ W g ❛♥❞ W f ❛r❡ r❡s♣❡❝t✐✈❡•② ❞❡♥s❡ ✐♥ Ω d 0 (E) ❛♥❞ C 0 (M, R)✳ ■♥ ❬✷✵( ❛r❡ ❛•s♦ ❣✐✈❡♥ ❛ ❢❡✇ ❡①❛♠♣•❡s ♦❢ ✉♥✐✈❡rs❛• ❦❡r♥❡•s ❛♥❞ ❜✉✐•❞✐♥❣ t❡❝❤♥✐q✉❡s ✐♥ t❤❡ ❣❡♥❡r❛• ❝❛s❡✳ •❛✉ss✐❛♥ ❦❡r♥❡•s ❛r❡ t❤❡ s✐♠♣•❡st ❡①❛♠♣•❡s✱ ✇❤✐❝❤ ❡①♣•❛✐♥s t❤❡✐r ✇✐❞❡ ✉s❡ ✐♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✳ ❚♦ s✉♠ ✉♣✱ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✷ ♣r♦✈✐❞❡s ❛ q✉✐t❡ ♥❛t✉r❛• ✭❜✉t ♥♦t ✉♥✐q✉❡✮ ✇❛② t♦ ❜✉✐•❞ ✺✻ ❦❡r♥❡•s ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❜② ♠❛❦✐♥❣ t❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ❦❡r♥❡•s ❞❡✜♥❡❞ s❡♣✲ ❛r❛t❡•② ✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ❞♦♠❛✐♥ ✭d✲❝✉rr❡♥ts ✐♥ E✮ ❛♥❞ ✐♥ t❤❡ s✐❣♥❛• ❞♦♠❛✐♥ ✭0✲ ❝✉rr❡♥ts ✐♥ M ✮✳ ◆♦t❡ t❤❛t ♥♦♥ ♣r♦❞✉❝t ❦❡r♥❡•s ❝♦✉•❞ ❛•s♦ ❜❡ ✉s❡❞ ❜✉t t❤❡ ♣r♦❞✉❝t s✐t✉❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛ss✉♠♣t✐♦♥ ❜❡t✇❡❡♥ s❤❛♣❡ ❛♥❞ ❢✉♥❝t✐♦♥❛• ✐♥❢♦r♠❛t✐♦♥ ✇❤✐❝❤ ✐s ♥❛t✉r❛• ✇❤❡♥ ♠♦❞❡•✐♥❣ t❤❡ r❡s✐❞✉❛• ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝✲ t✐♦♥❛• s❤❛♣❡s ❛s ♥♦✐s❡✳ ▼♦r❡♦✈❡r✱ t❤❡ ✉s❡ ♦❢ ♣r♦❞✉❝t ❦❡r♥❡•s •❡❛❞s t♦ s✐♠♣•❡r ❛♥❞ ❢❛st❡r ❝♦♠♣✉t❛t✐♦♥❛• s❝❤❡♠❡s✱ ❛s ✇❡ s❤❛•• ❡①♣•❛✐♥ •❛t❡r ♦♥✳ ❲❡ s❡❡ t❤❛t✱ ❡✈❡♥t✉❛••②✱ ❡✈❡r②t❤✐♥❣ r❡•✐❡s ♦♥ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ❦❡r♥❡•s ♦♥ E ❛♥❞ M ✳ ❑❡r♥❡•s ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ E ♦❜✈✐♦✉s•② ❞♦ ♥♦t r❛✐s❡ ❛♥② ❛❞❞✐t✐♦♥❛• ❞✐✣❝✉•t② ✐♥ ♦✉r ❛♣♣r♦❛❝❤ ❝♦♠♣❛r❡❞ t♦ ✉s✉❛• ❝✉rr❡♥t s❡tt✐♥❣s✳ ❆♠♦♥❣ ♦t❤❡rs✱ ✇❡ ❤❛✈❡ ❛•r❡❛❞② ♠❡♥✲ t✐♦♥❡❞ r❛❞✐❛• s❝❛•❛r ❦❡r♥❡•s ❞❡✜♥❡❞ ❢♦r x, y ∈ E ❜② K(x, y) = k(|x -y|).■❞ E ✇❤❡r❡ k ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ R + ❛♥❞ ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ❚❤✐s ❢❛♠✐•② ♦❢ ❦❡r♥❡•s ✐s t❤❡ ♦♥•② ♦♥❡ t❤❛t ✐♥❞✉❝❡s ❛ |❑❍❙ ♥♦r♠ ✐♥✈❛r✐❛♥t ❢♦r ❛✣♥❡ ✐s♦♠❡tr✐❡s✳ ❚❤❡ ♠♦st ✉s✉❛• ✐s t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡•✱ ✇❤✐❝❤ ✐s ❛•s♦ C 0 ✉♥✐✈❡rs❛• ❛s ✇❡ s❛✐❞ ❡❛r•✐❡r✱ ❞❡✜♥❡❞ ❜② K(x, y) = exp -|x-y| 2 σ 2 ■❞ E ✱ σ ❜❡✐♥❣ ❛ s❝❛•❡ ♣❛r❛♠❡t❡r t❤❛t ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ r❛♥❣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ♣♦✐♥t ♣♦s✐t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❦❡r♥❡• ♦♥ t❤❡ s✐❣♥❛• ♠❛♥✐❢♦•❞ M ✐s ♦❢t❡♥ ♠♦r❡ ✐♥✈♦•✈❡❞✳ ❍♦✇❡✈❡r✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t✱ ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ t❤✐s ✐ss✉❡ ✐s ❞r❛s✲ t✐❝❛••② s✐♠♣•✐✜❡❞ ❜❡❝❛✉s❡ ✇❡ ♦♥•② ♥❡❡❞ t♦ ❞❡✜♥❡ r❡❛•✲✈❛•✉❡❞ ❦❡r♥❡•s ♦♥ M ✳ ❚❤✐s ✐s ❝♦♥tr❛st✐♥❣✱ ❢♦r ✐♥st❛♥❝❡✱ ✇✐t❤ t❤❡ ✐❞❡❛ ♦❢ ♣r♦❞✉❝t s♣❛❝❡ ❝✉rr❡♥ts ♦❢ s✉❜s❡❝t✐♦♥ ✷✳✶✳✸✱ t❤❛t ✇♦✉•❞ r❡q✉✐r❡ ❦❡r♥❡•s •✐✈✐♥❣ ✐♥ t❤❡ ❡①t❡r✐♦r ♣r♦❞✉❝t ♦❢ t❤❡ ✜❜❡r ❜✉♥❞•❡ ♦❢ M ✳ ■❢ M ✐s ❛♥ ❡♠❜❡❞❞❡❞ ♠❛♥✐❢♦•❞✱ ♦♥❡ ❝♦✉•❞ s✐♠♣•② ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ tr✐✈✐❛• ♣r♦♣❡rt② ♦♥ ❦❡r♥❡•s ✿ Pr♦♣❡rt② ✷✳✷✳✶✳ ▲❡t k ❜❡ ❛ r❡❛•✲✈❛•✉❡❞ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ ❛ s❡t A✳ ❚❤❡♥✱ ✐❢ B ✐s ❛ s✉❜s❡t ♦❢ A✱ t❤❡ ❢✉♥❝t✐♦♥ k : B × B → R ♦❜t❛✐♥❡❞ ❜② r❡str✐❝t✐♦♥ ✐s ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ B✳ ■♥ ♦✉r ❝❛s❡✱ ✐❢ M ✐s ❛ s✉❜♠❛♥✐❢♦•❞ ♦❢ s♦♠❡ ✈❡❝t♦r s♣❛❝❡ F ✱ ♦❜t❛✐♥✐♥❣ r❡❛•✲✈❛•✉❡❞ ❦❡r✲ ♥❡•s ♦♥ M ❜❡❝♦♠❡s str❛✐❣❤t❢♦r✇❛r❞ ❜② r❡str✐❝t✐♦♥ t♦ M ♦❢ ❦❡r♥❡•s ❞❡✜♥❡❞ ♦♥ F ✳ ❋♦r ✈❡❝t♦r ❦❡r♥❡•s✱ t❤✐s ♣r♦❝❡ss ❞♦❡s ♥♦t ❛♣♣•② ❜❡❝❛✉s❡ ♦❢ t❤❡ ♥♦♥✲❡q✉❛• t❛♥❣❡♥t s♣❛❝❡s ❛t ❞✐✛❡r❡♥t ♣♦✐♥ts✳ ❆ s❡❝♦♥❞ ✐♠♣♦rt❛♥t ❢❛❝t ✐s t❤❛t t❤❡ C 0 ✲✉♥✐✈❡rs❛•✐t② ♣r♦♣❡rt② tr❛♥s❢❡rs ❜② r❡str✐❝t✐♦♥✱ ✐✳❡ ✿ Pr♦♣❡rt② ✷✳✷✳✷✳ ■❢ k ✐s ❛ r❡❛•✲✈❛•✉❡❞ ♣♦s✐t✐✈❡ C 0 ✲✉♥✐✈❡rs❛• ❦❡r♥❡• ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ A ❛♥❞ B ❛ ❝•♦s❡❞ s✉❜s❡t ♦❢ A t❤❡♥ t❤❡ r❡str✐❝t❡❞ ❦❡r♥❡• k ✐s ❛•s♦ C 0 ✲✉♥✐✈❡rs❛•✳ ❚❤✐s ✐s ♣r♦✈❡❞ ✐♥ ❬✷✵(✱ ❝♦r♦••❛r② ✸✳ ■♥ ♣r❛❝t✐❝❛• t❡r♠s✱ ♣r♦♣ ✷✳✷✳✷ ❣✉❛r❛♥t❡❡s t❤❛t ✇❡ ❝❛♥ ♦❜t❛✐♥ C 0 ✲✉♥✐✈❡rs❛• ❦❡r♥❡• ♦♥ M ✐❢ ✇❡ r❡str✐❝t ❢♦r ✐♥st❛♥❝❡ ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ✐♥ F ♦♥ M ✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❛❜str❛❝t s✐❣♥❛• ♠❛♥✐❢♦•❞ M ❢♦r ✇❤✐❝❤ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛• ❡♠❜❡❞❞✐♥❣ ✐♥t♦ ❛ ✈❡❝t♦r s♣❛❝❡✱ t❤❡ r❡s✉•t✐♥❣ ❦❡r♥❡•s ✇♦✉•❞ ❞❡♣❡♥❞ ♦♥ t❤❡ ✇❛② t❤❛t M ✐s ❡♠❜❡❞❞❡❞ ✐♥t♦ ❛ ❞❡t❡r♠✐♥❡❞ s♣❛❝❡✳ ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ ❦❡r♥❡• ♠❡tr✐❝s ✿ ❛ ❢❡✇ ❡①❛♠♣•❡s ❋r♦♠ ❡q✳✭✷✳✶✻✮✱ ✇❡ s❡❡ t❤❛t t❤❡ ♠❡tr✐❝ ❜❡t✇❡❡♥ t✇♦ ❉✐r❛❝ ❢❝✉rr❡♥ts ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❣❡♦♠❡tr② ❝♦♠♣❛r✐s♦♥ t❤r♦✉❣❤ t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ t✇♦ ❉✐r❛❝s ❛♥❞ t❤❡ •♦❝❛• ✈♦•✉♠❡ ❡•❡♠❡♥ts✱ ❛♥❞ s✐❣♥❛• ♣r♦①✐♠✐t② ❡✈❛•✉❛t❡❞ ❜② t❤❡ ❦❡r♥❡• k f ✳ ▼♦r❡ ❣❡♥❡r❛••②✱ ♦♥❡ ❝❛♥ ✇r✐t❡ t❤❡ ❞✐st❛♥❝❡ ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t❤❛t r❡s✉•ts ❢r♦♠ t❤❡s❡ ♠❡tr✐❝s✳ ■❢ (X, f ) ❛♥❞ (Y, g) ❛r❡ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t♦ ✇❤✐❝❤ ✇❡ ❛ss♦❝✐❛t❡ t❤❡✐r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ ✺✼ r❡♣r❡s❡♥t❛t✐♦♥s C (X,f ) ❛♥❞ C (Y,g) ✱ t❤❡♥ ✇❡ ❤❛✈❡ ✿ C (X,f ) -C (Y,g) 2 W ′ = C (X,f ) 2 W ′ + C (Y,g) 2 W ′ -2 C (X,f ) , C (Y,g) W ′ = X×X k f (f (x 1 ), f (x 2 )). K g (x 1 , x 2 ) ξX (x 1 ), ξX (x 2 ) dH d (x 1 )dH d (x 2 ) + Y ×Y k f (g(y 1 ), g(y 2 )). K g (y 1 , y 2 ) ξY (y 1 ), ξY (y 2 ) dH d (y 1 )dH d (y 2 ) -2 X×Y k f (f (x), g(y)). K g (x, y) ξX (x), ξY (y) dH d (x)dH d (y) ✭✷✳✶✼✮ ✇❤❡r❡ ξX (x) ✐s t❤❡ ✉♥✐t d✲✈❡❝t♦r ♦❢ Λ d E ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡ t♦ X ❛t x✳ ❲❡ ❝❛♥ ❛•s♦ ✇r✐t❡ t❤❡ ❡①❛❝t ❡q✉✐✈❛•❡♥t ♦❢ ❡q✳✭✷✳✶✼✮ ❢♦r ❞✐s❝r❡t❡ s❤❛♣❡s✳ ◆❛♠❡•② ✐❢ C (X,f ) = n i=1 δ ξ X i (x i ,f i ) ❛♥❞ C (Y,g) = m k=1 δ ξ Y k (y k ,g k ) ✱ t❤❡♥ ✿ C (X,f ) -C (Y,g) 2 W ′ = n i=1 n j=1 k f (f i , f j ). K g (x i , x j )ξ X i , ξ X j + m k=1 m l=1 k f (g k , g l ). K g (y k , y l )ξ Y k , ξ Y l -2 n i=1 m k=1 k f (f i , g k ). K g (x i , y k )ξ X i , ξ Y k ✭✷✳✶✽✮ ❚❤✐s •❛st ❡q✉❛t✐♦♥ ❣✐✈❡s ❛ r❛t❤❡r s✐♠♣•❡ ❝•♦s❡❞ ❢♦r♠✉•❛ ❢♦r ❝♦♠♣✉t✐♥❣ ❞✐st❛♥❝❡s ❜❡✲ t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❜❛s❡❞ ♦♥•② ♦♥ ❦❡r♥❡• ❡✈❛•✉❛t✐♦♥s✳ ❲❡ ❛•s♦ s❡❡ t❤❛t ❢♦r s✉❢✲ ✜❝✐❡♥t•② r❡❣✉•❛r ❦❡r♥❡•s✱ ✇❡ ♦❜t❛✐♥ ❞✐✛❡r❡♥t✐❛❜•❡ ❞✐st❛♥❝❡s ✇✐t❤ r❡s♣❡❝t t♦ ♣♦s✐t✐♦♥ ❛♥❞ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s✱ ✇❤✐❝❤ s❤❛•• ❜❡ ♣r❡s❡♥t❡❞ ✐♥ ♠♦r❡ ❞❡t❛✐•s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥s✳ ◆♦✇✱ ✇❡ tr② t♦ ❣✐✈❡ ❛ •✐tt•❡ ♠♦r❡ ✐♥s✐❣❤t ♦♥ ❤♦✇ t❤❡s❡ ♠❡tr✐❝s ❜❡❤❛✈❡ ❜② ❝♦♥s✐❞❡r✐♥❣ ♠♦r❡ s♣❡❝✐✜❝❛••② t❤❡ ❝❛s❡ ♦❢ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s M = R✳ ▲❡t✬s ❛•s♦ t❛❦❡ t❤❡ ❦❡r♥❡•s k f ❛♥❞ K g t♦ ❜❡ •❛✉ss✐❛♥ ❦❡r♥❡•s✱ ✐✳❡ k f (f 1 , f 2 ) = e - |f 2 -f 1 | 2 σ 2 f ❛♥❞ K g (x 1 , x 2 )ξ 1 , ξ 2 = e - |x 2 -x 1 | 2 σ 2 g ξ 1 , ξ 2 ✳ ❚❤❡ ♣❛r❛♠❡t❡rs σ g ❛♥❞ σ f ❛r❡ t❤❡ s❝❛•❡s ♦❢ t❤❡ ❦❡r♥❡•s r❡s♣❡❝t✐✈❡•② ✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ❛♥❞ s✐❣♥❛• ❞♦♠❛✐♥✳ ❚❤❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❞✐r❛❝s ❞❡❝r❡❛s❡s ❛s t❤❡② •✐❡ ✬❢❛rt❤❡r✬ ❢r♦♠ ❡❛❝❤ ♦t❤❡r ✇✐t❤ r❡s♣❡❝t t♦ σ g ♦r ✐❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ s✐❣♥❛•s t❤❡② ❝❛rr② ❜❡❝♦♠❡s •❛r❣❡r ❝♦♠♣❛r❡❞ t♦ σ f ✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ σ f → +∞✱ ✇❡ ❤❛✈❡ k f (f 1 , f 2 ) = 1 ❢♦r ❛•• f 1 , f 2 ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ✇❡ ❣❡t ✐s ❡①❛❝t•② t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s X ❛♥❞ Y ❣✐✈❡♥ ❜② ❝✉rr❡♥ts✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ ✇❡ •❡t σ g → +∞✱ ❛s ✐❢ ❛•• ♣♦✐♥ts ✇❡r❡ s❡❡♥ ❜② t❤❡ ❦❡r♥❡• ❛t t❤❡ s❛♠❡ •♦❝❛t✐♦♥ ❣❡♦♠❡tr✐❝❛••②✱ ❡q✳✭✷✳✶✼✮ ❛♥❞ ❡q✳✭✷✳✶✽✮ ❜❡❝♦♠❡ s♦♠❡❤♦✇ ❛ ♠❡tr✐❝ ❜❡t✇❡❡♥ t❤❡ s✐❣♥❛• ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ t✇♦ ♦❜•❡❝ts✳ ❍♦✇❡✈❡r✱ ✇❡ s❡❡ t❤❛t ✐t st✐•• t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t s♦♠❡ ♦❢ t❤❡ ❣❡♦♠❡tr② t❤r♦✉❣❤ t❤❡ •♦❝❛• ✈♦•✉♠❡ ❡•❡♠❡♥ts ❛♥❞ •✉st ❛s ❢♦r ✉s✉❛• ❝✉r✲ r❡♥ts✱ t❤❡r❡ ❛r❡ s♦♠❡ ❝❛♥❝❡••❛t✐♦♥ ❡✛❡❝ts ♦❝❝✉rr✐♥❣ ❡ss❡♥t✐❛••② ❜❡❝❛✉s❡ ♦❢ ♦r✐❡♥t❛t✐♦♥✱ ✇❤✐❝❤ ❝❛✉s❡s t❤❡s❡ ♠❡tr✐❝s t♦ ❜❡ s❡♥s✐t✐✈❡ ♦♥•② t♦ ❜♦✉♥❞❛r✐❡s ♦❢ •❡✈❡• s❡ts✳ ❲❡ s❤❛•• ❜❡ ❞❡❛•✐♥❣ ✐♥ ♠♦r❡ ❞❡t❛✐•s ✇✐t❤ t❤❡s❡ q✉❡st✐♦♥s ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ s✉❜❝❛s❡ t❤❛t ❣✐✈❡s ❛ s✐♠♣•❡r ✈✐s✐♦♥ ♦❢ s✉❝❤ ♠❡tr✐❝s ✐s t❤❡ ✈❡r② s♣❡❝✐❛• ♦♥❡ ♦❢ ✷❉ ✐♠❛❣❡s✳ ■❢ Ω ∈ R 2 ✐s t❤❡ ✜①❡❞ ❞❡✜♥✐t✐♦♥ ❞♦♠❛✐♥ ♦❢ ✐♠❛❣❡s✱ ♦♥❡ ❝❛♥ r❡♣r❡s❡♥t ❛♥ ✐♠❛❣❡ I : Ω → R ❛s ❛ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t C (Ω,I) ∈ Ω 2 0 (R 2 , R) ′ ✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❞♦♠❛✐♥ ✐s ✢❛t✱ t❤❡r❡ ❛r❡ ♥♦ ❝✉rr❡♥ts✬ ❝❛♥❝❡••❛t✐♦♥ ❡✛❡❝ts ❛s ✇❡ ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✱ ❛♥❞ •♦❝❛• ❣❡♦♠❡tr② ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ s❛♠❡ 2✲✈❡❝t♦r ❛t ❡❛❝❤ ✺✽ ♣♦✐♥t✱ ✇❤✐❝❤ ✐s s✐♠♣•② ❛ s✉r❢❛❝❡ ❛r❡❛ ❛♥❞ t❤❡ ❦❡r♥❡• ♦♥ t❤❡ ❣❡♦♠❡tr✐❝❛• s♣❛❝❡ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ r❡❛•✲✈❛•✉❡❞ ❦❡r♥❡• k g ✳ ❚❤❡ ❞✐st❛♥❝❡ ♦❢ ❡q✳✭✷✳✶✼✮ t❤❡♥ ❜❡❝♦♠❡s ✿ C (Ω,f ) -C (Ω,g) 2 W ′ = Ω×Ω k g (x, y).[k f (I 1 (x), I 1 (y)) + k f (I 2 (x), I 2 (y)) -2k f (I 1 (x), I 2 (y))]dxdy ✭✷✳✶✾✮ ✕ ❋♦r t✇♦ •❛✉ss✐❛♥ ❦❡r♥❡•s k g (x, y) = e - |x-y| 2 σ 2 g ❛♥❞ k f (g, h) = e - |g-h| 2 σ 2 f ✱ ✇❡ s❡❡ t❤❛t t❤❡ ❞✐st❛♥❝❡ ✇❡ ❣❡t ❝♦♠❜✐♥❡s ❣❡♦♠❡tr✐❝ ♣r♦①✐♠✐t② ❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s ❛t s❝❛•❡ σ g ❛♥❞ ♣r♦①✐♠✐t② ♦❢ t❤❡ ✐♥t❡♥s✐t✐❡s ❛t s❝❛•❡ σ f ✱ ✇❤✐❝❤ ✐s ❛❝t✉❛••② ❝•♦s❡ t♦ t❤❡ ✐❞❡❛ ♦❢ ❜✐•❛t❡r❛• ✜•t❡r✐♥❣ ✐♥ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣ ❬✼✻(✳ ✕ ■❢ ♦♥❡ t❛❦❡s ❢♦r k f t❤❡ •✐♥❡❛r ❦❡r♥❡•✱ k f (g, h) = gh✱ ❛♥❞ •❡t σ g → 0✱ ❡q✳✭✷✳✶✾✮ ❜❡❝♦♠❡s ✿ C (Ω,f ) -C (Ω,g) 2 W ′ = Ω I 1 (x) 2 + I 2 (x) 2 -2I 1 (x)I 2 (x)dx = Ω (I 1 (x) -I 2 (x)) 2 dx ❛♥❞ t❤✐s ✐s t❤❡ ✉s✉❛• L 2 ✲❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✐♠❛❣❡s✳ ✕ ◆♦✇✱ ✐❢ ✇❡ •♦♦❦ ❛t ✐♠❛❣❡s ❛t ✈❡r② •❛r❣❡ ❣❡♦♠❡tr✐❝❛• s❝❛•❡ ❜② •❡tt✐♥❣ σ g → +∞✱ ❡q✳✭✷✳✶✾✮ ✐s ❛ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡♥s✐t✐❡s✬ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ t✇♦ ✐♠❛❣❡s ❀ ♠♦r❡ ♣r❡❝✐s❡•②✱ ✐t ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ♠❡tr✐❝ ❜❡t✇❡❡♥ t❤❡ ♠❡❛s✉r❡s µ 1 = λ•I -1 1 ❛♥❞ µ 2 = λ • I -1 2 ✱ ✇❤❡r❡ λ ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ R 2 ✳ ■♥❞❡❡❞✱ ❜② ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ✐♥ t❤❡ ✐♥t❡❣r❛•s ✿ C (Ω,f ) -C (Ω,g) 2 W ′ = Ω×Ω k f (I 1 (x), I 1 (y))dλ(x)dλ(y) + Ω×Ω k f (I 2 (x), I 2 (y))dλ(x)dλ(y) -2 Ω×Ω k f (I 1 (x), I 2 (y))dλ(x)dλ(y) = R×R k f (u, v)dµ 1 (u)dµ 1 (v) + R×R k f (u, v)dµ 2 (u)dµ 2 (v) -2 R×R k f (u, v)dµ 1 (u)dµ 2 (v) ■♥ t❤❡ ❞✐s❝r❡t❡ s❡tt✐♥❣✱ r❡♣r❡s❡♥t✐♥❣ ❡✈❡r② ♣✐①❡• ❛s ❛ ❉✐r❛❝ ❢❝✉rr❡♥t δ (x i ,I(x i )) ✱ t❤❡s❡ ♠❡❛s✉r❡s ❝❛♥ ❜❡ t❤♦✉❣❤t ❛s ✐♥t❡♥s✐t② ❤✐st♦❣r❛♠s ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♣✐①❡•s ❤❛✈✐♥❣ ❛ ♣❛rt✐❝✉•❛r ✐♥t❡♥s✐t②✳ ❚❤✉s t❤❡ ♠❡tr✐❝s ✇❡ ♦❜t❛✐♥ ❛t ✐♥✜♥✐t❡ ❣❡♦♠❡tr✐❝❛• s❝❛•❡s ❛r❡ ♠❡❛s✉r✐♥❣ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ✐♥t❡♥s✐t✐❡s ❤✐st♦❣r❛♠s ♦❢ t❤❡ ✐♠❛❣❡s✳ ❋♦r ✐♥st❛♥❝❡✱ ✐❢ k f ✐s •❛✉ss✐❛♥ ❛♥❞ ✇❡ •❡t σ f → 0✱ ♦♥❡ ❝❛♥ ✈❡r✐❢② t❤❛t t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥ ❣✐✈❡s ❡①❛❝t•② t❤❡ L 2 ✲❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✐♠❛❣❡s✬ ❤✐st♦❣r❛♠s✳ ■♥ s✉♠♠❛r②✱ t❤❡s❡ •❛st ❡①❛♠♣•❡s s❤♦✇ t❤❛t ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ r❡♣r❡s❡♥t❛t✐♦♥ ❝♦♠✲ ❜✐♥❡❞ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❦❡r♥❡•s ♣r♦✈✐❞❡ ❛ ✈❡r② ✇✐❞❡ r❛♥❣❡ ♦❢ ♣♦ss✐❜•❡ ♠❡tr✐❝s ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ❢r♦♠ ❞✐st❛♥❝❡s ❜❛s❡❞ ♦♥ ♣✉r❡ ❣❡♦♠❡tr② ♦♥ ♦♥❡ ❡①tr❡♠❡ t♦ ❞✐st❛♥❝❡s t❤❛t s✐♠♣•② ❝♦♠♣❛r❡ ❤✐st♦❣r❛♠s ♦❢ s✐❣♥❛•s ♦♥ t❤❡ ♦t❤❡r✳ ■t ❛•s♦ ❡♥❛❜•❡s ❛♥❛•②s✐s ❛t ✈❛r✐♦✉s s❝❛•❡s ❜♦t❤ ✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ❞♦♠❛✐♥ ❛s ✇❡•• ❛s ♠✉•t✐✲s❝❛•❡ ♠❡tr✐❝s ✿ ♦♥❡ s✐♠♣•❡ ✇❛② ❜❡✐♥❣ t♦ ♠❛❦❡ s✉♠s ♦❢ •❛✉ss✐❛♥ ❦❡r♥❡•s ✇✐t❤ ❞✐✛❡r❡♥t σ f ❛♥❞ σ g ✳ ✺✾ ✷✳✷✳✸ Pr♦♣❡rt✐❡s ♦❢ ❦❡r♥❡• ♥♦r♠s ♦♥ ❢❝✉rr❡♥ts ❲❡ ♥♦✇ t✉r♥ t♦ ❣❡♥❡r❛• ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐st❛♥❝❡s ✇❡ ❤❛✈❡ ❡①❤✐❜✐t❡❞ ♣r❡✈✐♦✉s•② ❛♥❞ s❤♦✇ t❤❡ t❤❡♦r❡t✐❝❛• ❜❡♥❡✜ts ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r✐❣✐♥❛• ♣r♦❜•❡♠ ✇❡ s❡t ♦✉t t♦ ❤❛♥❞•❡ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t✇♦ ❦❡r♥❡•s K g ❛♥❞ k f ❛r❡ ❣✐✈❡♥ r❡s♣❡❝t✐✈❡•② ♦♥ s♣❛❝❡ E ❛♥❞ ♠❛♥✐❢♦•❞ M ✱ ♣r♦✈✐❞✐♥❣ t✇♦ |❑❍❙ W g ❛♥❞ W f ✳ ❲❡ ✇✐•• ❛•s♦ ❛ss✉♠❡✱ ✐♥ ❛•• t❤✐s s❡❝t✐♦♥✱ t❤❛t ❜♦t❤ ❦❡r♥❡•s s❛t✐s❢② t❤❡ r❡q✉✐r❡♠❡♥ts ♦❢ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✷✳ ❈♦♥tr♦• ❜② L 1 ✲♥♦r♠ ♦♥ ❛ ✜①❡❞ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ❲❡ st❛rt ❜② t❤❡ ❢♦••♦✇✐♥❣ ✈❡r② s✐♠♣•❡ •❡♠♠❛ ✿ ▲❡♠♠❛ ✷✳✷✳✷✳ ❋♦r ❛•• x 1 , x 2 ∈ E✱ ξ 1 , ξ 2 ∈ Λ d E ❛♥❞ m 1 , m 2 ∈✱ t❤❡ ❢♦••♦✇✐♥❣ ✐♥✲ ❡q✉❛•✐t② ❤♦•❞s ✿ δ ξ 2 (x 2 ,m 2 ) -δ ξ 1 (x 1 ,m 1 ) W ′ ≤ δ m 1 W ′ f δ ξ 2 x 2 -δ ξ 1 x 1 W ′ g + δ ξ 2 x 2 W ′ g δ m 2 -δ m 1 W ′ f . ✭✷✳✷✵✮ Pr♦♦❢✳ ❚❤✐s ✐s •✉st ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② s✐♥❝❡ ✿ δ ξ 2 (x 2 ,m 2 ) -δ ξ 1 (x 1 ,m 1 ) W ′ ≤ δ ξ 2 (x 2 ,m 2 ) -δ ξ 2 (x 2 ,m 1 ) W ′ + δ ξ 2 (x 2 ,m 1 ) -δ ξ 1 (x 1 ,m 1 ) W ′ ❛♥❞ ❜② t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❦❡r♥❡• ❡q✳✭✷✳✶✹✮✱ δ ξ 2 (x 2 ,m 2 ) -δ ξ 2 (x 2 ,m 1 ) 2 W ′ = K g (x 2 , x 2 )ξ 2 , ξ 2 . (k f (m 2 , m 2 ) -2k f (m 1 , m 2 ) + k f (m 1 , m 1 )) = δ ξ 2 x 2 2 W ′ g δ m 2 -δ m 1 2 W ′ f ❛♥❞ s✐♠✐•❛r•② ✇❡ ❤❛✈❡ δ ξ 2 (x 2 ,m 1 ) -δ ξ 1 (x 1 ,m 1 ) W ′ = δ m 1 W ′ f δ ξ 2 x 2 -δ ξ 1 x 1 W ′ g ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ |❑❍❙ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣✉♥❝t✉❛• ❢❝✉rr❡♥ts ✐s ❞♦♠✐♥❛t❡❞ ❜♦t❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡✐r ❣❡♦♠❡tr✐❝❛• ♣❛rts ❛♥❞ ♦❢ t❤❡✐r ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s✳ ❚❤✐s ✐s t❤❡ ❣❡♥❡r❛• ✐❞❡❛ ✇❡ ✇✐•• ❢♦r♠✉•❛t❡ ✐♥ ❛ ♠♦r❡ ♣r❡❝✐s❡ ✇❛② ✇✐t❤ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s✳ ❲❡ ❞❡♥♦t❡ ❜② d M t❤❡ ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡ ✐♥❞✉❝❡❞ ♦♥ M ❜② ✐ts |✐❡✲ ♠❛♥♥✐❛♥ str✉❝t✉r❡✳ ❚❤❡ ♥❡①t ♣r♦♣♦s✐t✐♦♥ ❡①❛♠✐♥❡s t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✐s ❛ ✜①❡❞ s✉♣♣♦rt X ❛♥❞ s❤♦✇s t❤❛t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ W ′ ✲♥♦r♠ ✐s t❤❡♥ ❞♦♠✐♥❛t❡❞ ❜② t❤❡ L 1 ♥♦r♠ ♦♥ X✳ ❚❤❡♦r❡♠ ✷✳✷✳✶✳ ▲❡t X ❜❡ ❛ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ f 1 : X → M ❛♥❞ f 2 : X → M t✇♦ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X t❛❦✐♥❣ ✈❛•✉❡ ✐♥ M ✳ ❲❡ ❛ss✉♠❡ t❤❛t W f ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (M, R)✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t β s✉❝❤ t❤❛t ✿ C (X,f 1 ) -C (X,f 2 ) W ′ ≤ β X d M (f 1 (x), f 2 (x))dH d (x) Pr♦♦❢✳ ❲❡ r❡❝❛•• t❤❛t ❢♦r ❛ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t t❤❡r❡ ❡①✐sts ❢♦r H d ✲❛•♠♦st ❡✈❡r② ♣♦✐♥t x ∈ X ❛ ✉♥✐t d✲✈❡❝t♦r ξ(x) ❛♥❞ ❜② ❞❡✜♥✐t✐♦♥ ✿ C (X,f ) (ω) = X ω (x,f (x)) ( ξ(x))dH d (x) = X δ ξ(x) (x,f (x)) (ω)dH d (x) ◆♦✇ ✇❡ ❤❛✈❡ ❜② tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ♦♥ . W ′ ✿ C (X,f 1 ) -C (X,f 2 ) W ′ ≤ X δ ξ(x) (x,f 1 (x)) -δ ξ(x) (x,f 2 (x)) W ′ dH d (x) . ✭✷✳✷✶✮ ✻✵ ❋r♦♠ •❡♠♠❛ ✷✳✷✳✷ ❛♥❞ ❡q✳✭✷✳✷✵✮✱ δ ξ(x) (x,f 1 (x)) -δ ξ(x) (x,f 2 (x)) W ′ ≤ δ ξ(x) x W ′ g δ f 2 (x) - δ f 1 (x) W ′ f ✳ ◆♦✇✱ ❢♦r ❛♥② m 1 ✱ m 2 ∈ M ❛♥❞ h ∈ W f ✇❡ ❤❛✈❡ ✿ |(δ m 1 -δ m 2 )(h)| = |h(m 1 ) -h(m 2 )| ≤ Dh ∞ d M (m 1 , m 2 ) ≤ ❈t❡ h W f d M (m 1 , m 2 ) t❤❡ •❛st ✐♥❡q✉❛•✐t② r❡s✉•t✐♥❣ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉♦✉s ❡♠❜❡❞❞✐♥❣ W f ֒→ C 1 0 (M, R)✳ ❚❤❡r❡✲ ❢♦r❡ ✇❡ ❣❡t δ f 2 (x) -δ f 1 (x) W ′ f ≤ ❈t❡ d M (f 1 (x), f 2 (x)) . ▼♦r❡♦✈❡r✱ s✐♥❝❡ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❦❡r♥❡• K g ✐s ❜♦✉♥❞❡❞✱ ✇❡ ❛•s♦ ❤❛✈❡ δ ξ(x) x W ′ g ≤ ❈t❡✳ ❇❛❝❦ t♦ ❡q✉❛t✐♦♥ ❡q✳✭✷✳✷✶✮✱ ✇❡ ❣❡t ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❞❡r✐✈❛t✐♦♥s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦♥st❛♥t β > 0 s✉❝❤ t❤❛t ✿ C (X,f 1 ) -C (X,f 2 ) W ′ ≤ β X d M (f 1 (x), f 2 (x))dH d (x) ✇❤✐❝❤ ♣r♦✈❡s t❤❡ st❛t❡❞ r❡s✉•t✳ ❆ str❛✐❣❤t❢♦r✇❛r❞ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡♦r❡♠ ✷✳✷✳✶ ❛♥❞ t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡✲ ♦r❡♠ ✐s t❤❛t ✐❢ f n ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♦♥ X t❤❛t ❝♦♥✈❡r❣❡s ♣♦✐♥t✇✐s❡•② t♦ ❛ ❢✉♥❝t✐♦♥ f ✱ t❤❡♥ C (X,fn) → C (X,f ) ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s✐❣♥❛• ✐♠♣•✐❡s ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ W ′ ✲♥♦r♠✳ ❲❡ s❤❛•• s❡❡✱ ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts✱ t❤❛t t❤❡ r❡✈❡rs❡ ❞♦♠✐♥❛t✐♦♥ ❞♦❡s ♥♦t ❤♦•❞ ✐♥ ❣❡♥❡r❛•✳ ❈♦♥tr♦• ❜② ❞❡❢♦r♠❛t✐♦♥s✬ ♠❡tr✐❝s ❚❤❡ ♣✉r♣♦s❡ ✐s ♥♦✇ t♦ •✐♥❦ t❤❡ W ′ ✲♠❡tr✐❝s ✇✐t❤ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✬ ❞❡❢♦r♠❛t✐♦♥s✳ ❲❡ s❤♦✇ t❤❛t ✇❡ ❝❛♥ ❜❛s✐❝❛••② ❝♦♥tr♦• t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❢❝✉rr❡♥ts ❜❡t✇❡❡♥ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❛♥❞ ✐ts ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ❜② t❤❡ ❛♠♦✉♥t ♦❢ ❞❡❢♦r♠❛t✐♦♥✳ ▲❡t✬s ✜rst ❢♦❝✉s t❤❡ s✐♠♣•❡st tr❛♥s♣♦rt ♠♦❞❡• ❣✐✈❡♥ ❜② ✷✳✶ ❛♥❞ ❛ss✉♠❡ t❤❛t ❞✐✛❡♦♠♦r♣❤✐s♠s φ ❛r❡ C 1 ❛♥❞ s✉❝❤ t❤❛t |φ(x) -x| → 0 ❛♥❞ |d x φ -■❞| → 0| ❛s x → ∞✳ ❚❤❡ ❝❡♥tr❛• r❡s✉•t ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t❤❡ ❢♦••♦✇✐♥❣ ✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳ ▲❡t X ❜❡ ❛ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ f : X → M ❛ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥✳ ❆ss✉♠❡ t❤❛t W g ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (E, Λ d E * )✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t γ > 0 ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ t❤❡ ❦❡r♥❡•s K g ❛♥❞ k f s✉❝❤ t❤❛t✱ ❢♦r ❛•• ❞❡❢♦r♠❛t✐♦♥ φ ✿ C (X,f ) -C (φ(X),f •φ -1 ) W ′ ≤ γH d (X) φ -■❞ 1,∞ Pr♦♦❢✳ ❋♦r t❤❡ ❡♥t✐r❡ ♣r♦♦❢✱ ✇❡ s❤❛•• ✉s❡ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥ ❈t❡ ❢♦r ✇r✐t✐♥❣ t❤❡ s✉❝✲ ❝❡ss✐✈❡ ❞✐✛❡r❡♥t ✬✉♥✐✈❡rs❛•✬ ❝♦♥st❛♥ts ✭✐✳❡ ♥♦t ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ s❤❛♣❡ (X, f ) ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥ φ✮✳ ❙t❛rt✐♥❣ ❛s ✐♥ t❤❡♦r❡♠ ✷✳✷✳✶✱ ✇❡ ❤❛✈❡ ✿ C (X,f ) -C (φ(X),f •φ -1 ) W ′ ≤ X δ ξ(x) (x,f (x)) -δ dxφ. ξ(x) (φ(x),f (x)) W ′ dH d (x) ✭✷✳✷✷✮ ✇❤❡r❡ ❢♦r t❤❡ ✈♦•✉♠❡ ❡•❡♠❡♥t ξ(x) = ξ1 (x) ∧ ... ∧ ξd (x)✱ d x φ. ξ(x) ✐s t❤❡ tr❛♥s♣♦rt❡❞ ✈♦•✉♠❡ ❡•❡♠❡♥t ❜② φ✱ ❡q✉❛• t♦ d x φ. ξ(x) = dφ x ( ξ1 (x))∧...∧dφ x ( ξd (x))✳ ❋r♦♠ ❡q✳✭✷✳✷✵✮ ✇❡ ❣❡t ✿ δ ξ(x) (x,f (x)) -δ dxφ. ξ(x) (φ(x),f (x)) W ′ ≤ δ f (x) W ′ f δ dxφ. ξ(x) φ(x) -δ ξ(x) x W ′ g ✻✶ ❛♥❞ s✐♥❝❡ t❤❡ ❦❡r♥❡• k f ✐s ❜♦✉♥❞❡❞ ✇❡ ❤❛✈❡ δ f (x) W ′ f ≤ ❈t❡✳ ▼♦r❡♦✈❡r ✿ δ dxφ. ξ(x) φ(x) -δ ξ(x) x W ′ g ≤ δ dxφ. ξ(x) φ(x) -δ ξ(x) φ(x) W ′ g + δ ξ(x) φ(x) -δ ξ(x) x W ′ g . ✭✷✳✷✸✮ ❋♦❝✉s✐♥❣ ✜rst ♦♥ t❤❡ s❡❝♦♥❞ t❡r♠✱ ✇❡ ❝❛♥ ✇r✐t❡ ❢♦r ❛♥② ω ∈ W g ✱ |(δ ξ(x) φ(x) -δ ξ(x) x )(ω)| = |ω φ(x) ( ξ(x)) -ω x ( ξ(x))| ≤ |dω| ∞ .|φ(x) -x|.| ξ(x)|. ❲❡ ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ | ξ(x)| = 1✳ W g ❜❡✐♥❣ ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥ C 1 0 (E, Λ d E * )✱ |dω| ∞ ≤ ❈t❡ ω Wg ✳ ❚❤✉s✱ ✇❡ ❞❡❞✉❝❡ t❤❛t δ ξ(x) φ(x) -δ ξ(x) x W ′ g ≤ ❈t❡ φ -■❞ ∞ ✳ ❆s ❢♦r t❤❡ ✜rst t❡r♠ ✐♥ ❡q✳✭✷✳✷✷✮✱ δ dxφ. ξ(x) φ(x) -δ ξ(x) φ(x) W ′ g = δ dxφ. ξ(x)-ξ(x) φ(x) W ′ g ≤ ❈t❡|d x φ. ξ(x) -ξ(x)| s✐♥❝❡ K g ✐s ❜♦✉♥❞❡❞ ❛♥❞✱ ❜② ❞❡✜♥✐t✐♦♥✱ |d x φ. ξ(x)-ξ(x)| ≤ |dφ-■❞| ∞ .| ξ| = |dφ-■❞| ∞ ✳ ■t r❡s✉•ts t❤❛t ✇❡ ❣❡t ❡✈❡♥t✉❛••② ✿ C (X,f ) -C (φ(X),ψ•f •φ -1 ) W ′ ≤ ❈t❡. φ -■❞ 1,∞ X dH d (x) = ❈t❡.H d (X) φ -■❞ 1,∞ ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢✳ ❚❤✐s ✐s s♦♠❡❤♦✇ t❤❡ ❛♥❛•♦❣♦✉s t♦ t❤❡♦r❡♠ ✷✳✷✳✶ ❢♦r ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✿ ✐t s❤♦✇s t❤❛t t❤❡ |❑❍❙ ♥♦r♠ ✐s ▲✐♣s❝❤✐t③ r❡❣✉•❛r ✇✐t❤ r❡s♣❡❝t t♦ ❞❡❢♦r✲ ♠❛t✐♦♥s✳ ❚❤✐s ♣r♦♣❡rt② ✐s ❛ ♥❛t✉r❛• ♦♥❡ t♦ ❡①♣❡❝t ❢r♦♠ ❛ ❞✐st❛♥❝❡ ✐♥ ♦✉r ❝♦♥t❡①t✱ ❜✉t ♥♦t❡ t❤❛t ♠❛♥② ✉s✉❛• ♠❡tr✐❝s ❞♦ ♥♦t s❛t✐s❢② ✐t✳ ❚❤✐s ✐s ♥♦t❛❜•② ✇❤❛t ❤❛♣♣❡♥s ❢♦r L p ❞✐st❛♥❝❡s✳ ■♥❞❡❡❞✱ ✐❢ ✇❡ s♣❡❝✐❢② t❤❡ ♣r❡✈✐♦✉s t♦ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡ ✇❤❡r❡ φ ✐s ❛ s♠❛•• ❞❡❢♦r♠❛t✐♦♥ t❤❛t •❡❛✈❡s X ❣•♦❜❛••② ✐♥✈❛r✐❛♥t ✭φ(X) = X✮ ❛♥❞ ✇❡ ✇✐s❤ t♦ ❝♦♠♣❛r❡ t❤❡ ✐♥✐t✐❛• ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ✇✐t❤ t❤❡ ❞❡❢♦r♠❡❞ ♦♥❡ (φ(X), f • φ -1 ) = (X, f • φ -1 )✱ t❤❡♥ ❜② t❤❡♦r❡♠ ✷✳✷✳✷✱ ✇❡ ❦♥♦✇ t❤❛t✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ✱ t❤❡ ❢❝✉rr❡♥t✬s ❞✐st❛♥❝❡ r❡✲ ♠❛✐♥s s♠❛•• ✐❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ φ ✐s s♠❛••✳ ■t ✐s ♥♦ •♦♥❣❡r tr✉❡ ✐❢ ✇❡ ❝♦♠♣✉t❡ ✐♥st❡❛❞ X |f -f • φ -1 | p ✱ t❤❡ L p ❞✐st❛♥❝❡ ♦♥ X ✭0 < p ≤ ∞✮✳ ❚❤✐s ✐s ❡❛s✐•② s❡❡♥ ✐❢ ✇❡ ❝❤♦♦s❡ ❢♦r X t❤❡ ✉♥✐t ❝✐r❝•❡ S 1 ❛♥❞ ❝♦♥s✐❞❡r ❝r❡♥❡•❛t❡❞ s✐❣♥❛•s ❛s ✐♥ ✜❣✉r❡ ✷✳✽✳ ■♥tr♦❞✉❝✐♥❣ t❤❡ ♦♣❡r❛t♦r τ dθ t❤❛t ❛❝ts ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❜② r♦t❛t✐♦♥ ♦❢ ❛♥ ❛♥❣•❡ dθ✱ ✇❡ s❡❡ ✐♥❞❡❡❞ t❤❛t ✿ sup f ∈L p (S 1 ), f L p ≤1 S 1 |f -f • τ -1 dθ | p = 1 ✇❤❡r❡❛s✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡♦r❡♠ ✷✳✷✳✷ sup f ∈L p (S 1 ), f L p ≤1 C (X,f ) -C (X,f •τ -1 dθ ) W ′ = O(dθ) . ❚❤✐s ❡✈❡♥t✉❛••② s❤♦✇s t❤❛t W ′ ♥♦r♠ ❛♥❞ L p ♥♦r♠ ♦♥ ❛ ✜①❡❞ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ❛r❡ ♥♦t ❡q✉✐✈❛•❡♥t ✐♥ ❣❡♥❡r❛•✳ ❚❤✐s s✐♠♣•❡ ❡①❛♠♣•❡ ❛•s♦ r❡✈❡❛•s t❤❡ •✐♠✐ts ♦❢ s♦♠❡ ♣r❡✈✐♦✉s ❛♣♣r♦❛❝❤❡s ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❝♦♠♣❛r✐s♦♥ t❤❛t r❡•② ♦♥ ❡st✐♠❛t✐♥❣ ❝♦rr❡✲ s♣♦♥❞❡♥❝❡ ♠❛♣s ❜❡t✇❡❡♥ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rts ❜❡❢♦r❡ ❝♦♠♣❛r✐♥❣ s✐❣♥❛•s✳ |❑❍❙ ♥♦r♠s ♦♥ ❢❝✉rr❡♥ts s✉❝❝❡ss❢✉••② ❛✈♦✐❞ s✉❝❤ ✐ss✉❡s ❜❡❝❛✉s❡ s❤❛♣❡ ❛♥❞ ❢✉♥❝t✐♦♥ ❛r❡ ❝♦♠♣❛r❡❞ t❤r♦✉❣❤ ♦♥❡ s✐♥❣•❡ ♥♦r♠ t❤❛t ❤❛s ❡♥♦✉❣❤ s♣❛t✐❛• r❡❣✉•❛r✐t② t♦ ❝♦♠♣❛r❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ❛t ♣♦✐♥ts ✐♥ ❛ ❝❡rt❛✐♥ ♥❡✐❣❤❜♦r❤♦♦❞✳ ✻✷ ❋✐❣✉r❡ ✷✳✽✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❢❝✉rr❡♥t✬s ♥♦r♠ ❛♥❞ t❤❡ L p ♥♦r♠ ♦♥ ❛ ✜①❡❞ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✿ ❡①❛♠♣•❡ ♦❢ ❝r❡♥❡•❛t✐♦♥s ♦♥ t❤❡ ✉♥✐t ❝✐r❝•❡✳ ❲❡ ❝❛♥ ❛•s♦ ❞r❛✇ ❛ ❢❡✇ ♠♦r❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡♦r❡♠ ✷✳✷✳✷✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❛s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ✐♥ ❝❤❛♣t❡r ✶✱ ✇❡ ❝❛♥ ♠♦❞❡• ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ E ❛s ✢♦✇s ♦❢ t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞s✳ ◆❛♠❡•②✱ ✇❡ ❝♦♥s✐❞❡r ✈❡❝t♦r ✜❡•❞s v(t, x) s✉❝❤ t❤❛t ❢♦r ❛•• t ∈ [0, 1]✱ v(t, .) ✐s ❛ C 1 ✈❡❝t♦r ✜❡•❞ ♦❢ E s✉❝❤ t❤❛t u ❛♥❞ ✐ts ✜rst ♦r❞❡r ❞❡r✐✈❛t✐✈❡s ❛r❡ ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ❲❡ ❞❡✜♥❡ |v| 1,∞ = |v| ∞ +|dv| ∞ ✳ ❚❤❡♥✱ χ 1 ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛•• s✉❝❤ t✐♠❡✲ ✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞s s✉❝❤ t❤❛t 1 0 |v(t, .)| 1,∞ dt < +∞ ❛♥❞ ✐t ✐s ♥❛t✉r❛••② ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ u χ 1 = 1 0 |u(t, .)| 1,∞ dt✳ ❆s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ✐♥ ❝❤❛♣t❡r ✶ ❛♥❞ ❬✽✻(✱ ❛♥② u ∈ χ 1 ❣✐✈❡s ❛ ✢♦✇ ❛♣♣•✐❝❛t✐♦♥ φ v t : x → φ v t (x) s♦•✈✐♥❣ t❤❡ ❖❉❊ dy ds = v(s, y) ✇✐t❤ ✐♥✐t✐❛• ❝♦♥❞✐t✐♦♥ y(0) = x ∈ E✱ ❛♥❞ ❢♦r ❛•• t ∈ [0, 1], φ v t ✐s ❛ C 1 ✲❞✐✛❡♦♠♦r♣❤✐s♠ ♦❢ E s✉❝❤ t❤❛t |φ v t (x) -x| → 0 ❛♥❞ |d x φ v t -■❞| → 0| ❛s x → ∞✳ ❲❡ ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ ✭✇❡ r❡❢❡r t♦ ❬✹✶( ♦r ❬✽✻( ❢♦r t❤❡ ❞❡t❛✐•s✮ ✿ ▲❡♠♠❛ ✷✳✷✳✸✳ ❋♦r ❛•• R > 0 t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t C(R) > 0 s✉❝❤ t❤❛t✱ ❢♦r ❛•• u ∈ χ 1 ✇✐t❤ u χ 1 R ❛♥❞ t ∈ [0, 1] ✿ φ v t -Id ∞ C(R). v χ 1 , dφ v t -Id ∞ C(R). v χ 1 . ❋r♦♠ t❤❡♦r❡♠ ✷✳✷✳✷ ❛♥❞ •❡♠♠❛ ✷✳✷✳✸✱ ✇❡ ♦❜t❛✐♥ str❛✐❣❤t❢♦r✇❛r❞•② t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥✲ tr♦• ✿ ❈♦r♦••❛r② ✷✳✷✳✶✳ ▲❡t X ❜❡ ❛ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ f : X → M ❛ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥✳ ❆ss✉♠❡ t❤❛t W g ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (E, Λ d E * )✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t γ > 0 ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ t❤❡ ❦❡r♥❡•s K g ❛♥❞ k f s✉❝❤ t❤❛t✱ ❢♦r ❛•• v ∈ χ 1 ✇✐t❤ u χ 1 s♠❛•• ❡♥♦✉❣❤ ✿ C (X,f ) -C (φ v 1 (X),f •(φ v 1 ) -1 ) W ′ ≤ γH d (X) v χ 1 ■♥ ❛❞❞✐t✐♦♥✱ ❛ss✉♠✐♥❣ s✉✣❝✐❡♥t s♣❛t✐❛• r❡❣✉•❛r✐t② ♦♥ t❤❡ ✈❡❝t♦r ✜❡•❞ v✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❡ ✐♠♣♦rt❛♥t ♣r♦♣❡rt② ♦❢ ✇❡❛❦ ❝♦♥t✐♥✉✐t② ♦❢ W ′ ♥♦r♠s ✇✐t❤ r❡s♣❡❝t t♦ v✳ ■t r❡•✐❡s ♦♥ t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛✱ t❤❛t ✐s ♣r♦✈❡❞ ✐♥ ❬✹✶( ❝❤❛♣✳ ✶ ♦r ✐♥ ❬✽✻( ✿ ▲❡♠♠❛ ✷✳✷✳✹✳ ▲❡t V ❜❡ ❛ ❍✐•❜❡rt s♣❛❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s ♦♥ E s✉❝❤ t❤❛t V ✐s ❝♦♥✲ t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 2 0 (E, E)✳ ❚❤❡♥✱ ✐❢ v n ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s ✐♥ L 2 ([0, 1], V ) = L 2 V s✉❝❤ t❤❛t v n → v ✇❡❛❦•② ✐♥ L 2 V ✱ φ vn t ❛♥❞ dφ vn t ❝♦♥✈❡r❣❡ t♦✇❛r❞s φ v t ❛♥❞ dφ v t ✉♥✐❢♦r♠•② ♦♥ ❡✈❡r② ❝♦♠♣❛❝t ♦❢ E✳ ✻✸ ❲❡ ♦❜t❛✐♥ t❤❡r❡❢♦r❡ ✿ ❈♦r♦••❛r② ✷✳✷✳✷✳ ■❢ V ✐s ❛ ❍✐•❜❡rt s♣❛❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 2 0 (E, E) t❤❡♥✱ ❢♦r ❛♥② ✜①❡❞ ❛♥❞ ❜♦✉♥❞❡❞ r❡❝t✐✜❛❜•❡ s❡t X ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ ♠❡❛✲ s✉r❛❜•❡ s✐❣♥❛• f ♦♥ X✱ t❤❡ ❛♣♣•✐❝❛t✐♦♥ v → C (φ v 1 (X),f •(φ v 1 ) -1 ) ✐s ✇❡❛❦•② ❝♦♥t✐♥✉♦✉s ❢r♦♠ L 2 V t♦ W ′ ✳ Pr♦♦❢✳ ▲❡t v n ✐s ❛ s❡q✉❡♥❝❡ t❤❛t ✇❡❛❦•② ❝♦♥✈❡r❣❡s t♦✇❛r❞ v ✐♥ L 2 V ✳ ❲✐t❤ •❡♠♠❛ ✷✳✷✳✹✱ ✇❡ ❦♥♦✇ t❤❛t φ vn 1 ❛♥❞ dφ vn 1 ❝♦♥✈❡r❣❡ r❡s♣❡❝t✐✈❡•② t♦ φ v 1 ❛♥❞ dφ v 1 ✉♥✐❢♦r♠•② ♦♥ ❡✈❡r② ❝♦♠♣❛❝t ♦❢ E✱ ✐✳❡ φ vn 1 -φ v 1 1,∞ → 0 ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ X✳ ❚❤❡♥✱ t❤❛♥❦s t♦ t❤❡♦r❡♠ ✷✳✷✳✷✱ ✇❡ ❛r❡ ❛••♦✇❡❞ t♦ ❝♦♥❝•✉❞❡ t❤❛t C (φ vn 1 (X),f •(φ vn 1 ) -1 ) W ′ --→ C (φ v 1 (X),f •(φ v 1 ) -1 ) ✱ ✇❤✐❝❤ ♣r♦✈❡s t❤❡ ✇❡❛❦ ❝♦♥t✐♥✉✐t②✳ ❚♦ ❝♦♥❝•✉❞❡ ♦♥ t❤✐s t♦♣✐❝✱ ✇❡ ❣✐✈❡ ❛ •✐tt•❡ ♠♦r❡ ❣❡♥❡r❛• r❡s✉•t t❤❛t ❡♥❝♦♠♣❛ss❡s t❤❡♦r❡♠s ✷✳✷✳✶ ❛♥❞ ❝♦r♦••❛r② ✷✳✷✳✶ ❜② ❝♦♥tr♦••✐♥❣ ❢❝✉rr❡♥ts✬ ♥♦r♠s ❜♦t❤ ✇✐t❤ r❡s♣❡❝t t♦ ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ s✉♣♣♦rt ❛♥❞ t❤❡ s✐❣♥❛• ✈❛•✉❡s✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❣❡♥❡r❛• ❛❝t✐♦♥ ♦❢ ❛♣♣•✐❝❛t✐♦♥s ψ(x, m) = (ψ g (x), ψ f (x, m)) ♣r❡s❡♥t❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ✷✳✶✳✷ ❛♥❞ ✐♥ ❡q✳✭✷✳✺✮✳ ❆s ✇❡ ❞✐❞ ♣r❡✈✐♦✉s•②✱ ✇❡ ❝❛♥ ❣❡♥❡r❛t❡ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s ❜② ✢♦✇✐♥❣ ✈❡❝t♦r ✜❡•❞s ♦❢ t❤❡ ❢♦r♠ (v g (x), v f (x, m))✳ ❲❡ ❛ss✉♠❡ t❤❛t v g ∈ χ 1 ❛♥❞ t❤❛t v f ✐s ❛ t✐♠❡✲✈❛r②✐♥❣ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ❢♦r ❛•• t ∈ [0, 1]✱ v f t ✐s ❛ •♦❝❛••② ▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ❢r♦♠ E × M t♦ T M ❛♥❞ t❤❛t 1 0 v f t ∞ dt < ∞✳ ❲❡ s❤❛•• ❞❡♥♦t❡ ❜② v χ 0 . = 1 0 v t ∞ ❚❤❡♥✱ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✿ ❚❤❡♦r❡♠ ✷✳✷✳✸✳ ▲❡t X ❜❡ ❛ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ♦❢ ✜♥✐t❡ ✈♦•✉♠❡ ❛♥❞ f : X → M ❛ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥✳ ❆ss✉♠❡ t❤❛t W g ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (E, Λ d E * ) ❛♥❞ t❤❛t W f ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (M, R)✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t γ > 0 ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ t❤❡ ❦❡r♥❡•s K g ❛♥❞ k f s✉❝❤ t❤❛t✱ ❢♦r ❛•• v = (v g , v f )✱ ✇✐t❤ v g ❛ χ 1 ✈❡❝t♦r ✜❡•❞ ♦❢ E ❛♥❞ v f ❛ •♦❝❛••② ▲✐♣s❝❤✐t③ ❜♦✉♥❞❡❞ ✈❡❝t♦r ✜❡•❞ ❢r♦♠ E × M t♦ T M ✱ ✇❡ ❤❛✈❡ ✿ C (X,f ) -C ψ.(X,f ) W ′ ≤ γ❱♦•(X) v g χ 1 + v f χ 0 Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ❡ss❡♥t✐❛••② ❝♦♠❜✐♥❡s t❤❡ t✇♦ ♣r❡✈✐♦✉s t❤❡♦r❡♠s✳ ❲❡ r❡♠✐♥❞ t❤❛t ψ.(X, f ) = (ψ g (X), ψ f ((ψ g ) -1 ), f • (ψ g ) -1 )✳ ❚♦ •✐❣❤t❡♥ t❤❡ ❡①♣r❡ss✐♦♥s✱ ✇❡ s❤❛•• ❞❡♥♦t❡ ❜② ψ f .f t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ X ❜② (ψ f .f )(x) . = ψ f (x, f (x)) ❲❡ ❤❛✈❡✱ ❜② tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ✿ C (X,f ) -C ψ.(X,f ) W ′ ≤ C (X,f ) -C (X,ψ f .f ) W ′ + C (X,ψ f .f ) -C ψ.(X,f ) W ′ ≤ C (X,f ) -C (X,ψ f .f ) W ′ + C (X,ψ f .f ) -C (ψ g (X),ψ f .f •(ψ g ) -1 ) W ′ ✭✷✳✷✹✮ ❯s✐♥❣ t❤❡♦r❡♠ ✷✳✷✳✷✱ t❤❡ s❡❝♦♥❞ t❡r♠ ✐s ❞♦♠✐♥❛t❡❞ ❜② ✿ C (X,ψ f .f ) -C (ψ g (X),ψ f .f •(ψ g ) -1 ) W ′ ≤ ❈t❡.❱♦•(X) v g χ 1 ✭✷✳✷✺✮ ✻✹ ❋♦r t❤❡ ✜rst t❡r♠✱ ❛♣♣•②✐♥❣ t❤❡♦r❡♠ ✷✳✷✳✶✱ ✇❡ ✜♥❞ t❤❛t ✿ C (X,f ) -C (X,ψ f .f ) W ′ ≤ ❈t❡ X d M ((ψ f .f )(x), f (x))dH d (x) . ❯s✐♥❣ ♥♦✇ ✉s✉❛• r❡s✉•t ♦♥ t❤❡ ✢♦✇ ♦❢ t❤❡ ✈❡❝t♦r ✜❡•❞ v = (v g , v f )✱ ✇❡ ❝❛♥ ❜♦✉♥❞✱ ❢♦r ❛•• (x, m) ∈ E × M ✱ d M (ψ f (x, m), m) ❜② (v g , v f ) χ 0 ≤ v g χ 0 + v f χ 0 ❛♥❞ t❤✉s ✇❡ ❤❛✈❡ ✿ C (X,f ) -C (X,ψ f .f ) W ′ ≤ ❈t❡.❱♦•(X). v g χ 0 + v f χ 0 ✭✷✳✷✻✮ ❙✐♥❝❡ v g χ 0 ≤ v g χ 1 ✱ ❡q✳✭✷✳✷✺✮ ❛♥❞ ✭✷✳✷✻✮ ❣✐✈❡ t❤❡ ❞❡s✐r❡❞ r❡s✉•t✳ ❱❛r✐❛t✐♦♥s✬ ❢♦r♠✉•❛ ❢♦r t❤❡ ♠❡tr✐❝s ❲❡ ♥♦✇ ❣♦ ♦♥❡ st❡♣ ❢✉rt❤❡r ❛♥❞ ❣✐✈❡ ❛ ♠♦r❡ ♣r❡❝✐s❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ✐♥✜♥✐t❡s✐♠❛• ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ W ′ ✲♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❣❡♦♠❡tr✐❝❛• ♣♦s✐t✐♦♥s ❛♥❞ s✐❣♥❛• ✈❛•✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❚❤✐s s❤❛•• •❡❛❞ t♦ ❛ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ ❈❛rt❛♥✬s ❢♦r♠✉•❛ t❤❛t ✇❡ st❛t❡❞ ❢♦r ✉s✉❛• ❝✉rr❡♥ts ✐♥ t❤❡♦r❡♠ ✶✳✸✳✷ ♦❢ s❡❝t✐♦♥ ✶✳✸✳✹✳ ■♥ ❛•• t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❛ss✉♠❡ t❤❛t M ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ t❤❛t ❛•• ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ C 1 r❡❣✉•❛r✐t②✳ ❙✐♠✐•❛r•② t♦ ✶✳✸✳✹✱ ✜①✐♥❣ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f )✱ ✇❡ ✇❛♥t t♦ ❡①♣r❡ss t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ C (X,f ) (ω) ❢♦r ❛♥② ω ∈ Ω d 0 (E, M )✱ ✇✐t❤ r❡s♣❡❝t t♦ X ❛s ✇❡•• ❛s t❤❡ s✐❣♥❛• ✈❛•✉❡s f ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥tr♦❞✉❝❡ s♦♠❡ s♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ✈❡❝t♦r ✜❡•❞ v ✐♥ E × M ✇❤✐❝❤✱ ❛s ♣r❡✈✐♦✉s•②✱ ✐s s✉❝❤ t❤❛t v(x, m) = (v g (x), v f (x, m)) ❛♥❞ ✐ts ✢♦✇ ❛t t✐♠❡ t✱ ψ t (x, m) = (ψ g (x), ψ f (x, m))✳ ❲❤❛t ✇❡ •♦♦❦ ❢♦r ✐s t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ C (X,f ) (ω) ✇❤❡♥ ❞❡❢♦r♠✐♥❣ (X, f ) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ v✱ ✐✳❡ ✿ d dt t=0 C ψt.(X,f ) (ω) = d dt t=0 C (X,f ) (ψ * t ω) ✇❤❡r❡ ψ * t ω ✐s ❞❡✜♥❡❞ ❛s ✐♥ ❡q✳✭✷✳✶✸✮✳ ❚❤❡ s✐♠♣•❡st ✇❛② t♦ ♣r♦❝❡❡❞ ✐s t♦ ✉s❡ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❝✉rr❡♥ts ✐♥ E × M ❛♥❞ t❤❡♥ ❞❡❞✉❝❡ ❛♥ ❡q✉✐✈❛•❡♥t r❡s✉•t ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts t❤r♦✉❣❤ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✐♥tr♦❞✉❝❡❞ ✐♥ ✷✳✷✳✶✳ ▲❡t✬s ❝❛•• T G t❤❡ ❝✉rr❡♥t ✐♥ Ω d 0 (E × M ) ′ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❣r❛♣❤ G ♦❢ (X, f ) ✐♥ E × M ✳ ❲❡ ❛ss✉♠❡ t❤❛t X ✐s ❛ C 1 ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ ❛♥❞ t❤❛t f ✐s C 1 s♦ t❤❛t G ✐s ✐ts❡•❢ ❛ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ ✐♥ E × M ✳ ❋♦r ❛♥② ❞✐✛❡r❡♥t✐❛• ❢♦r♠ η ∈ Ω d 0 (E × M )✱ ✇❡ ❤❛✈❡ T ψt(G) (η) = T G (ψ ♯ t η) = G ψ ♯ t η ❛♥❞ ❜② ❈❛rt❛♥✬s ❢♦r♠✉•❛ ✿ d dt t=0 T ψt(G) (η) = G ı v (dη) + ∂G ı v (η) ✭✷✳✷✼✮ ✇❤❡r❡ ı v ✐s ♦♥❝❡ ❛❣❛✐♥ t❤❡ ❝♦♥tr❛❝t✐♦♥ ♦♣❡r❛t♦r ♦♥ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ✭✐♥ E × M ✮✳ ❚♦ ❣❡t ❜❛❝❦ t♦ ❢✉♥❝t✐♦♥❛• ❢♦r♠s ❛♥❞ ❝✉rr❡♥ts✱ ✇❡ r❡❝❛•• ❢r♦♠ ✷✳✷✳✶ t❤❛t ✇❡ ❤❛✈❡ ❛♥ ✐♥•❡❝t✐✈❡ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥ p : Ω d 0 (E, M ) → Ω d 0 (E × M ) s✉❝❤ t❤❛t p(ω) (x,m) (ζ) = ω (x,m) (l(ζ)) ✇✐t❤ l t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ Λ d (E × M ) ♦♥ Λ d (E) ❞❡✜♥❡❞ ✐♥ ✷✳✷✳✶✳ ▲❡♠♠❛ ✷✳✷✳✺✳ ❲✐t❤ t❤❡ ♣r❡✈✐♦✉s ♥♦t❛t✐♦♥s✱ ✇❡ ❤❛✈❡ ✿ C (X,f ) (ω) = T G (p(ω)) ✭✷✳✷✽✮ ✻✺ Pr♦♦❢✳ ❯♣ t♦ ✉s✐♥❣ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦❢ X✱ ✇❡ ❝❛♥ ❛ss✉♠❡✱ ✇✐t❤♦✉t •♦ss ♦❢ ❣❡♥❡r✲ ❛•✐t②✱ t❤❛t X ❛❞♠✐ts ❛ C 1 ✲♣❛r❛♠❡tr✐③❛t✐♦♥ γ : U → E ✇✐t❤ U ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d ✳ ❚❤❡♥ Γ : U → E × M ❞❡✜♥❡❞ ❜② Γ(u) = (γ(u), f • γ(u)) ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ G ❛♥❞ ✿ T G (p(ω)) = U p(ω) (γ(u),f •γ(u)) ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du 1 ...du d = U ω (γ(u),f •γ(u)) l ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du 1 ...du d ❇② ❞❡✜♥✐t✐♦♥ ♦❢ l ❛♥❞ Γ✱ ✇❡ s❡❡ t❤❛t l ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d ✳ ❆♥❞ t❤✉s ✿ T G (p(ω)) = U ω (γ(u),f •γ(u)) ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d du 1 ...du d = C (X,f ) (ω) ❲❡ ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♠♠✉t❛t✐✈✐t② ♣r♦♣❡rt② ✿ ▲❡♠♠❛ ✷✳✷✳✻✳ ❋♦r ❛♥② ω ∈ Ω d 0 (E, M )✱ p(ψ * t ω) = ψ ♯ t (p(ω)) ✭✷✳✷✾✮ Pr♦♦❢✳ ❋♦r ❛♥② (x, m) ∈ E ×M ❛♥❞ ζ = ζ 1 ∧...∧ζ d ∈ Λ d (E ×M )✱ ✇❡ •❡t ξ 1 ∧...∧ξ d = l(ζ 1 ∧ ... ∧ ζ d )✳ ❚❤❡♥ ✿ p(ψ * t ω) (x,m) (ζ) = (ψ * t ω) (x,m) (ξ 1 ∧ ... ∧ ξ d ) = ω ψt(x,m) (∂ x ψ g t (ξ 1 ) ∧ ... ∧ ∂ x ψ g t (ξ d )) ◆♦✇✱ s✐♥❝❡ ∂ m ψ g t = 0✱ ✇❡ ❤❛✈❡ ✐♥ ❢❛❝t ∀i ∈ {1, .., d}, ∂ x ψ g t (ξ i ) = dψ g t (ζ i ) ❛♥❞ ∂ x ψ g t (ξ 1 ) ∧ ... ∧ ∂ x ψ g t (ξ d ) = l(dψ t (ζ 1 ) ∧ ... ∧ dψ t (ζ d ))✱ ✇❤✐❝❤ ❣✐✈❡s ✿ p(ψ * t ω) (x,m) (ζ) = ω ψt(x,m) (l(dψ t (ζ 1 ) ∧ ... ∧ dψ t (ζ d ))) = ψ ♯ t p(ω) (x,m) (ζ) ❲✐t❤ ❡q✳✭✷✳✷✼✮✱✭✷✳✷✽✮ ❛♥❞ ✭✷✳✷✾✮✱ ✇❡ ♥♦✇ ❞❡❞✉❝❡ ✿ d dt t=0 C ψt.(X,f ) (ω) = d dt t=0 C (X,f ) (ψ * t ω) = d dt t=0 T G (p(ψ * t ω)) = d dt t=0 T G (ψ ♯ t p(ω)) = G ı v (d[p(ω)]) + ∂G ı v (p(ω)) ❲❡ ❤❛✈❡ t❤✉s s❤♦✇♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✿ ✻✻ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✸✳ ❋♦r X ❛ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ ♦❢ E✱ f ❛ C 1 ❢✉♥❝t✐♦♥ ♦♥ X ❛♥❞ ❢♦r ❛♥② ✈❡❝t♦r ✜❡•❞ ♦❢ t❤❡ ❢♦r♠ v(x, m) = (v g (x), v f (x, m)) ✇❤♦s❡ ✢♦✇ ψ t ❡①✐sts ❛t t✐♠❡ 0 < t < ǫ✱ ✇❡ ❤❛✈❡ ❢♦r ❛•• C 1 ❢✉♥❝t✐♦♥❛• ❢♦r♠ ω ∈ Ω d 0 (E, M ) ✿ d dt t=0 C ψt.(X,f ) (ω) = G ı v (d[p(ω)]) + ∂G ı v (p(ω)) ✭✷✳✸✵✮ ❲❡ s❡❡ t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡s ✇❡ •♦♦❦ ❢♦r ❡✈❡♥t✉❛••② ❝♦♥s✐st ✐♥ t❛❦✐♥❣ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ♣r♦❞✉❝t ❝✉rr❡♥t T G ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❣r❛♣❤ ❜✉t r❡str✐❝t❡❞ t♦ ♣❛rt✐❝✉•❛r ❞✐✛❡r❡♥✲ t✐❛• ❢♦r♠s p(ω) ♦❢ E × M ✳ ❲❡ ❛•r❡❛❞② ❝❛♥ ❞r❛✇ ❛ ❢❡✇ ❝♦♥s❡q✉❡♥❝❡s ❢r♦♠ t❤✐s✱ r❡•②✐♥❣ ♦♥ ✇❤❛t ✇❛s ♣r❡✈✐♦✉s•② ❞♦♥❡ ✇✐t❤ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❝✉rr❡♥ts ✭t❤❡♦r❡♠ ✶✳✸✳✷✮✳ ■♥ t❤❛t ❝❛s❡✱ ✐t s❤♦✇s t❤❛t ✐♥ E × M ✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❢❝✉rr❡♥ts✬ ♠❡tr✐❝s ❛t ❡❛❝❤ ♣♦✐♥t ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ X ✐s ♦rt❤♦❣♦♥❛• t♦ t❤❡ ❣r❛♣❤ ♦❢ (X, f )✳ ■t ❛•s♦ ✐♥✈♦•✈❡s ❛ t❡r♠ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ G ✇❤❡r❡ ♣♦✐♥ts ❤❛✈❡ ❣r❛❞✐❡♥ts t❤❛t ❛r❡ ♦rt❤♦❣♦♥❛• t♦ t❤✐s ❜♦✉♥❞❛r② ✐♥ E × M ✳ ■❢ f ✐s ❛ C 1 ❢✉♥❝t✐♦♥✱ ∂G ✐s s✐♠♣•② t❤❡ s❡t ♦❢ {(x, f (x))} ❢♦r x ∈ ∂X ❜✉t ✇❡ s❡❡ t❤❛t t❤✐s ❝♦✉•❞ ❜❡ ❣❡♥❡r❛•✐③❡❞ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ ❢♦r ✐♥st❛♥❝❡ f ✐s ♣✐❡❝❡✇✐s❡ C 1 ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ s♦♠❡ ♥❡✇ ❜♦✉♥❞❛r✐❡s ❛♣♣❡❛r ♦♥ t❤❡ ③♦♥❡s ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ t❤❡ s✐❣♥❛•✳ ❚❤❡ ♣r❡✈✐♦✉s r❡s✉•t ✐s ❤♦✇❡✈❡r ✐♥tr✐♥s✐❝❛••② ✇r✐tt❡♥ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝✉rr❡♥ts ✐♥ E × M ✳ ❙t❛rt✐♥❣ ❢r♦♠ ❡q✳✭✷✳✸✵✮✱ ✇❡ ❝❛♥ ❛❝t✉❛••② ❣♦ ❛ •✐tt•❡ ❢✉rt❤❡r ❜② r❡✐♥t❡r♣r❡t✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❡r♠s ♦♥ t❤❡ ❣r❛♣❤ G ❛s ✐♥t❡❣r❛•s ♦♥ X ❛♥❞ s❡♣❛r❛t❡ t❤❡ ✈❛r✐❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② t❤❡ ✜❡•❞ v g ❢r♦♠ t❤❡ ♦♥❡ ✐♥❞✉❝❡❞ ❜② v f ✳ ❋♦r ❛♥② ❝♦♦r❞✐♥❛t❡ s②st❡♠s (x 1 , ..., x n ) ❛♥❞ (m 1 , ..., m p ) ♦❢ E ❛♥❞ M ✱ ❛ ❢✉♥❝t✐♦♥❛• ❢♦r♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✿ ω . = 1≤i 1 <...<i d ≤n a i 1 ,...,i d (x, m)dx i 1 ∧ ... ∧ dx i d ❛♥❞ t❤❡♥ p(ω) ❤❛s t❤❡ ❡①❛❝t s❛♠❡ ❡①♣r❡ss✐♦♥ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ (x 1 , ..., x n , m 1 , ..., m p ) ♦❢ E × M ✳ ❲❡ s❤❛•• ❞❡♥♦t❡ ✐♥ s❤♦rt ω = a I (x, m)dx I ❢♦r t❤❡ s✉♠ ♦♥ ❛•• ♠✉•t✐✲✐♥❞✐❝❡s ✐♥ ❛ ❣✐✈❡♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠✳ ◆♦✇✱ s✐♥❝❡ ✇❡ ❤❛✈❡ ❛ss✉♠❡❞ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f ✐s C 1 ✿ ∂G = {(x, f (x)) ∈ E × M/ x ∈ ∂X} ✇❤✐❝❤ ✐s t❤❡ ❣r❛♣❤ ♦❢ t❤❡ (d -1)✲❞✐♠❡♥s✐♦♥❛• ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (∂X, f |∂X )✳ ■❢ η : O → E ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ∂X ✭O ❜❡✐♥❣ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d-1 ✮✱ η : u → (η(u), f • η(u)) ∈ E × M ✐s ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ ∂G ❛♥❞ ✿ ∂G ı v (p(ω)) = O p(ω) (η(u),f •η(u)) (v(η(u), f • η(u)) ∧ ∂ η ∂u 1 ... ∧ ∂ η ∂u d-1 ) = O ω (η(u),f •η(u)) (v g (η(u)) ∧ ∂η ∂u 1 ... ∧ ∂η ∂u d-1 ) = ∂X ı vg ω ❚❤❡ ❝❛s❡ ♦❢ t❤❡ ✐♥t❡❣r❛• ♦♥ G ✐s ❛ •✐tt•❡ ♠♦r❡ ❞✐✣❝✉•t✳ ▲❡t✬s ❝♦♥s✐❞❡r s♦♠❡ •♦❝❛• ❝♦♦r❞✐♥❛t❡ s②st❡♠s (x 1 , ..., x n ) ♦❢ E s✉❝❤ t❤❛t (dx 1 , .., dx d ) ✐s ❛ ❜❛s✐s ♦❢ T x X * ❛♥❞ (dx d+1 , .., dx n ) ✐s ❛ ❜❛s✐s ♦❢ (T x X ⊥ ) * ✳ ■♥ t❤❡ s❛♠❡ s♣✐r✐t✱ ✇❡ ✇✐•• ❞❡♥♦t❡ ❜② v ⊤ g (x) t❤❡ ❝♦♠♣♦♥❡♥t ♦❢ v g t❛♥❣❡♥t t♦ X ❛t x ❛♥❞ v ⊥ g (x) t❤❡ ♥♦r♠❛• ❝♦♠♣♦♥❡♥t✳ ❲❡ t❤❡♥ ❞❡❝♦♠♣♦s❡ t❤❡ ✈❡❝t♦r ✜❡•❞ ❛s v = (v ⊥ g , 0) + (v ⊤ g , 0) + (0, v f ) ❇② •✐♥❡❛r✐t② ♦❢ t❤❡ ❝♦♥tr❛❝t✐♦♥ ♦♣❡r❛t♦r✱ ✇❡ ❤❛✈❡ ✿ G ı v dp(ω) = G ı (v ⊥ g ,0) dp(ω) + G ı (v ⊤ g ,0) dp(ω) + G ı (0,v f ) dp(ω) . ✭✷✳✸✶✮ ✻✼ ◆♦✇✱ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝♦♦r❞✐♥❛t❡ s②st❡♠✱ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❡①t❡r✐♦r ❞❡r✐✈❛t✐✈❡ ♦❢ p(ω) ❛s ❛ ❞✐✛❡r❡♥t✐❛• ❢♦r♠ ♦❢ E × M ✿ d[p(ω)] = n j=1 ∂a I ∂x j dx j ∧ dx I + p l=1 ∂a I ∂m l dm l ∧ dx I = d x ω + d m ω ❛♥❞ ✐♥tr♦❞✉❝✐♥❣ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ X✱ γ : U → E ❛♥❞ Γ : u → (γ(u), f • γ(u)) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ G✱ ✇❡ ❤❛✈❡✱ ❢♦r t❤❡ ✜rst ✐♥t❡❣r❛• ♦❢ ❡q✳✭✷✳✸✶✮ ✿ G ı (v ⊥ g ,0) dp(ω) = U (d x ω + d m ω) (v ⊥ g (γ(u)), 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du ✇✐t❤ ∂Γ ∂u i = ∂γ ∂u i , df • ∂γ ∂u i . ✭✷✳✸✷✮ ❲❡ r❡❝❛••✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ❡①t❡r✐♦r ♣r♦❞✉❝t ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✱ t❤❛t ❢♦r ❛•• l ∈ {1, .., p} ❛♥❞ u ∈ U ✿ dm l ∧ dx I (v ⊥ g (γ(u)), 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = dm l (v ⊥ g (γ(u)), 0)dx I ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d + d i=1 (-1) i dm l ∂Γ ∂u i dx I v ⊥ g ∧ ∂Γ ∂u 1 ∧ .. ∂Γ ∂u i .. ∧ ∂Γ ∂u d ❚❤❡ ✜rst t❡r♠ ❡q✉❛•s ③❡r♦ ❜❡❝❛✉s❡ (v ⊥ g (γ(u)), 0) ❤❛s ♦♥•② ③❡r♦ ❝♦♠♣♦♥❡♥ts ✐♥ M ✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❛•s♦ ♥✉•• s✐♥❝❡ ❛•• ❞✐✛❡r❡♥t✐❛• ❢♦r♠s dx I ✐♥ ♦✉r ❝❛s❡ ❤❛✈❡ ♦♥•② ❝♦♠♣♦♥❡♥ts dx i ✐♥ T x X * ✇❤❡r❡❛s v ⊥ g (x) ∈ T x X ⊥ ❢♦r ❛•• x ∈ X✳ ❚❤❡r❡❢♦r❡ ✿ G ı (v ⊥ g ,0) dp(ω) = U d x ω (v ⊥ g (γ(u)), 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du = U d x ω v ⊥ g (γ(u)) ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d du = X ı v ⊥ g d x ω ✭✷✳✸✸✮ t❤❡ s❡❝♦♥❞ ❡q✉❛•✐t② r❡s✉•t✐♥❣ ❢r♦♠ ❡q✳✭✷✳✸✷✮ ❛♥❞ t❤❡ ❢❛❝t t❤❡ (d + 1)✲❞✐✛❡r❡♥t✐❛• ❢♦r♠ d x ω ❤❛s ♥♦ ❝♦♠♣♦♥❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢✉♥❝t✐♦♥❛• ❝♦♦r❞✐♥❛t❡s dm 1 , .., dm p ✳ ◆♦✇ ❢♦r t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛• ♦❢ ❡q✳✭✷✳✸✶✮✱ ✇❡ ✇r✐t❡ ✿ G ı (v ⊤ g ,0) dp(ω) = U (d x ω + d m ω) (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du .
❖♥ t❤❡ ✜rst ❤❛♥❞✱ ✇❡ ❤❛✈❡ ✿

d x ω (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = n j=1 ∂a I ∂x j dx j ∧ dx I (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = n j=1 ∂a I ∂x j dx j ∧ dx I v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d ✻✽ ❛♥❞ s✐♥❝❡ ∂γ ∂u 1 , ..., ∂γ ∂u d ✐s ❛ ❜❛s✐s ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T γ(u) X✱ v ⊤ g ✐s •✐♥❡❛r•② ❞❡♣❡♥❞❡♥t t♦ t❤❡s❡ ✈❡❝t♦rs✳ ❚❤✉s v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = 0 ❢♦r ❛•• u ❛♥❞ G ı (v ⊤ g ,0) d x ω = 0 . ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ d m ω (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = p l=1 ∂a I ∂m l dm l ∧ dx I (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d ∆ l . ❉❡✈❡•♦♣✐♥❣ ∆ l ❣✐✈❡s ✿ ∆ l = dm l (v ⊤ g , 0)dx I ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d + d i=1 (-1) i dm l ∂Γ ∂u i dx I (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u i ∧ ... ∧ ∂Γ ∂u d = d i=1 (-1) i dm l ∂Γ ∂u i dx I (v ⊤ g , 0) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u i ∧ ... ∧ ∂Γ ∂u d = d i=1 (-1) i df l ∂γ ∂u i dx I v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u i ∧ ... ∧ ∂γ ∂u d ❖♥❡ ❝❛♥ ✇r✐t❡ df l ✐♥ t❤❡ s❛♠❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛s df l = d s=1 ∂f l ∂x s dx s ✱ ✇❤✐❝❤ ❣✐✈❡s ✿ ∆ l = d s=1 ∂f l ∂x s d i=1 (-1) i dx s ∂γ ∂u i dx I v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u i ∧ ... ∧ ∂γ ∂u d = d s=1 ∂f l ∂x s dx s ∧ dx I v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d - d s=1 ∂f l ∂x s dx s (v ⊤ g )dx I ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = -df l (v ⊤ g )dx I ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d . ■♥❞❡❡❞ v ⊤ g ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = 0 s✐♥❝❡ v ⊤ g ∈ T γ(u) X ❛♥❞ ∂γ ∂u 1 , ..., ∂γ ∂u d ✐s ❛ ❜❛s✐s ♦❢ T γ(u) X✳ ❊✈❡♥t✉❛••②✱ ✇❡ ♦❜t❛✐♥ t❤❛t ✿ G ı (v ⊤ g ,0) dp(ω) = U - n l=1 ∂a I ∂m l df l (v ⊤ g )dx I ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = U - n l=1 ∂a I ∂m l dm l (df (v ⊤ g ))dx I ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = U - n l=1 ∂a I ∂m l dm l ∧ dx I df (v ⊤ g ) ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = X -ı df (v ⊤ g ) d m ω ✭✷✳✸✹✮ ✻✾ ❋✐❣✉r❡ ✷✳✾✿ •r❛❞✐❡♥t ♦❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❲❡ r❡♣r❡s❡♥t t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ ♣♦✐♥t ♣♦s✐t✐♦♥s✳ ❚❤❡ ❣r❛❞✐❡♥t ✐s ♦rt❤♦❣♦♥❛• t♦ t❤❡ ❝✉r✈❡ ♦♥ t❤❡ ❛r❡❛s ♦❢ ❝♦♥st❛♥t s✐❣♥❛•s ❜✉t ♥♦t ✐♥ ③♦♥❡s ✇❤❡r❡ s✐❣♥❛• ✐s ✈❛r②✐♥❣✳ ◆♦t❡ ❛•s♦ t❤❡ ♣❛rt✐❝✉•❛r ❜❡❤❛✈✐♦r ♦♥ ❜♦✉♥❞❛r✐❡s ❝r❡❛t❡❞ ❜② ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ s✐❣♥❛•✳ ❋✐♥❛••②✱ ❢♦r t❤❡ •❛st ✐♥t❡❣r❛• ♦❢ ❡q✳✭✷✳✸✶✮✱ ✇✐t❤ t❤❡ s❛♠❡ ❦✐♥❞ ♦❢ r❡❛s♦♥✐♥❣✱ ✇❡ ♦❜t❛✐♥ ❡❛s✐•② t❤❛t ✿ G ı (0,v f ) dp(ω) = U d m ω (0, v f ) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d du ❛♥❞ •✉st ❛s ♣r❡✈✐♦✉s•② ✇❡ ❝❛♥ ✇r✐t❡ ❢♦r ❛•• u ∈ U ✿ d m ω (0, v f ) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = p l=1 ∂a I ∂m l dm l ∧ dx I (0, v f ) ∧ ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = p l=1 ∂a I ∂m l dm l (v f ).dx I ∂Γ ∂u 1 ∧ ... ∧ ∂Γ ∂u d = p l=1 ∂a I ∂m l dm l (v f ).dx I ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d = p l=1 ∂a I ∂m l dm l ∧ dx I (0, v f ) ∧ ∂γ ∂u 1 ∧ ... ∧ ∂γ ∂u d . ❚❤❡ s❡❝♦♥❞ ❛♥❞ t❤❡ •❛st ❡q✉❛•✐t② r❡s✉•t ❛❣❛✐♥ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t dx i (0, v f ) = 0 ❢♦r ❛•• i ∈ {1, .., n}✳ ❈♦♥s❡q✉❡♥t•② ✿ G ı (0,v f ) dp(ω) = X ı v f d m ω . ✭✷✳✸✺✮ •r♦✉♣✐♥❣ ❡q✳✭✷✳✸✸✮ ✭✷✳✸✹✮ ❛♥❞ ✭✷✳✸✺✮ t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡ ❡✈❡♥t✉❛••② s❤♦✇♥ t❤❡ ❢♦••♦✇✐♥❣ ✈❛r✐❛t✐♦♥✬s r❡s✉•t ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ ♠❡tr✐❝s ✿ ❚❤❡♦r❡♠ ✷✳✷✳✹✳ ❋♦r ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ✇✐t❤ X ❛ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜✲ ♠❛♥✐❢♦•❞ ♦❢ E ❛♥❞ f ❛ C 1 ❢✉♥❝t✐♦♥✱ ❛♥❞ ❢♦r ❛♥② ✈❡❝t♦r ✜❡•❞ ♦❢ t❤❡ ❢♦r♠ v(x, m) = ✼✵ (v g (x), v f (x, m)) ✇❤♦s❡ ✢♦✇ ψ t ❡①✐sts ❛t t✐♠❡ 0 < t < ǫ✱ ✇❡ ❤❛✈❡ ❢♦r ❛•• C 1 ❢✉♥❝t✐♦♥❛• d✲❢♦r♠ ω ✿ d dt t=0 C ψt.(X,f ) (ω) = X ı v ⊥ g d x ω + ı (v f -df (v ⊤ g )) d m ω + ∂X ı v g ω ❋♦r t❤❡ ❝♦rr❡❝t ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢♦r♠✉•❛✱ •❡t✬s r❡♠✐♥❞ t❤❛t ✇❡ ❤❛✈❡ ❛❞♦♣t❡❞ t❤❡ s❤♦rt❝✉t ♥♦t❛t✐♦♥s ı u d x ω ❢♦r ı (u,0) d x ω ✭✇❤❡♥ u ✐s ❛ ✈❡❝t♦r ✜❡•❞ ✇✐t❤ ✈❛•✉❡s ✐♥ E✮ ❛♥❞ ı u d m ω ❢♦r ı (0,u) d m ω ✭✇❤❡♥ u ✐s ❛ ✈❡❝t♦r ✜❡•❞ ✇✐t❤ ✈❛•✉❡s ✐♥ M ✮✳ ❚❤✐s •❛st r❡s✉•t ❣✐✈❡s t❤❡ ✈❛r✐❛t✐♦♥ ❜❛s❡❞ ♦♥•② ♦♥ ✐♥t❡❣r❛•s ♦♥ X ❛♥❞ ∂X ♦❢ s❡✈❡r❛• ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ♦❜t❛✐♥❡❞ ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ ω✳ ■t ♣r♦✈✐❞❡s ❛ ♠✉❝❤ ♠♦r❡ ♣r❡❝✐s❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♠❡tr✐❝s t❤❛♥ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✸✳ ■♥❞❡❡❞✱ ✇❡ ✜rst s❡❡ t❤❛t t❤❡ ✈❛r✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❡❝t♦r ✜❡•❞ v g ✐♥✈♦•✈❡s ❜♦t❤ ✐ts ♦rt❤♦❣♦♥❛• ❛♥❞ t❛♥❣❡♥t✐❛• ❝♦♠♣♦♥❡♥ts ✿ t❤✉s t❤❡ ❣r❛❞✐❡♥t ♦❢ C (X,f ) (ω) ✇✐t❤ r❡s♣❡❝t t♦ ❞✐s♣•❛❝❡♠❡♥ts ♦❢ X ✐s ♥♦t ♥❡❝❡ss❛r✐•② ♦rt❤♦❣♦♥❛• t♦ X ❛s ❢♦r ✉s✉❛• ❝✉rr❡♥ts✳ ❍♦✇❡✈❡r✱ ✇❡ s❡❡ t❤❛t t❤❡ ♦♥•② t❡r♠ ✐♥✈♦•✈✐♥❣ v ⊤ g ❛❝t✉❛••② ❞❡♣❡♥❞s ♦♥•② ✐♥ df (v ⊤ g )✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐t ✈❛♥✐s❤❡s ✇❤❡♥❡✈❡r ∇f = 0 ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t t❤❡ ❣r❛❞✐❡♥t ♦❢ C (X,f ) (ω) ✇✐•• ❜❡ ♦rt❤♦❣♦♥❛• t♦ X ✐♥ r❡❣✐♦♥s ✇❤❡r❡ f ✐s ❝♦♥st❛♥t✳ ❚❤✐s ✐s ✐••✉str❛t❡❞ ❜② ✜❣✉r❡ ✷✳✾✳ ❆ s❡❝♦♥❞ ✐♠♣♦rt❛♥t ❡•❡♠❡♥t ✐s t❤❡ t❡r♠ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ X t❤❛t ♦♥•② ❛✛❡❝ts t❤❡ ✈❛r✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❣❡♦♠❡tr② s✐♥❝❡ ✐t ♦♥•② t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❝♦♠♣♦♥❡♥t v g ✳ ❖♥ ♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r②✱ t❤❡ ❣r❛❞✐❡♥t ✐s ♦rt❤♦❣♦♥❛• t♦ t❤❡ ❜♦✉♥❞❛r② ❜✉t ♥♦t ♥❡❝❡ss❛r✐•② t♦ X✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ✐♥t❡❣r❛•s ♦♥ ∂X ✐♥ t❤❡ ✈❛r✐❛t✐♦♥ ✐s ♦❢ ❝r✉❝✐❛• ✐♠♣♦rt❛♥❝❡ ♥✉♠❡r✐❝❛••② ✇❤❡♥ ❝♦♠♣✉t✐♥❣ ❣r❛❞✐❡♥ts ♦❢ ❢❝✉rr❡♥ts ❞✐st❛♥❝❡s ❜❡❝❛✉s❡ t❤❡② ✐♥❞✉❝❡ s✐♥❣✉•❛r✐t✐❡s ♦♥ t❤❡ ❜♦✉♥❞❛r②✱ ❛s ✇❡ st✉❞② ✐♥ ♠♦r❡ ❞❡t❛✐•s ✐♥ ✹✳✷✳✶✳ ❆s ✇❡ ♠❡♥t✐♦♥❡❞ ❡❛r•✐❡r ❛❢t❡r ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✸✱ ✇❡ ❝♦✉•❞ ❣❡♥❡r❛•✐③❡ t❤❡♦r❡♠ ✷✳✷✳✹ t♦ ♣✐❡❝❡✇✐s❡ C 1 ❢✉♥❝t✐♦♥s f ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ❛❞❞✐t✐♦♥❛• ❜♦✉♥❞❛r✐❡s ❛r❡ ❝r❡❛t❡❞ ❜② ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ t❤❡ s✐❣♥❛•✱ ✐♥❞✉❝✐♥❣ ❛❣❛✐♥ ❣r❛❞✐❡♥t s✐♥❣✉•❛r✐t✐❡s ❛t t❤♦s❡ ♣♦✐♥ts ❛s ♦♥❡ ❝❛♥ s❡❡ ♦♥ ✜❣✉r❡ ✷✳✾✳ ✷✳✸ ❆•❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❲❡ ♥♦✇ ✐••✉str❛t❡ ❤♦✇ t❤❡ t❤❡♦r❡t✐❝❛• ❝♦♥❝❡♣t ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥tr♦❞✉❝❡❞ ❛❜♦✈❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❦❡r♥❡• ♠❡tr✐❝s ♦✛❡rs ❛ ❣❡♥✉✐♥❡ s♦•✉t✐♦♥ t♦ t❤❡ s✐♠✉•t❛♥❡♦✉s ♣r♦❝❡ss✲ ✐♥❣ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ s✐❣♥❛• ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ ❛•❣♦r✐t❤♠s ✇❡ ♣r❡s❡♥t ✇✐•• ❢♦❝✉s ❡①❝•✉s✐✈❡•② ♦♥ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s ❛•t❤♦✉❣❤✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞✱ ✐t ❝♦✉•❞ ❜❡ ❛❞❛♣t❡❞ t♦ ❞✐✛❡r❡♥t t②♣❡s ♦❢ s✐❣♥❛• s♣❛❝❡s✳ ❲❡ ❤❛✈❡ ❡①♣•❛✐♥❡❞✱ ❢r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❤♦✇ ❍✐•❜❡rt ♥♦r♠s ❜❛s❡❞ ♦♥ ❦❡r♥❡•s s❤❛r❡ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ✇✐t❤ r❡s♣❡❝t t♦ ❣❡♦♠❡tr✐❝❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ♣❡rt✉r❜❛t✐♦♥s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡ ♦♣❡♥s t❤❡ ✇❛② ❢♦r ✈❛r✐♦✉s ♣r♦❝❡ss✐♥❣ ❛•❣♦r✐t❤♠s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥t❡♥❞ t♦ s❤❡❞ •✐❣❤t ♦♥ t❤❡ ♣♦t❡♥t✐❛• ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❢r❛♠❡✇♦r❦✳ ❚❤❡ ✜rst ❛♣♣•✐❝❛t✐♦♥ ✐••✉str❛t❡s t❤❡ ❢✉•• ♣♦t❡♥t✐❛• ♦❢ t❤❡ ❍✐•❜❡rt str✉❝t✉r❡ ✇✐t❤ t❤❡ ❞❡s✐❣♥ ♦❢ r❡❞✉♥❞❛♥❝② r❡❞✉❝t✐♦♥ ♦r ❝♦♠♣r❡ss✐♦♥ ❛•❣♦r✐t❤♠s ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡ r❡♣r❡s❡♥t❛t✐♦♥s t❤r♦✉❣❤ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t s❝❤❡♠❡s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡✱ ♠♦r❡ ❞✐r❡❝t•② r❡•❛t❡❞ t♦ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠②✬s ✐ss✉❡s ❡①♣♦s❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝❤❛♣t❡r✱ ✐s t❤❡ ❞❡✲ s✐❣♥ ♦❢ ❛ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ ♠❛t❝❤✐♥❣ ❛•❣♦r✐t❤♠ ❢♦r t❤❡ s✐♠✉•t❛♥❡♦✉s ❣❡♦♠❡tr✐❝ ❛♥❞ ❢✉♥❝t✐♦♥❛• r❡❣✐str❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐❝ tr❛♥s♣♦rt✳ ❲❡ ♣♦st✲ ♣♦♥❡ ❛♣♣•✐❝❛t✐♦♥s t♦ ♠❡❞✐❝❛• ✐♠❛❣✐♥❣ ❞❛t❛s❡ts t♦ ❝❤❛♣t❡r ✹✱ ✇❤✐❝❤ ✇✐•• ❣❡♥❡r❛•✐③❡ t❤❡ ❢r❛♠❡✇♦r❦ t♦ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ✼✶ ✷✳✸✳✶ ❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ▼❛t❝❤✐♥❣ P✉rs✉✐t ♦♥ ❢❝✉rr❡♥ts ▲❡t ✉s st❛rt ✇✐t❤ t❤❡ ✐ss✉❡ ♦❢ t❤❡ r❡❞✉♥❞❛♥❝② ♦❢ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥s✳ ❚❤❡ ❞✐s✲ ❝r❡t✐③❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✐♥ ♣✉♥❝t✉❛• ❢❝✉rr❡♥ts ❣✐✈❡♥ ❜② ❡q✳✭✷✳✾✮ ✇✐•• ♣r♦✈✐❞❡ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ ❛ ♥✉♠❜❡r ♦❢ ❡•❡♠❡♥ts t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✐♥✐t✐❛• s❛♠♣•✐♥❣ ♦❢ t❤❡ s❤❛♣❡✳ ❋♦r ❛ s✉r❢❛❝❡✱ t❤✐s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♥✉♠❜❡r ♦❢ tr✐❛♥❣•❡s ✐♥ t❤❡ ♠❡s❤✳ •❡♥❡r❛••②✱ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❝♦✉•❞ ❜❡ ❝•❡❛r•② r❡❞✉❝❡❞ s✐♥❝❡✱ ❢♦r s✐♠♣•❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ •♦✇✲❢r❡q✉❡♥❝② ❣❡♦♠❡tr✐❡s ❛♥❞ s✐❣♥❛•s✱ ♦♥•② ❛ ❢❡✇ t❡r♠s s❤♦✉•❞ ❝❛♣t✉r❡ ♠♦st ♦❢ t❤❡ s❤❛♣❡✳ ❍♦✇❡✈❡r✱ t❤❡ q✉❛•✐t② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♥❡❡❞s t♦ ❜❡ q✉❛♥t✐✜❡❞ ✐♥ ❛ ♠❡❛♥✐♥❣❢✉• ✇❛②✱ ❡s♣❡❝✐❛••② ✇❤❡♥ t❤❡ ❢✉♥❝t✐♦♥❛• ♣❛rt ✐s ❛•s♦ ✐♥✈♦•✈❡❞✱ t❤r♦✉❣❤ ❛♥ ❛♣♣r♦♣r✐❛t❡ ♥♦r♠ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❛ ♥❛t✉r❛• ❝❛♥❞✐❞❛t❡ ❣✐✈❡♥ ❜② t❤❡ ❦❡r♥❡• ❍✐•❜❡rt str✉❝t✉r❡✳ ❚❤✐s ✐ss✉❡ ♦❢ r❡❞✉♥❞❛♥❝② r❡❞✉❝t✐♦♥ ♦r ❝♦♠♣r❡ss✐♦♥ ✐s ✐♠♣♦rt❛♥t ❢♦r ✐♥✲ st❛♥❝❡ ✇❤❡♥ ♠❛❦✐♥❣ ❝r✉❞❡ ♠❡❛♥s ♦❢ ❝✉rr❡♥ts ❜❡❝❛✉s❡ ✇✐t❤♦✉t ❢✉rt❤❡r tr❡❛t♠❡♥t✱ t❤❡ ♥✉♠❜❡r ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts ✐♥✈♦•✈❡❞ ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥ ✇♦✉•❞ ✐♥❝r❡❛s❡ ❞r❛♠❛t✐❝❛••②✳ (❡t✱ ❛s ✇❡ s❤❛•• s❡❡ ✐♥ ❝❤❛♣t❡r ✹✱ ✇✐t❤ ♠♦r❡ r❡❝❡♥t ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡s✱ ❝♦♠♣r❡ss✐♦♥ ✐s ❢❛r •❡ss ❞❡❝✐s✐✈❡ ✐♥ ♣r❛❝t✐❝❡✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❤✐❣❤❡r ♦r❞❡r st❛t✐st✐❝s ❢♦r t❤❡ ❛♥❛•②s✐s ♦❢ ✈❛r✐❛❜✐•✐t② ❛r♦✉♥❞ ❛ ♠❡❛♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ t❤r♦✉❣❤ ❝♦♠♣✉t❛t✐♦♥s ♦❢ W ′ ✲✐♥♥❡r ♣r♦❞✉❝ts ✐♥ ▲❉❆ ♦r ❙❱▼ ❢♦r ✐♥st❛♥❝❡✱ ❤❛✈✐♥❣ r❡❞✉❝❡❞ ✜rst t❤❡ ♥✉♠❜❡r ♦❢ ❉✐r❛❝s ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❞❛t❛ ❝❛♥ •❡❛❞ t♦ s✉❜st❛♥t✐❛• ❣❛✐♥ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡✳ ❲❤❛t ✇❡ ♣r♦♣♦s❡ ❤❡r❡ ✐s ♦♥❡ ✇❛② t♦ ❛❞❞r❡ss t❤❡ ♣r♦❜•❡♠ ❜② ♣❡r❢♦r♠✐♥❣ ❝♦♠♣r❡ss✐♦♥ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✿ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ❦❡❡♣ ✐♥ ♠✐♥❞ t❤❛t t❤✐s ✐s ♥♦t ❡q✉✐✈❛•❡♥t t♦ ♠♦r❡ ❝•❛ss✐❝❛• ♠❡s❤ r❡❞✉❝t✐♦♥ ♦r ✈❡rt✐❝❡s✬s ❞❡❝✐♠❛t✐♦♥ ❢♦r ✐♥st❛♥❝❡ ❜❡❝❛✉s❡ ♦✉r ♦✉t♣✉t ✐s ❛ ❣❡♥❡r❛• ❢❝✉r✲ r❡♥t ✭❛ s❡t ♦❢ ❉✐r❛❝s✮ t❤❛t ❞♦❡s ♥♦t ❤❛✈❡ ❛♥ ✉♥❞❡r•②✐♥❣ s❤❛♣❡ str✉❝t✉r❡✳ ❚❤✐s •♦ss ♦❢ str✉❝t✉r❡ ❛••♦✇s ✐♥ ❝♦♠♣❡♥s❛t✐♦♥ ❤✐❣❤ ❝♦♠♣r❡ss✐♦♥ r❛t❡s ✇✐t❤ ❛♥ ❛❝❝✉r❛❝② t❤❛t ✐s ❞✐r❡❝t•② r❡•❛t❡❞ t♦ t❤❡ ♠❡tr✐❝ ♦♥ ❢❝✉rr❡♥ts t❤❛t ✇❡ ✉s❡✳ ❆s ♦♥❡ ❝❛♥ ❡①♣❡❝t✱ t❤❡ ♣r♦❜•❡♠ ♦❢ r❡❞✉♥❞❛♥❝② r❡❞✉❝t✐♦♥ ♦r ❝♦♠♣r❡ss✐♦♥ ✐s ❞❡❡♣•② s✐♠♣•✐✜❡❞ t❤❛♥❦s t♦ t❤❡ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡ t❤❛t ❤❛s ❜❡❡♥ ❞❡✜♥❡❞ ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ■♥❞❡❡❞✱ ❝•❛ss✐❝❛• ♠❛t❝❤✐♥❣✲♣✉rs✉✐t ❛•❣♦r✐t❤♠s ✐♥ ❣❡♥❡r❛• ❍✐•❜❡rt s♣❛❝❡s ❤❛✈❡ ❛•r❡❛❞② ❜❡❡♥ st✉❞✐❡❞ ❜② ▼❛••❛t ❛♥❞ )❤❛♥❣ ✐♥ ❬✺✾( ❛♥❞ •❛t❡r ❛❞❛♣t❡❞ t♦ ❝✉r✲ r❡♥ts ✐♥ ❬✸✺(✳ ❲❡ ❝❛♥ ♣r♦❝❡❡❞ ✐♥ ❛ s✐♠✐•❛r ✇❛② ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❈♦♥s✐❞❡r✐♥❣ ❛ ❞✐s❝r❡t✐③❡❞ ❢❝✉rr❡♥t C = i=1..N δ ξ i (x i ,m i ) ∈ W ′ ✳ ◆✱ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦♠❡♥t❛✱ ✐s ❛✉t♦♠❛t✐❝❛••② ❣✐✈❡♥ ❜② t❤❡ ♠❡s❤ ♦♥ t❤❡ s✉❜♠❛♥✐❢♦•❞ ✭♥✉♠❜❡r ♦❢ ❡❞❣❡s ❢♦r ❝✉r✈❡s✱ ♥✉♠❜❡r ♦❢ tr✐❛♥❣•❡s ❢♦r s✉r❢❛❝❡s✱✳✳✳✮✳ ❚❤✐s s✉❜♠❛♥✐❢♦•❞ ♠✐❣❤t ❤❛✈❡ s♦♠❡ ✈❡r② r❡❣✉•❛r r❡❣✐♦♥s ✇✐t❤ •♦✇✲❢r❡q✉❡♥❝② ❣❡♦♠❡tr✐❝❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ✈❛r✐❛t✐♦♥s✱ ✇❤✐❝❤ r❡s✉•ts ✐♥ ❛ ✈❡r② r❡❞✉♥❞❛♥t r❡♣r❡s❡♥t❛t✐♦♥ ❜② ❢❝✉rr❡♥ts ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ♠❛♥② ❛❞•❛❝❡♥t ♥♦❞❡s ♣r❡s❡♥t t❤❡ s❛♠❡ •♦❝❛• ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛•✳ ❚❤❡ ❣♦❛• ♦❢ ♠❛t❝❤✐♥❣✲♣✉rs✉✐t ✐s t♦ ✜♥❞ ❛ ♠♦r❡ ❛♣♣r♦♣r✐❛t❡ ❛♥❞ r❡❞✉❝❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ C ✐♥ t❡r♠s ♦❢ ❡•❡♠❡♥t❛r② ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ •✐✈❡♥ ❛ ❝❡rt❛✐♥ t❤r❡s❤♦•❞ ε > 0✱ ✇❡ ✇❛♥t t♦ ✜♥❞ Π n (C) s✉❝❤ t❤❛t C = Π n (C) + R n (C) ❛♥❞ R n (C) W ′ ε✳ R n (C) ✇✐•• ❜❡ ❝❛••❡❞ t❤❡ r❡s✐❞✉❛• ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❙♦♠❡❤♦✇✱ t❤✐s ✐s •✐♥❦❡❞ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ ✜♥❞✐♥❣ t❤❡ ❜❡st ♣r♦•❡❝t✐♦♥ ♦❢ C ♦♥ ❛ n✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ♦❢ W ′ ✳ ❚❤✐s ♣r♦❜•❡♠ ✐s ❤♦✇❡✈❡r t♦♦ t✐♠❡✲❝♦♥s✉♠✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛••② ❢♦r ✉s✉❛• ❛♣♣•✐❝❛t✐♦♥s✳ ■♥st❡❛❞✱ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t ✐s ❛ ❣r❡❡❞② ❛•❣♦r✐t❤♠ t❤❛t ❝♦♥str✉❝ts ❛ ❢❛♠✐•② ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ ✈❡❝t♦rs st❡♣ ❜② st❡♣✳ ❚❤❡ r❡s✉•t ✐s ❛ s✉❜♦♣t✐♠❛• ❢❝✉rr❡♥t t❤❛t ❛♣♣r♦①✐♠❛t❡s t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t C ✇✐t❤ ❛ r❡s✐❞✉❛• ✇❤♦s❡ ❡♥❡r❣② ✐s ❜❡•♦✇ t❤r❡s❤♦•❞✳ ❚❤❡ ❛•❣♦r✐t❤♠ ❜❛s✐❝❛••② ♣r♦❝❡❡❞s ❛s ❢♦•✲ •♦✇s✳ ❲❡ ♥❡❡❞ t♦ s♣❡❝✐❢② ❛ ✬❞✐❝t✐♦♥❛r②✬ D ♦❢ ❡•❡♠❡♥ts ✐♥ W ′ ✳ ■♥ ♦✉r ❝❛s❡✱ ✇❡ t②♣✐❝❛••② ❝♦♥s✐❞❡r t❤❡ s❡t ♦❢ ❛•• ❡•❡♠❡♥t❛r② ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts {δ ξ (x,m) } ✇✐t❤ ξ ❛ ✉♥✐t ✈❡❝t♦r ✼✷ ✐♥ Λ d E✳ ❚❤❡ ✜rst st❡♣ ♦❢ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t ❛•❣♦r✐t❤♠ ✐s t♦ ✜♥❞ δ ξ ′ 1 (x ′ 1 ,m ′ 1 ) ∈ D t❤❛t ✐s ❜❡st ❝♦rr❡•❛t❡❞ t♦ ❈✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ tr② t♦ ♠❛①✐♠✐③❡✱ ✇✐t❤ r❡s♣❡❝t t♦ x, m, ξ✱ t❤❡ q✉❛♥t✐t② ✿ C, δ ξ (x,m) W ′ = ξ T N i=1 K((x, m), (x i , m i ))ξ i . ✭✷✳✸✻✮ ❙✐♥❝❡ ξ ✐s t❛❦❡♥ ❛♠♦♥❣ ✉♥✐t ✈❡❝t♦rs✱ t❤❡ ♣r♦❜•❡♠ ✐s ❡q✉✐✈❛•❡♥t t♦ ♠❛①✐♠✐③✐♥❣ | i=1..N K((x, m), (x i , m i ))ξ i | = |γ(x, m)| ✇✐t❤ r❡s♣❡❝t t♦ (x, m) ❛♥❞ t❛❦❡ ξ ❛s t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ γ(x, m)✳ ❲❡ ❣❡t ❛ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ C ✿ C = Π 1 (C) + R 1 (C) . ✇❤❡r❡ Π 1 (C) = C, δ ξ ′ 1 (x ′ 1 ,m ′ 1 ) W ′ δ ξ ′ 1 (x ′ 1 ,m ′ 1 ) ✳ ❚❤❡ ❛•❣♦r✐t❤♠ t❤❡♥ ❛♣♣•✐❡s t❤❡ s❛♠❡ ♣r♦✲ ❝❡❞✉r❡ t♦ t❤❡ r❡s✐❞✉❛• R 1 (C)✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s ❛ s❡❝♦♥❞ ✈❡❝t♦r δ ξ ′ 2 (x ′ 2 ,m ′ 2 ) ∈ D✱ ❛♥❞ ❛ r❡s✐❞✉❛• R 2 (C)✳ ❚❤❡ ❛•❣♦r✐t❤♠ ✐s st♦♣♣❡❞ ✇❤❡♥ t❤❡ |❑❍❙ ♥♦r♠ ♦❢ t❤❡ r❡s✐❞✉❛• ❞❡❝r❡❛s❡s ❜❡•♦✇ t❤❡ ❣✐✈❡♥ t❤r❡s❤♦•❞ ε✳ ■♥ ♠♦st ❝❛s❡s✱ ✐t ❛♣♣❡❛rs t❤❛t t❤❡ ❝♦♠♣r❡ss✐♦♥ ✐s ❜❡tt❡r ✇✐t❤ t❤❡ ♦rt❤♦❣♦♥❛• ✈❡r✲ s✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s s❝❤❡♠❡✱ ✐♥ ✇❤✐❝❤ t❤❡ ❢❛♠✐•② ♦❢ ✈❡❝t♦rs ✐s ♦rt❤♦♥♦r♠❛•✐③❡❞ ❛t ❡❛❝❤ st❡♣✱ ✐♥ ♦r❞❡r t♦ ❢♦r❝❡ t❤❡ ♣r♦•❡❝t✐♦♥ ❛♥❞ t❤❡ r❡s✐❞✉❛• t♦ ❜❡ ♦rt❤♦❣♦♥❛• ✐♥ W ′ ✳ ❚❤❡ ❝•❛ss✐❝❛• ❛•❣♦r✐t❤♠ ✐s ❜❛s❡❞ ♦♥ ❛ •r❛♠✲❙❝❤♠✐❞t ♦rt❤♦♥♦r♠❛•✐③❛t✐♦♥ ❛t ❡❛❝❤ st❡♣✳ ■♥ ♦✉r ❝❛s❡✱ ❢♦••♦✇✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ❬✸✺(✱ ✐t ✐s ♣♦ss✐❜•❡ t♦ ♦rt❤♦❣♦♥❛•✐③❡ ♠♦r❡ ❡✣❝✐❡♥t•② ❜② ❦❡❡♣✐♥❣ t❤❡ ✈❛•✉❡s ♦❢ (x ′ i , m ′ i ) ❢♦✉♥❞ ❞✉r✐♥❣ ♣r❡✈✐♦✉s st❡♣s ❛♥❞ s✐♠♣•② ♠♦❞✐❢② t❤❡ ✈❡❝t♦rs ξ ′ i ✳ ❚❤✐s ✐s ❞♦♥❡ ❜② ✐♠♣♦s✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ ♦rt❤♦❣♦♥❛•✐t② ❝♦♥❞✐t✐♦♥✳ ▲❡t (e k ) ❜❡ t❤❡ ❝❛♥♦♥✐❝❛• ❜❛s✐s ♦❢ t❤❡ ✈❡❝t♦r s♣❛❝❡ Λ d E✳ ■❢ C = Π n (C) + R n (C) ❛♥❞ Π n (C) = i=1..n δ α n i (x ′ i ,m ′ i ) ✱ ✇❡ ✇✐•• ❛❞❞ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ❝♦♥str❛✐♥t ✿ δ e k (x ′ i ,m ′ i ) ⊥R n (C) ⇐⇒ C, δ e k (x ′ i ,m ′ i ) W * = Π n (C), δ e k (x ′ i ,m ′ i ) W * ❢♦r ❛•• ❜❛s✐s ✈❡❝t♦rs e k ❛♥❞ ❢♦r ❛•• i ∈ {1, .., n}✳ ■t ✐s t❤❡♥ str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝❤❡❝❦ t❤❛t t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛•❡♥t t♦ t❤❡ ❢♦••♦✇✐♥❣ s②st❡♠ ♦❢ •✐♥❡❛r ❡q✉❛t✐♦♥s t♦ ✜♥❞ t❤❡ α n i ✿ ∀i ∈ {1, .., n}, n j=1 K((x ′ i , m ′ i ), (x ′ j , m ′ j ))α n j k = γ(x ′ i , m ′ i ) k . ✭✷✳✸✼✮ ❲❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ♥♦r♠ ♦❢ t❤❡ r❡s✐❞✉❛• R n (C) ♠♦♥♦t♦♥✐❝❛••② ❞❡❝r❡❛s❡s t♦ ③❡r♦ ❛s n → ∞✳ ❍❡♥❝❡ t❤❡ ❛•❣♦r✐t❤♠ ❝♦♥✈❡r❣❡s ❛♥❞ ❡✈❡♥t✉❛••② ✇❤❡♥ t❤❡ r❡s✐❞✉❛• ❣♦❡s ❜❡•♦✇ t❤❡ ❣✐✈❡♥ t❤r❡s❤♦•❞ ❛t ❛ ❝❡rt❛✐♥ st❡♣ n✱ ✇❡ ♦❜t❛✐♥ ❛ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ C ✇✐t❤ n ♦rt❤♦❣♦♥❛• ❞✐r❛❝ ❢❝✉rr❡♥ts ✭✇✐t❤ ❣❡♥❡r❛••② n ≪ N ✱ ❛s ✇❡ s❤❛•• s❡❡ ♦♥ t❤❡ ❝♦♠✐♥❣ ❡①❛♠♣•❡s✮✳ ▲❡t✬s ❣✐✈❡ ❛ ❢❡✇ ❝♦♠♠❡♥ts ♦♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• ❝♦♠♣•❡①✐t② ♦❢ t❤❡ ❛•❣♦r✐t❤♠✱ ❛•✲ t❤♦✉❣❤ ♥✉♠❡r✐❝❛• ✐ss✉❡s r❡•❛t❡❞ t♦ ❢❝✉rr❡♥ts ✇✐•• ❜❡ ❞❡✈❡•♦♣❡❞ ✐♥ ✷✳✸✳✸✳ ❆t ❡❛❝❤ st❡♣✱ ✇❡ ❤❛✈❡ t♦ s♦•✈❡ ❛ •✐♥❡❛r s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❡q✳✭✷✳✸✼✮ ✇❤✐❝❤ ✐s 3n × 3n ✐❢ t❤❡ ❡♠✲ ❜❡❞❞✐♥❣ s♣❛❝❡ ✐s R 3 ✿ s✐♥❝❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❉✐r❛❝s ♦❢ t❤❡ ❝♦♠♣r❡ss❡❞ ✈❡rs✐♦♥ ♦❢ C✱ ✐t r❡♠❛✐♥s ❣❡♥❡r❛••② s♠❛••✳ ■♥st❡❛❞✱ t❤❡ ♠♦r❡ t❡❝❤♥✐❝❛• ❛♥❞ t✐♠❡✲❝♦♥s✉♠✐♥❣ ✼✸ ♣❛rt ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ✐s t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ |γ(x, m)| ✇✐t❤ r❡s♣❡❝t t♦ (x, m)✳ ■♥✲ ❞❡❡❞✱ t❤❡ ♦♣t✐♠✉♠ ♥❡❡❞ ♥♦t ❜❡ ♦♥❡ ♦❢ t❤❡ (x i , m i ) ❛♥❞ t❤✉s ✐t r❡q✉✐r❡s t♦ ❝♦♠♣✉t❡ γ(x, m) ❢♦r ♣♦✐♥ts ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ✐♥ E × M ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✭❜❡❝❛✉s❡ ✉s✐♥❣ •❛✉ss✐❛♥ ❦❡r♥❡•s ❢♦r ✐♥st❛♥❝❡ ♠❛❦❡s γ ❞❡❝r❡❛s❡s t♦ ③❡r♦ ✇❤❡♥ (x, m) ❣♦❡s ❢❛r ❢r♦♠ t❤❡ (x i , f i )✮✳ ❚❤✐s r❡❞✉❝❡s t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ s✉♠s ♦❢ ❦❡r♥❡•s ❢♦r ❡❛❝❤ ♣♦✐♥t (x, m)✱ ✇❤✐❝❤ ❤❛s ❛ ❜r✉t❡ ❢♦r❝❡ ❝♦♠♣•❡①✐t② ♦❢ n t✐♠❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❝❤♦s❡♥ ♥❡✐❣❤❜♦r❤♦♦❞✳ ❯s✉❛••②✱ ♦♥❡ ❝❤♦♦s❡s ❛ ❣r✐❞ ♦❢ ♣♦✐♥ts ✐♥ E × M ❛r♦✉♥❞ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✇❤♦s❡ st❡♣s ❞❡♣❡♥❞ ♦♥ t❤❡ s❝❛•❡s ♦❢ t❤❡ ❦❡r♥❡•s σ g ❛♥❞ σ f ✳ ❙✉❝❤ ❣r✐❞s ❝❛♥ ❤❛✈❡ ♥✉♠❡r♦✉s ♣♦✐♥ts ❡s♣❡❝✐❛••② ❢♦r s♠❛•• ❦❡r♥❡•s ✇❤✐❝❤ ♠❛❦❡s ❜r✉t❡ ❢♦r❝❡ ❝♦♠♣✉t❛t✐♦♥ ♥❡❛r•② ✐♠♣♦ss✐❜•❡ ✐♥ s♦♠❡ ❝❛s❡s✳ ■♥ ❬✷✽✱ ✸✺( ✇❛s ♣r♦♣♦s❡❞✱ ❢♦r ✉s✉❛• ❝✉rr❡♥ts✱ ❛♥ ❛•t❡r♥❛t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ ♠❡t❤♦❞✱ ❡①♣r❡ss✐♥❣ ❡✈❡r②t❤✐♥❣ ❛s ❝♦♥✈♦•✉t✐♦♥s ♦♥ t❤❡ ❣r✐❞ ♦❢ ♣♦✐♥ts✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t•② ✉s✐♥❣ ❋❋❚s✳ ❋♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ s✉❝❤ ♠❡t❤♦❞s st✐•• s✉✛❡r ❜♦t❤ ✐♥ ♠❡♠♦r② ❛♥❞ s♣❡❡❞ ❢r♦♠ t❤❡ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ❞✉❡ t♦ t❤❡ ❛❞•✉♥❝t✐♦♥ ♦❢ s✐❣♥❛•s✳ ❆ ♣♦ss✐❜•❡ ✇❛② t♦ ❤❛♥❞•❡ ✐t ✐s t♦ ♠❛❦❡ ❛•t❡r♥❛t✐✈❡ ♠❛①✐♠✐③❛t✐♦♥ ❜❡t✇❡❡♥ x ❛♥❞ m ✐♥ ❡q✳✭✷✳✸✻✮✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞♦♥❡ ❜② ✉s✐♥❣ s✉❝❝❡ss✐✈❡•② ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ❛♥❞ ♣✉r❡•② ❢✉♥❝t✐♦♥❛• ❣r✐❞s✳ ❲❡ r❡❢❡r t♦ ❢♦••♦✇✐♥❣ s❡❝t✐♦♥ ✷✳✸✳✸ ❢♦r ❛ ♠♦r❡ ❝♦♠♣•❡t❡ ❞✐s❝✉ss✐♦♥ ♦♥ t❤❡ ♥✉♠❡r✐❝❛• t♦♣✐❝✳ ❍❡r❡ ❛r❡ ♥♦✇ ❛ ❢❡✇ ✐••✉str❛t✐✈❡ ❡①❛♠♣•❡s ❢♦r r❡❛• ✈❛•✉❡❞ ❞❛t❛ ♦♥ ❝✉r✈❡s ♦r s✉r❢❛❝❡s✳ ❲❡ ✇✐•• ❛•✇❛②s ❝♦♥s✐❞❡r ❦❡r♥❡•s ♦♥ ❢❝✉rr❡♥ts t❤❛t ❛r❡ t❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ❛ •❛✉s✲ s✐❛♥ ❦❡r♥❡• ✐♥ R 3 ♦❢ s❝❛•❡ ♣❛r❛♠❡t❡r σ g ✇✐t❤ ❛ r❡❛• •❛✉ss✐❛♥ ❦❡r♥❡• ✐♥ t❤❡ s✐❣♥❛• s♣❛❝❡ ♦❢ s❝❛•❡ ♣❛r❛♠❡t❡r σ f ✳ ■♥ ✜❣✉r❡ ✷✳✶✵ ❛♥❞ ✷✳✶✶✱ ✇❡ ❡♠♣❤❛s✐③❡ t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ❜♦t❤ ❦❡r♥❡• s✐③❡s ♦♥ t❤❡ ❝♦♠♣r❡ss✐♦♥ ❢❛❝t♦r ❛s ✇❡•• ❛s ♦♥ t❤❡ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ♦❢ t❤❡ ❝♦♠♣r❡ss❡❞ s❤❛♣❡✳ ❚❤❡ ❜✐❣❣❡r t❤❡ ♣❛r❛♠❡t❡r σ g ✱ t❤❡ ❝♦❛rs❡r t❤❡ s❝❛•❡ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❛♥❞ ❢❡✇❡r ♣✉♥❝t✉❛• ❢❝✉rr❡♥ts ❛r❡ t❤❡r❡❢♦r❡ ♥❡❡❞❡❞ t♦ ❝♦♠♣r❡ss s❤❛♣❡s ❜✉t ♠♦r❡ s♠❛••❡r ❢❡❛t✉r❡s ❛r❡ •♦st✳ ■♥ ✜❣✉r❡ ✷✳✶✷✱ ✇❡ ❢♦❝✉s ♠♦r❡ ♣r❡❝✐s❡•② ♦♥ t❤❡ ❝♦♠♣r❡ss✐♦♥✬s ❜❡❤❛✈✐♦r ✇❤❡♥ ❝♦♠♣✉t✐♥❣ ♠❛t❝❤✐♥❣✲♣✉rs✉✐t ♦♥ ❛ s✐♠✉•❛t❡❞ ✜❜❡r ❜✉♥❞•❡ ♦❢ ✷❉ ❝✉r✈❡s ❝❛rr②✐♥❣ ❞✐✛❡r❡♥t s✐❣♥❛•s✳ ❚❤❡ s❝❛•❡ σ g ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ ✜❣✉r❡s ❜✉t ✇❡ s❤♦✇ t❤❡ r❡s✉•ts ♦❢ ♠❛t❝❤✐♥❣✲♣✉rs✉✐t ❢♦r t✇♦ r❛❞✐❝❛••② ❞✐✛❡r❡♥t ✈❛•✉❡s ♦❢ σ f ✳ ■♥ ❜♦t❤ ❝❛s❡s✱ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t ♣r♦✈✐❞❡s ❛♥ ❛❝❝✉r❛t❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥ ✭❛❝❝♦r❞✐♥❣•② t♦ t❤❡ ❦❡r♥❡• ♥♦r♠✮ ✇✐t❤ ❛ ✈❡r② •✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ❉✐r❛❝s ❝♦♠✲ ♣❛r❡❞ t♦ t❤❡ ♦r✐❣✐♥❛• s❛♠♣•✐♥❣✳ ❍♦✇❡✈❡r✱ ♥♦t❡ t❤❡ ✐♠♣♦rt❛♥t ✐♥✢✉❡♥❝❡ ♦❢ σ f ✿ t❛❦✐♥❣ ❛ •❛r❣❡r ✈❛•✉❡ ❢♦r t❤✐s ♣❛r❛♠❡t❡r ♠❡❛♥s t❤❛t t❤❡ r❛❞✐✉s ♦❢ ❛✈❡r❛❣✐♥❣ ❢♦r t❤❡ s✐❣♥❛• ♣❛rt ✐s ❤✐❣❤❡r✳ ■♥ ❝♦♥❝•✉s✐♦♥✱ t❤❡s❡ ✜rst ❡①❛♠♣•❡s ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ♣r♦❝❡ss✐♥❣ ✇❡r❡ ♠❡❛♥t t♦ ❤✐❣❤•✐❣❤t t❤❛t t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❢❝✉rr❡♥t✬s r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ t❤❡ ✉s❡ ♦❢ |❑❍❙ ♠❡tr✐❝s ♣r♦✈✐❞❡s ❛♥ ❡❛s② s♦•✉t✐♦♥ t♦ ❛❞❞r❡ss t❤❡ ✐ss✉❡ ♦❢ r❡❞✉♥❞❛♥❝② ❛♥❞ ❝♦♠♣r❡ss✐♦♥✳ ❚❤❡ ♠❡t❤♦❞ ♣r♦✈✐❞❡s ✐♠♣♦rt❛♥t ❝♦♠♣r❡ss✐♦♥ ❢❛❝t♦rs ❛♥❞ ❡♥❛❜•❡s s❝❛•❡ ❛♥❛•②s✐s ♦♥ ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛• t❤r♦✉❣❤ t❤❡ ❦❡r♥❡• ♣❛r❛♠❡t❡rs σ g ❛♥❞ σ f ✳ ✼✹ σ g = 0.04, σ f = 0.4,
❋✐❣✉r❡ ✷✳✶✵✿ ▼❛t❝❤✐♥❣ ♣✉rs✉✐t ♦♥ ❛ ✏♣❛✐♥t❡❞✑ ❜✉♥♥② ✇✐t❤ ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡rs σ g ❛♥❞ σ f ✳ •❡♦♠❡tr✐❝❛••②✱ t❤❡ s✉r❢❛❝❡ ❤❛s 0.16 × 0.22 × 0.12 ❡①t❡♥s✐♦♥ ✐♥ t❤❡ ✸❉ s♣❛❝❡ ❛♥❞ t❤❡ s✐❣♥❛• ❣♦❡s ❢r♦♠ ✈❛•✉❡ ③❡r♦ ✭❜•✉❡✮ t♦ ♦♥❡ ✭r❡❞✮✳ ❚❤❡ ♦r✐❣✐♥❛• s❛♠♣•✐♥❣ ♦❢ t❤❡ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ❤❛s ✻✾✹✺✶ ❉✐r❛❝s ❛♥❞ ✇❡ ❝❤♦♦s❡ ❛ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ❢♦r t❤❡ ❛•❣♦r✐t❤♠ ♦❢ ǫ = 5%✳ ❚❤❡ r❡s✉•t✐♥❣ ❉✐r❛❝ ❢❝✉rr❡♥ts δ ξ k (x k ,m k ) ❛r❡ ❤❡r❡ r❡♣r❡s❡♥t❡❞ ❛s ❝♦•♦r❡❞ ✈❡❝t♦rs ❛❝❝♦r❞✐♥❣•② t♦ t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s m k ✳ ❱❡❝t♦rs ❛r❡ ❛•• ♦❢ s❛♠❡ •❡♥❣t❤ ❝♦✈❡r✐♥❣ ❛♥ ❛r❡❛ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ♥♦r♠ ♦❢ ξ k ✳ ◆♦t✐❝❡ t❤❛t t❤❡ s❛♠♣•✐♥❣ ✐♥❝r❡❛s❡s ❛s σ g ✐s s♠❛••❡r ✇❤✐•❡ t❤❡ ✈❡❝t♦r✬s ❝♦•♦rs ❛r❡ ♠♦r❡ ❛❝❝✉r❛t❡ ✇❤❡♥ σ f ✐s s♠❛••❡r✳ σ g = 0.04, σ f = 0.4 σ g = 0.01, σ f = 0.1 ❋✐❣✉r❡ ✷✳✶✶✿ ❈•♦s❡ ✉♣ ♦♥ t✇♦ ♦❢ t❤❡ ♣r❡✈✐♦✉s r❡s✉•ts✳ ✷✳✸✳✷ |❡❣✐str❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❆ s❡❝♦♥❞ ❝•❛ss ♦❢ ❛♣♣•✐❝❛t✐♦♥s t❤❛t ❝❛♥ ❜❡ ❡❛s✐•② ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❢✉♥❝✲ t✐♦♥❛• ❝✉rr❡♥ts ✐s t❤❡ ♣r♦❜•❡♠ ♦❢ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❚❤✐s ✐s ❛ r❡❝✉rr❡♥t ✐ss✉❡ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② t❤❛t ✐s ♦❢t❡♥ ❛ ♣r❡•✐♠✐♥❛r② st❡♣ ❢♦r ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s ❛♥❞ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ t❤❡ ✈❛r✐❛❜✐•✐t② ✭❝❢ ❝❤❛♣t❡r ✹✮✳ ❚❤❡ ❞✐st❛♥❝❡s r❡s✉•t✐♥❣ ❢r♦♠ ❦❡r♥❡• ♠❡tr✐❝s ♦♥ ❢❝✉rr❡♥ts t❤❛t ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❝❛♥ ♦❜✈✐♦✉s•② ♣•❛② t❤❡ r♦•❡ ♦❢ ❛ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ ♦❜•❡❝ts ❛♥❞ ❜❡ ✉s❡❞ ✐♥ ❜❛s✐❝❛••② ❛♥② ✐♥❡①❛❝t ♠❛t❝❤✐♥❣ ❛♣♣r♦❛❝❤✳ ❚❤❡ ♦♥❡ ✇❡ ❢♦❝✉s ♦♥✱ ✐♥ t❤✐s t❤❡s✐s✱ ✼✺ ❋✐❣✉r❡ ✷✳✶✷✿ ▼❛t❝❤✐♥❣ ♣✉rs✉✐t ♦♥ ❛ ✷❉ ✜❜❡r ❜✉♥❞•❡✱ ❡❛❝❤ ✜❜❡r ❝❛rr②✐♥❣ ♦♥❡ ✈❛•✉❡ ♦❢ s✐❣♥❛• r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❝♦•♦r✳ ❖♥ t♦♣✱ t❤❡ ✐♥✐t✐❛• ♦❜•❡❝t ❝♦♥s✐st✐♥❣ ♦❢ ✸✵✵ ✜❜❡rs✳ ❇❡•♦✇✱ ✇❡ s❤♦✇ t✇♦ r❡s✉•ts ♦❢ ♠❛t❝❤✐♥❣✲♣✉rs✉✐t ✇✐t❤ t❤❡ s❛♠❡ σ g ❜✉t t✇♦ ❞✐✛❡r❡♥t ✈❛•✉❡s ❢♦r σ f ✿ σ f = 200 ❢♦r t❤❡ •❡❢t ✜❣✉r❡✱ σ f = 20 ❢♦r t❤❡ r✐❣❤t ♦♥❡✳ ✐s t❤❡ ▲❉❉▼▼ s❡tt✐♥❣ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✳ ❚❤✐s ✇✐•• •❡❛❞ t♦ ❛ ❣❡♥❡r❛•✐③❛✲ t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ❛♥❞ s✉r❢❛❝❡ r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ ❝✉rr❡♥ts t♦ t❤❡ ✇✐❞❡r ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❆•t❤♦✉❣❤✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞ s❡✈❡r❛• t✐♠❡s✱ t❤❡r❡ ❛r❡ ♥♦ t❤❡♦r❡t✐❝❛• •✐♠✐t❛t✐♦♥s t♦ ❛♣♣•② ❢❝✉rr❡♥ts ✐♥ ✈❛r✐♦✉s ❝❛s❡s ♦❢ s✐❣♥❛•s✱ t❤❡ ♥✉♠❡r✐❝❛• ✐♠♣•❡♠❡♥t❛t✐♦♥s t❤❛t ❤❛✈❡ ❜❡❡♥ ❞♦♥❡ ❛r❡ r❡str✐❝t❡❞ t♦ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s ✐♥ ❜❛s✐✲ ❝❛••② t✇♦ s✐t✉❛t✐♦♥s ✿ ♣✉r❡•② ❞✐✛❡♦♠♦r♣❤✐❝ r❡❣✐str❛t✐♦♥ ♦❢ t❤❡ s✉♣♣♦rt ✭❜❛s❡❞ ♦♥ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♠♦❞❡• ♦❢ ❡q✳✭✷✳✶✮✮ ❛♥❞ ✇✐t❤ ❛❞•✉♥❝t✐♦♥ ♦❢ ❛ r❡s✐❞✉❛• s✐❣♥❛• ✭♠♦❞❡•❡❞ ❛s ✐♥ ❡q✳✭✷✳✷✮✮✳ ❚❤❡ ❡ss❡♥t✐❛• ♣r❛❝t✐❝❛• ❞✐✛❡r❡♥❝❡ ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠ ♦♥ ❝✉r✲ r❡♥ts ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♥❡✇ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ❛♥❞ t❤❡✐r ❣r❛❞✐❡♥ts✳ ❲❡ ❞❡t❛✐• t❤❛t ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣❛r❛❣r❛♣❤✳ ❉✐st❛♥❝❡s ❛♥❞ ❣r❛❞✐❡♥ts ✐♥ t❤❡ ❞✐s❝r❡t❡ s❡tt✐♥❣ ▲❡t✬s ❝♦♥s✐❞❡r ❞✐s❝r❡t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✿ ❛ s♦✉r❝❡ (X, f ) ❣✐✈❡♥ ❜② t❤❡ s❡t ♦❢ ♣♦✐♥ts (q, f ) = (q i , f i ) i=1,..,n ✱ q i ∈ E✱ f i ∈ M = R ❛♥❞ ❛ ♠❡s❤ ❝♦♥t❛✐♥✐♥❣ t❤❡ •✐st ♦❢ ❢❛❝❡s ✭s❡❣♠❡♥ts ❢♦r ❝✉r✈❡s✱ tr✐❛♥❣•❡s ❢♦r s✉r❢❛❝❡s✮ ✇❤✐❝❤ ❝❛♥ ❜❡ t❛❦❡♥✱ ✐♥ ♣r❛❝t✐❝❡✱ t♦ ❜❡ ❛ •✐st ♦❢ (d + 1)✲♦r✐❡♥t❡❞ s✐♠♣•❡①❡s (α 1 i , .., α d+1 i ) i=1,..,m ✇✐t❤ α k i ✐♥❞❡①❡s ✐♥ {1, .., n}✱ ❛ t❛r❣❡t (Y, g) ✇✐t❤ ♣♦✐♥ts (y j , g j ) j=1,..,r ❣✐✈❡♥ ❛•s♦ ✇✐t❤ ♠❡s❤ (β 1 i , .., β d+1 i ) i=1,..,s ✳ ❋r♦♠ t❤✐s✱ ♦♥❡ ❤❛s ✜rst t♦ ❝♦♠♣✉t❡ t❤❡ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥s C (X,f ) = m i=1 δ ξ i (q i , fi ) ❛♥❞ ✼✻ C (Y,g) = s j=1 δ η j (ỹ j ,g j ) ♦❢ ❡❛❝❤ s❤❛♣❡✳ ■♥❞❡❡❞✱ t❤❡ ✈❛r✐❛❜•❡s qi ✱ fi ❛♥❞ ξ i ❛r❡ t❤❡♠s❡•✈❡s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♣♦✐♥ts (q i ) ❛♥❞ s✐❣♥❛•s (f i ) ♦❢ (X, f ) r❡s♣❡❝t✐✈❡•② ❛s t❤❡ ❝❡♥t❡rs✱ ✐♥t❡r♣♦•❛t❡❞ s✐❣♥❛• ✈❛•✉❡ ❛♥❞ ♥♦r♠❛• ✈❡❝t♦r ♦❢ t❤❡ i th ❢❛❝❡✳ ▼♦r❡ ❡①♣•✐❝✐t•②✱ ❢♦r ❛♥② i = 1, .., m ✿          qi = 1 d + 1 (q α 1 i + ... + q α d+1 i ) fi = 1 d + 1 (f α 1 i + ... + f α d+1 i ) ξ i = Λ d h=1 (q α h i -q α d+1 i ) ✭✷✳✸✽✮ ❛♥❞ s✐♠✐•❛r ❡q✉❛t✐♦♥s ❤♦•❞ ❢♦r t❤❡ t❛r❣❡t s❤❛♣❡ (Y, g)✳ ◆♦✇✱ t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ t❤❛t s❤❛•• ❜❡ ✉s❡❞ ✐♥ t❤❡ ❛•❣♦r✐t❤♠s ✐s ♥❛t✉r❛••② C (X,f ) -C (Y,g) 2 W ′ ✇❤✐❝❤ ✇r✐t❡s✱ ❛s ✐♥ ❡q✳✭✷✳✶✽✮ ✿ C (X,f ) -C (Y,g) 2 W ′ = m i=1 m j=1 k f ( fi , fj ). K g (q i , qj )ξ i , ξ j + s k=1 s l=1 k f (g k , gl ). K g (ỹ k , ỹl )η k , η l -2 m i=1 s k=1 k f ( fi , gk ). K g (q i , ỹk )ξ i , η k ✭✷✳✸✾✮ ❈♦♥s✐❞❡r✐♥❣ ♦♥•② t❤❡ ❞❡♣❡♥❞❡♥❝② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s♦✉r❝❡ ❢s❤❛♣❡✱ t❤❡ ♣r❡✈✐♦✉s ❞✐st❛♥❝❡ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛❜•❡s qi , fi , ξ i t❤❛t ✇❡ ✇✐•• ❝❛•• Ã✳ ❇② ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❡q✳✭✷✳✸✽✮✱ ✇❡ s❡❡ t❤❛t Ã(q, f , ξ) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ q ❛♥❞ f t❤❛t ✇❡ ✇✐•• ❞❡♥♦t❡ A(q, f )✳ ❋♦r r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠✱ ♦♥❡ ❛•s♦ ♥❡❡❞s t❤❡ ❣r❛❞✐❡♥ts ♦❢ A ✇✐t❤ r❡s♣❡❝t t♦ q ❛♥❞ f ✳ ❚❤♦s❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✉s✐♥❣ ❝❤❛✐♥ r✉•❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ s✐♥❝❡ ✿        ∂ q k A = n i=1 ∂ qi à • ∂ q k qi + ∂ ξ i à • ∂ q k ξ i ∂ f k A = n j=1 ∂ fi à • ∂ f k fi ✭✷✳✹✵✮ ◆♦✇✱ t❤❡ ❣r❛❞✐❡♥ts ∂ qi Ã, ∂ fi à ❛♥❞ ∂ ξ i à ❛r❡ ❝♦♠♣✉t❡❞ ❡❛s✐•② ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ ❡q✳✭✷✳✸✾✮✱ ✇❤✐•❡ t❤❡ ∂ q k qi ✱ ∂ f k fi ❛♥❞ ∂ q k ξ i ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ❡q✳✭✷✳✸✽✮✳ ❖♥❡ ♥♦t❛❜•❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❡①♣r❡ss✐♥❣ t❤❡ ❣r❛❞✐❡♥ts ✐♥ t❤✐s ✇❛② ✐s t❤❛t ✐t s❡♣❛r❛t❡s ✇❤❛t ✐♥✈♦•✈❡s t❤❡ ♣❛ss❛❣❡ t♦ ❢❝✉rr❡♥ts✬ ✈❛r✐❛❜•❡s ❢r♦♠ ✇❤❛t ✐♥✈♦•✈❡s ♦♥•② ❦❡r♥❡•s✬ ❝♦♠♣✉t❛t✐♦♥s✱ ❛••♦✇✐♥❣ t♦ ✇r✐t❡ ♥✉♠❡r✐❝❛• ❝♦❞❡s ✇❤✐❝❤ ❛r❡ ✈❡r② ❛❞❛♣t❛❜•❡ t♦ ✈❛r✐♦✉s ❞✐♠❡♥s✐♦♥s ❛♥❞ t♦ t❤❡ ✉s❡ ♦❢ ❞✐✛❡r❡♥t ❦❡r♥❡•s✳ |❡❣✐str❛t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝❛• ❞❡❢♦r♠❛t✐♦♥s ◆♦✇ t❤❛t t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❛♥❞ ✐ts ❣r❛❞✐❡♥ts ❛r❡ ❡①♣•✐❝✐t✱ t❤❡ ✇❤♦•❡ ▲❉❉▼▼ ♠❛❝❤✐♥❡r② ❞❡t❛✐•❡❞ ✐♥ ❝❤❛♣t❡r ✶ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞•②✳ ■♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ s❤❛•• ❜❡ ❞❡❛•✐♥❣ ✇✐t❤ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ ✇❡ ✐♥t❡♥❞ t♦ ❞❡❢♦r♠ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ♦♥•② ✇✐t❤ ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ E✱ ✇❤✐❝❤ ✐s t❤❡ ♠♦❞❡• ♦❢ tr❛♥s♣♦rt ❣✐✈❡♥ ✐♥ ❡q✳✭✷✳✶✮✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ✐♥ ▲❉❉▼▼ ✇r✐t❡s ✐♥ ❣❡♥❡r❛• ✿ min v∈L 2 V J(v) . = 1 0 v t 2 V dt + γ C (φ 1 (X),f •(φ 1 ) -1 ) -C (Y,g) 2 W ′ . ✭✷✳✹✶✮ ✼✼ ❋✐❣✉r❡ ✷✳✶✸✿ ❊①❛♠♣•❡ ♦❢ r❡❣✐str❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥❛• ❝✉r✈❡s ✭t♦♣ •❡❢t✮ ✇✐t❤ ❜✐♥❛r② s✐❣♥❛• ✭❜•✉❡ ✐s ③❡r♦ ❛♥❞ r❡❞ ✐s ♦♥❡✮✳ ❖♥ t♦♣ r✐❣❤t✱ ✇❡ s❤♦✇ t❤❡ ❝•❛ss✐❝❛• ♠❛t❝❤✐♥❣ ✇✐t❤ ❝✉rr❡♥ts ♦♥ t❤❡ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ❝✉r✈❡s✳ ❖♥ ❜♦tt♦♠ •❡❢t✱ t❤❡ s❛♠❡ ❝✉r✈❡s ❛r❡ ♠❛t❝❤❡❞ ✇✐t❤ ♦✉r ❡①t❡♥s✐♦♥ ♦❢ ▲❉❉▼▼ t♦ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ■♥ ❜♦t❤ ❝❛s❡s✱ t❤❡ ❞❡❢♦r♠❡❞ ❝✉r✈❡ ✜ts ❝•♦s❡•② t♦ t❤❡ t❛r❣❡t ♦♥❡ ❜✉t ♥♦t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ✜❡•❞ ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥t✬s ❛♣♣r♦❛❝❤✳ ❋✐♥❛••②✱ ♦♥ t❤❡ r✐❣❤t✱ ✇❡ s❤♦✇ t❤❡ r❡s✉•t ♦❢ ♠❛t❝❤✐♥❣ ✇❡ ♦❜t❛✐♥ ❛❣❛✐♥ ✇✐t❤ ❢❝✉rr❡♥ts✬ ▲❉❉▼▼ ❜✉t ✇✐t❤ ❛ ❜✐❣ ✈❛•✉❡ ♦❢ σ f ❝♦♠♣❛r❡❞ t♦ t❤❡ s✐❣♥❛•✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ t❤❡ ♠❛t❝❤✐♥❣ ✐s ♥❡❛r•② s✐♠✐•❛r t♦ t❤❡ ❝✉rr❡♥t ♠❛t❝❤✐♥❣✳ ✼✽ ❇❛s❡❞ ♦♥ t❤❡ ❝♦♥tr♦• r❡s✉•t ♦❢ ♣r❡✈✐♦✉s s❡❝t✐♦♥s✱ ♦♥❡ ❝❛♥ s❤♦✇ ❡①✐st❡♥❝❡ ♦❢ s♦•✉t✐♦♥s t♦ t❤✐s ✈❛r✐❛t✐♦♥❛• ♣r♦❜•❡♠ ✿ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✶✳ ■❢ W g ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 1 0 (E, Λ d E * ) ❛♥❞ V ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 2 0 (E, E) t❤❡♥✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡s (X, f ) ❛♥❞ (Y, g)✱ t❤❡r❡ ❡①✐sts ❛ ♠✐♥✐♠✉♠ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ ❡q✳✭✷✳✹✶✮✳ Pr♦♦❢✳ ❚❤❛♥❦s t♦ t❤❡ r❡s✉•t ♦❢ ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✶ ❢r♦♠ ❝❤❛♣t❡r ✶✱ ✇❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❛♣♣•✐❝❛t✐♦♥ v → C (φ v 1 (X),f •(φ v 1 ) -1 ) -C (Y,g) 2 W ′ . = A(φ v 1 ) ✐s ✇❡❛❦•② ❝♦♥t✐♥✉♦✉s ♦♥ L 2 V ✳ ❇✉t t❤✐s ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❝♦r♦••❛r② ✷✳✷✳✷ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ◆♦✇✱ ✐♥ t❤❡ ❞✐s❝r❡t❡ s❡tt✐♥❣✱ ❛s ✇❡ ♣r❡s❡♥t❡❞ ✐♥ ✶✳✸✳✷✱ t❤✐s ♣r♦❜•❡♠ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❛♥ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r t❤❡ ✐♥✐t✐❛• ♠♦♠❡♥t✉♠ p 0 ✱ ✐✳❡ ✿ min p 0 J(p 0 ) . = E(p 0 ) + γ C (φ 1 (X),f •(φ 1 ) -1 ) -C (Y,g) 2 W ′ . ✭✷✳✹✷✮ ✇❤❡r❡ E(p 0 ) ✐s t❤❡ ❡♥❡r❣② ✭♦r ❡q✉✐✈❛•❡♥t•② t❤❡ r❡❞✉❝❡❞ ❍❛♠✐•t♦♥✐❛♥✮ ♦❢ t❤❡ ❞❡❢♦r✲ ♠❛t✐♦♥ ✇❤✐❝❤ ❡q✉❛•s H r (q, p 0 ) = p T 0 K V (q, q)p 0 ✐❢ t❤❡ ✈❡❝t♦r ✜❡•❞s ❛r❡ ❣❡♥❡r❛t❡❞ ❜② ❛ ❦❡r♥❡• K V ✱ φ 1 ✐s t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❛t t✐♠❡ ✶ ♦❜t❛✐♥❡❞ ❜② ✢♦✇✐♥❣ ❢♦r✇❛r❞ t❤❡ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ❡q✉❛t✐♦♥s ❛♥❞ γ ✐s t❤❡ tr❛❞❡✲♦✛ ♣❛r❛♠❡t❡r✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♣❛rt ✭❜❛s✐❝❛••② t❤❡ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❞✐✛❡r✲ ❡♥t✐❛• ❡q✉❛t✐♦♥s✮ r❡♠❛✐♥ t♦t❛••② ✐❞❡♥t✐❝❛• t♦ t❤❡ ✉s✉❛• ❝✉rr❡♥ts✬ ❝❛s❡ ❜❡❝❛✉s❡ ✐♥ t❤✐s ♠♦❞❡• ♦❢ ❞❡❢♦r♠❛t✐♦♥✱ t❤❡ tr❛♥s♣♦rt ♦❢ ❢✉♥❝t✐♦♥❛• ✐♥❢♦r♠❛t✐♦♥ ✐s tr✐✈✐❛• ✭✐✳❡ f t ✐s ❝♦♥✲ st❛♥t ❛♥❞ ❡q✉❛• t♦ f ❛•♦♥❣ t❤❡ ❣❡♦❞❡s✐❝✮✳ ◆♦t❡ t❤❛t ❢♦r ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞ ❛❝t✐♦♥s •✐❦❡ t❤❡ ♦♥❡ ♦♥ ✈❡❝t♦r ✜❡•❞s ♦r t❡♥s♦rs✱ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥s ✇♦✉•❞ ❛•s♦ ❛✛❡❝t t❤❡ s✐❣♥❛• ♣❛rt✳ ❆♥ ❛❞❛♣t✐✈❡ st❡♣ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ✐s t❤❡♥ ❞♦♥❡ ♦♥ t❤❡ ♠♦♠❡♥t✉♠ p 0 = (p 0,i ) i=1,..,n ✳ ❚❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ p 0 ✐s ♦❜t❛✐♥❡❞ ❜② ✜rst ✢♦✇✐♥❣ t❤❡ ❢♦r✇❛r❞ ❍❛♠✐•t♦♥✐❛♥ ❡q✉❛t✐♦♥ ✇❤✐❝❤ ❣✐✈❡s t❤❡ ♣♦✐♥t ♣♦s✐t✐♦♥s q t ❛♥❞ t❤❡ ♠♦♠❡♥t✉♠ p t ❛t ❛•• t✐♠❡s t ∈ [0, 1]✱ t❤❡ s✐❣♥❛•s f t r❡♠❛✐♥✐♥❣ ❝♦♥st❛♥t•② ❡q✉❛•s t♦ f ✳ ❚❤❡♥✱ ✇❡ ♦❜t❛✐♥ ✿ ∇ p 0 J = K V (q, q)p 0 -P 0 ✇✐t❤ P 0 ❛♥ ❛❞•♦✐♥t ✈❛r✐❛❜•❡ ♦❜t❛✐♥❡❞ ❜② ✢♦✇✐♥❣ ❜❛❝❦✇❛r❞ ✐♥ t✐♠❡ t❤❡ ❝♦✉♣•❡❞ ❡q✉❛✲ t✐♦♥ ✿    Qt = -(∂ 2 q,p H r (q t , p t )) * Q t + (∂ 2 q,q H r (q t , p t )) * P t Ṗt = -(∂ 2 p,p H r (q t , p t )) * Q t + (∂ 2 q,p H r (q t , p t )) * P t Q 1 = -∇ q 1 A(q 1 , f ), P 1 = 0 ✭✷✳✹✸✮ ✇❤❡r❡ ∇ q 1 A(q 1 , f ) ✐s ❜❛s✐❝❛••② t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❜❡✲ t✇❡❡♥ t❤❡ ❞❡❢♦r♠❡❞ s♦✉r❝❡ ❢s❤❛♣❡ ❛♥❞ t❤❡ t❛r❣❡t✱ ❝♦♠♣✉t❡❞ ❛s ✐♥ ✷✳✸✳✷✳ ❚❤❡ ✇❤♦•❡ ♣r♦❝❡❞✉r❡ ✇❛s ❝♦❞❡❞ ✐♥ ▼❆❚▲❆❇ ❢♦r ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ ✇✐t❤ s♦♠❡ ♣❛rts ✐♥ ❈ ❛♥❞ •P❯✱ ❛s ✇❡ ✇✐•• ❡①♣•❛✐♥ ✐♥ ✷✳✸✳✸✳ ❲❡ ♣r❡s❡♥t ❛ ❢❡✇ r❡s✉•ts ♦❢ t❤✐s ❛•❣♦r✐t❤♠✳ ❆s ✇❡ ❝❛♥ ❡①♣❡❝t ✇✐t❤ t❤✐s ♠♦❞❡•✱ t❤❡ r❡s✉•t✐♥❣ ♠❛t❝❤✐♥❣ ✐s ❞r✐✈❡♥ ❜♦t❤ ❜② t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ s❤❛♣❡s ❛♥❞ ❜② t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s t❤❡② ❝❛rr②✱ ✇❤✐❝❤ ✇❡ ✜rst s❤♦✇ ♦♥ t❤❡ ✈❡r② s✐♠♣•❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✷✳✶✸✳ ❲❡ ❛•s♦ s❡❡ ✐♥ t❤❛t ✜❣✉r❡ t❤❛t t❤❡ ❛tt❛❝❤♠❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛•s ✐s ❝♦♥tr♦••❡❞ ❜② t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡r♥❡• k f ✿ ✐♥❞❡❡❞✱ ❛t ✈❡r② ❝♦❛rs❡ s❝❛•❡ ❢♦r σ f ✱ t❤❡ ❢✉♥❝t✐♦♥❛• ❛r❡ ♥♦ ♠♦r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❛♥❞ t❤❡ ❛♣♣r♦❛❝❤ ✐s t❤✉s ✐❞❡♥t✐❝❛• t♦ ✉s✉❛• ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ❝✉rr❡♥ts✳ ◆♦✇✱ ✇❡ ❞r❛✇ ❛tt❡♥t✐♦♥ t♦ t❤❡ ✐♠♣♦rt❛♥t ❞✐✛❡r❡♥❝❡s t❤❛t ❡①✐sts ✇❤❡♥ ♠❛t❝❤✐♥❣ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥st❡❛❞ ♦❢ t❤❡ ♦t❤❡r ♣♦ss✐❜•❡ s♣❛❝❡s ♦❢ ❝✉rr❡♥ts ❞✐s❝✉ss❡❞ ✐♥ ✷✳✶✳✸✳ ❚❤✐s ✐s ♣❛rt✐❝✉•❛r•② ❡❛s② t♦ ❤✐❣❤•✐❣❤t ✇✐t❤ ❝♦•♦r❡❞ ❝✉rr❡♥ts ✿ ✐❢ ✇❡ r❡❢❡r ❛❣❛✐♥ t♦ ✼✾ ❋✐❣✉r❡ ✷✳✶✹✿ ❇❛❝❦ t♦ t❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✷✳✻✱ t❤❡ ♠❛t❝❤✐♥❣ ✐s ❤❡r❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ t❤❡ s✉r❢❛❝❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✷✳✹✱ ✇❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥tr♦❞✉❝❡ ♥♦ ❜✐❛s ❜❡t✇❡❡♥ •♦✇ ❛♥❞ ❤✐❣❤ s✐❣♥❛•s ❛s ♦♣♣♦s❡❞ t♦ ❝♦•♦r❡❞ ❝✉rr❡♥ts✳ ■t ❛•s♦ ❣✐✈❡s s✐❣♥✐✜❝❛♥t•② ❞✐✛❡r❡♥t r❡s✉•ts ❢r♦♠ ❝✉rr❡♥ts ✐♥ t❤❡ ♣r♦❞✉❝t E×M ✐♥ ❝❡rt❛✐♥ s✐t✉❛t✐♦♥s✱ ❛s t❤❡ ❡①❛♠♣•❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✶✹✳ ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♣r♦✈✐❞❡ r♦❜✉st♥❡ss t♦ ❞✐s❝♦♥♥❡❝t✐♦♥s ♦❢ t❤❡ s❤❛♣❡ ❛♥❞ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ t❤❡ s✐❣♥❛•✳ ❚❤✐s ✐s ❛ ♣❛rt✐❝✉•❛r•② ✐♠♣♦rt❛♥t ♣r♦♣❡rt② ✇❤❡♥ ✇♦r❦✐♥❣ ✇✐t❤ ✜❜❡r ❜✉♥❞•❡ t②♣❡ ♦❢ ❞❛t❛ •✐❦❡ t❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✷✳✶✺✳ ❙♦♠❡ ❛❞❞✐t✐♦♥❛• r❡s✉•ts ♦♥ ♠❡❞✐❝❛• ✐♠❛❣✐♥❣ ❞❛t❛s❡ts ✇✐•• ❜❡ st✉❞✐❡❞ ✐♥ ❞❡t❛✐• ✐♥ ❝❤❛♣t❡r ✹✱ t❤❛t ❣❡♥❡r❛•✐③❡s t❤✐s s❡tt✐♥❣ t♦ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❛ ❣r♦✉♣ ♦❢ s✉❜•❡❝ts✳ |❡❣✐str❛t✐♦♥ ✇✐t❤ ❛❞❞✐t✐♦♥❛• ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s✬ ❡st✐♠❛t✐♦♥ ❚❤❡ ❞❡❢♦r♠❛t✐♦♥ ♠♦❞❡• ✉♥❞❡r•②✐♥❣ t❤❡ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠ ♦♥•② tr❛♥s♣♦rts ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ❜② ❞✐✛❡♦♠♦r♣❤✐s♠✳ ■t ✐s ❝•❡❛r t❤❛t ✐♥ ♠❛♥② s✐t✉❛t✐♦♥s✱ s♦♠❡ ✈❛r✐❛t✐♦♥s ✐♥ t❤❡ ♣❛tt❡r♥s ♦❢ t❤❡ s✐❣♥❛•s ♠✐❣❤t ❡①✐st ❜❡t✇❡❡♥ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ✇❤✐❝❤ ❛r❡ ♥♦t t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ♥♦r ❡st✐♠❛t❡❞ ❜② t❤❡ ❛•❣♦r✐t❤♠✳ ❚❤✐s ❝❛♥ ❛•s♦ ❞❡t❡r✐♦r❛t❡ t❤❡ ❣❡♦♠❡tr✐❝❛• ❞❡❢♦r♠❛t✐♦♥s t❤❡♠s❡•✈❡s s✐♥❝❡ r❡❣✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t s✐❣♥❛• ✈❛•✉❡s ✇♦✉•❞ ❤❛✈❡ ❛ t❡♥❞❡♥❝② t♦ ❜❡ ♠✐s♠❛t❝❤❡❞✳ ❖♥❡ s♦•✉t✐♦♥ t♦ ❛❞❞r❡ss s✉❝❤ s✐t✉❛t✐♦♥s ✐s t♦ ✐♥tr♦❞✉❝❡ ❛♥ ❡♥r✐❝❤❡❞ ❞❡❢♦r♠❛t✐♦♥ ❣r♦✉♣ t❤❛t ✇♦✉•❞ ✐♥✈♦•✈❡✱ ✐♥ ❝♦♠♣•❡♠❡♥t t♦ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ ❛❝t✐♥❣ ♦♥ t❤❡ s✉♣♣♦rt✱ ❛ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥ t❤❛t ❛❞❞s t♦ t❤❡ s♦✉r❝❡ s❤❛♣❡✳ ❲❡ ❝❛♥ ♠♦❞❡• t❤✐s ❡①❛❝t•② ❜② t❤❡ tr❛♥s♣♦rt ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❡①♣r❡ss❡❞ ✐♥ ❡q✳✭✷✳✷✮ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝t✐♦♥ ♦♥ ❢❝✉rr❡♥ts ♣r♦✈✐❞❡❞ ❜② ❡q✳✭✷✳✶✷✮✳ ■♥ t❤✐s s❡tt✐♥❣✱ t❤❡ ❣♦❛• ✇♦✉•❞ ❜❡ t♦ ❡st✐♠❛t❡ ❜♦t❤ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ φ ❛s ♣r❡✈✐♦✉s•②✱ ❛♥❞ t❤❡ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥ h✱ ❜② s♦•✈✐♥❣ t❤❡r❡❢♦r❡ ❛♥ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ♦❢ t❤❡ ❢♦r♠ ✿ min p 0,h E(p 0 ) + γ 1 C (φ 1 (X),(f +h)•(φ 1 ) -1 ) -C (Y,g) W ′ + γ 2 X h 2 (x)dH d (x) ✭✷✳✹✹✮ ❖❜s❡r✈❡ t❤❛t ❛ ♣❡♥❛•t② t❡r♠ ♦♥ t❤❡ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥ h ✐s ❛❞❞❡❞ t♦ t❤❡ ❝r✐t❡r✐♦♥ ✐♥ ♦r❞❡r t♦ ✐♠♣♦s❡ ❛ ❝♦♥str❛✐♥t ♦♥ t❤✐s ❢✉♥❝t✐♦♥ h ✇❡ ✐♥t❡♥❞ t♦ ❡st✐♠❛t❡✳ ❚❤✐s ✐s ✐♠♣♦r✲ t❛♥t t♦ ❛✈♦✐❞ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ ❛•• t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛r✐❛❜✐•✐t② ✇♦✉•❞ ❜❡ ❡①♣•❛✐♥❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ h ❛♥❞ ♥♦t ❛•s♦ ❜② t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❛❝t✐♥❣ ♦♥ t❤❡ s✉♣♣♦rt ✭✇❤✐❝❤ ❝❛♥♥♦t ♠♦❞✐❢② ❞✐r❡❝t•② t❤❡ ✈❛•✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• ❜✉t ❝❛♥ ❝❤❛♥❣❡ t❤❡ s❤❛♣❡ ♦❢ t❤❡ •❡✈❡• ✽✵ ❙♦✉r❝❡ ❛♥❞ t❛r❣❡t ❋❝✉rr❡♥t ♠❛t❝❤✐♥❣ ❋✐❣✉r❡ ✷✳✶✺✿ ❊①❛♠♣•❡ ♦❢ ♠❛t❝❤✐♥❣ ♦♥ t❤❡ ❝❛s❡ ♦❢ ❛ ✜❜❡r ❜✉♥❞•❡ ✇✐t❤ s✐❣♥❛•✳ ❖♥ t❤❡ •❡❢t ✜❣✉r❡✱ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❖♥ t❤❡ •❡❢t✱ t❤❡ r❡s✉•t✐♥❣ ♠❛t❝❤✲ ✐♥❣ ✇✐t❤ t❤❡ ❞❡❢♦r♠❡❞ s❤❛♣❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❣r✐❞ ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ s❡tt✐♥❣✳ s❡ts ♦❢ s✐❣♥❛•s✮✳ ❚❤❡ ❛❞❞✐t✐♦♥❛• ✇❡✐❣❤t ♣❛r❛♠❡t❡r γ 2 ♣r❡❝✐s❡•② ❝♦♥tr♦•s t❤✐s✳ ❍❡r❡✱ ✇❡ ❝❤♦♦s❡ ❛ r❛t❤❡r ✇❡❛❦ ♣❡♥❛•t② ❢✉♥❝t✐♦♥ t❤❛t ✐s t❤❡ L 2 ✲♥♦r♠ ♦❢ h ♦♥ t❤❡ s❤❛♣❡ X✳ ❉✐✛❡r❡♥t ❝❤♦✐❝❡s ❝♦✉•❞ ❜❡ ♠❛❞❡ t♦ ✐♠♣♦s❡ ♠♦r❡ r❡❣✉•❛r✐t② t♦ t❤❡ ❢✉♥❝t✐♦♥ h ❜✉t ✇✐•• ♥♦t ❜❡ ❞✐s❝✉ss❡❞ ❤❡r❡✳ ◆❡✐t❤❡r ✇✐•• ❜❡ st✉❞✐❡❞ t❤❡ t❤❡♦r❡t✐❝❛• ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✉♠ t♦ ❡q✳✭✷✳✹✹✮ ✐♥ t❤❡ ❣❡♥❡r❛• s❡tt✐♥❣✱ ✇❤✐❝❤ ✐s ♥♦t ❛ ♣r✐♦r✐ ♦❜✈✐♦✉s t♦ ❞❡❞✉❝❡ ❢r♦♠ t❤❡ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ❝❛s❡ ❛♥❞ ✇♦✉•❞ ♣r♦❜❛❜•② r❡q✉✐r❡ ♣❛rt✐❝✉•❛r ❛tt❡♥t✐♦♥✳ ❚❤❡ ♥✉♠❡r✐❝❛• s❝❤❡♠❡ ✇❡ ♣r♦♣♦s❡ ❢♦r ♦♣t✐♠✐③✐♥❣ ❡q✳✭✷✳✹✹✮ ✐s ♥♦t ❢✉♥❞❛♠❡♥t❛••② ❞✐❢✲ ❢❡r❡♥t ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ✿ ✇❡ ♦♣t✐♠✐③❡ s✐♠✉•t❛♥❡♦✉s•② ✐♥ ❜♦t❤ ✈❛r✐❛❜•❡s p 0 ❛♥❞ h ❜② ❛❞❛♣t✐✈❡ ♠✉•t✐✲st❡♣ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✳ ❚❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ p 0 ✐s ❝♦♠♣✉t❡❞ ❡①❛❝t•② ❛s ✇❡ ❞✐❞ ❡❛r•✐❡r s✐♥❝❡ t❤❡ •❛st t❡r♠ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ✐♥ p 0 ✳ ❚❤❡ ♦♥•② ♥❡✇ ❝♦♠♣♦♥❡♥t ✐s t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ h✳ ■♥ t❤❡ ❞✐s❝r❡t❡ s❡tt✐♥❣✱ ✐❢ h i = h(q i ) ❛♥❞ hi = 1 d + 1 (h α 1 i + ... + h α d+1 i )✱ t❤❡♥ t❤❡ ♣❡♥❛•t② t❡r♠ ♦♥ h ✐s ✿ |h| L 2 = m i=1 h2 i ξ i ✭✷✳✹✺✮ ❛♥❞ t❤✉s t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ h ✐s ♦❜t❛✐♥❡❞ ❛s ✿ ∇ h k |h| L 2 = n i=1 2 hi .∂ h k hi ξ i ✭✷✳✹✻✮ ◆♦t❡ t❤❛t t❤❡ t❡r♠s ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉♠ ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦ ♦♥•② ✐❢ h k ✐s ❛ ✈❡rt❡① ♦❢ t❤❡ i th ❢❛❝❡ ♦❢ X s♦ t❤❛t t❤❡ s✉♠ ✐s ♥♦t ❛❝t✉❛••② ♦✈❡r ❛•• t❤❡ ❢❛❝❡s✳ |❡❣✐str❛t✐♦♥ ✇✐t❤ ❡st✐♠❛t✐♦♥ ♦❢ h ❝❛♥ ❜❡ t❤✉s ♣❡r❢♦r♠❡❞ ❛s s✉♠♠❡❞ ✉♣ ✐♥ ❛•❣♦r✐t❤♠ ✶✳ ❚❤✐s ❛•❣♦r✐t❤♠ s♦♠❡❤♦✇ ❣❡♥❡r❛•✐③❡s t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❜❡❝❛✉s❡ t❛❦✐♥❣ γ 2 → ∞ ✇♦✉•❞ ❡♥❢♦r❝❡ t❤❡ r❡s✐❞✉❛•s t♦ r❡♠❛✐♥ ③❡r♦✲✈❛•✉❡❞ ♦♥ ❛•• t❤❡ s♦✉r❝❡ s❤❛♣❡✳ ■♥ ✜❣✉r❡ ✷✳✶✻ ✐s s❤♦✇♥ ♦♥❡ r❡s✉•t ♦❢ s✉❝❤ ❛ ♠❛t❝❤✐♥❣ ✿ ✇❡ ❞r❛✇ s♣❡❝✐❛• ❛tt❡♥t✐♦♥ ♦♥ t❤❡ ❜❛•❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❢ ❡①✐st✐♥❣ s✐❣♥❛• ♦♥ t❤❡ s♦✉r❝❡ s❤❛♣❡ ✭❛s ✐♥ t❤❡ ♠❛t❝❤✐♥❣ ♠♦❞❡• ♦❢ s✉❜s❡❝t✐♦♥ ✷✳✸✳✷✮ ❛♥❞ t❤❡ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥ t❤❛t ✐s ❛❞❞❡❞ t♦ X✳ ❚❤✐s ❜❛•❛♥❝❡ ✐s ❝♦♥tr♦••❡❞ ❜② t❤❡ ♣♦♥❞❡r❛t✐♦♥ ♣❛r♠❛♠❡t❡rs γ 1 ❛♥❞ γ 2 ✳ ❚❤❛♥❦s t♦ t❤❡ ✉s❡ ♦❢ ❢❝✉rr❡♥ts✬ ✽✶ ❆•❣♦r✐t❤♠ ✶ |❡❣✐str❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ ❛❞•✉♥❝t✐♦♥ ♦❢ r❡s✐❞✉❛•s |❡q✉✐r❡✿ ❙♦✉r❝❡ s❤❛♣❡ (X, f ) = (q k , f k ) k=1,..,n ✱ t❛r❣❡t s❤❛♣❡ (Y, g)✱ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣ δ✳ ✶✿ s❡t p 0 = 0 ❛♥❞ h = 0✳ ✷✿ ✇❤✐•❡ ❈♦♥✈❡r❣❡♥❝❡ ❞♦ ✸✿ ❋•♦✇ (X, f ) ❜② ❢♦r✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ p 0 ❛♥❞ st♦r❡ t❤❡ ❞❡❢♦r♠❡❞ t❡♠♣•❛t❡ ❛t t✐♠❡ ✶ (φ p 0 , h).(X, f ) = (q k,1 , f k + h k )✳ ✹✿ ❈♦♠♣✉t❡ t❤❡ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ (Y, g) ❛♥❞ (φ p 0 , h).(X, f )✳ ✺✿ ❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥ts ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ A ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜♥❛• ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ q k,1 ❡q✳✭✷✳✹✵✮✳ ✻✿ ❆♣♣•② t❤❡ ❜❛❝❦✇❛r❞ ✢♦✇ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❡q✳✭✷✳✹✸✮ ❛♥❞ ❞❡❞✉❝❡ P 0 ✳ ✼✿ ❈♦♠♣✉t❡ ∇ p 0 J = K V (q, q)p 0 -P 0 ✳ ✽✿ ❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ∇ h A ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ A ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡s✐❞✉❛• ✈❛•✉❡s h k ❜② ❡q✳✭✷✳✹✵✮ ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ L 2 ♥♦r♠ ❜② ❡q✳✭✷✳✹✻✮✳ ✾✿ ❯♣❞❛t❡ ✈❛r✐❛❜•❡s ✿ p 0 ← p 0 + δ∇ p 0 J h ← h + δ∇ h J ✶✵✿ ❈♦♠♣✉t❡ ♥❡✇ ❢✉♥❝t✐♦♥❛• J(p 0 , h) ❛♥❞ ❛❞❛♣t st❡♣ δ ❜② •✐♥❡ s❡❛r❝❤✳ ✶✶✿ ❡♥❞ ✇❤✐•❡ ✶✷✿ r❡t✉r♥ ✐♥✐t✐❛• ♠♦♠❡♥t✉♠ ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ p 0 ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛• h✳ ♥♦r♠s t♦ ❞r✐✈❡ t❤❡ ❞❡❢♦r♠❛t✐♦♥s✱ r❡s✐❞✉❛•s ❝❛♥ ❜❡ ❝♦♥❝❡♥tr❛t❡❞ ✐♥ r❡❣✐♦♥s ✇❤❡r❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ s✐❣♥❛•s ❝❛♥♥♦t ❜❡ ❡①♣•❛✐♥❡❞ ❜② ❞❡❢♦r♠❛t✐♦♥s✳ ✷✳✸✳✸ ◆✉♠❡r✐❝❛• ❝♦♥s✐❞❡r❛t✐♦♥s ❚♦ ❝♦♥❝•✉❞❡ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❡①❛♠✐♥❡ ✐♥ ❞❡♣t❤ t❤❡ ♥✉♠❡r✐❝❛• ✐ss✉❡s t❤❛t ❛r❡ s♣❡❝✐❢✲ ✐❝❛••② ✐♥❞✉❝❡❞ ❜② t❤❡ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠s✱ ❡s♣❡❝✐❛••② ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ t❤❡ s❤❛♣❡s ❣❡ts •❛r❣❡✱ ❛♥❞ ❡✈❛•✉❛t❡ ❛ ❢❡✇ ♣♦ss✐❜•❡ s♦•✉t✐♦♥s t♦ ❛❞❞r❡ss t❤❡♠✳ ❑❡r♥❡• s✉♠s✬ ❝♦♠♣✉t❛t✐♦♥s ✿ ♥✉♠❡r✐❝❛• s❝❤❡♠❡s ❛♥❞ t❤❡♦r❡t✐❝❛• ❝♦♠♣•❡①✲ ✐t✐❡s ❚❤❡ ♦✈❡r✇❤❡•♠✐♥❣ ♣❛rt ♦❢ ♥✉♠❡r✐❝❛• ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❛•❣♦r✐t❤♠s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✐s t❛❦❡♥ ❜② ❡✈❛•✉❛t✐♦♥ ♦❢ ❦❡r♥❡•s ❛t ❡❛❝❤ ❞❡s❝❡♥t st❡♣✳ ❚❤✐s ♦❝❝✉rs ❛t ❡ss❡♥t✐❛••② t✇♦ s♣♦ts ✐♥ t❤❡ ♣r♦❝❡ss✳ ❋✐rst ❢♦r t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❞❡✲ ❢♦r♠❛t✐♦♥s t❤❡♠s❡•✈❡s ❜❡❝❛✉s❡ ✐t ✐s ♥❡❝❡ss❛r② t♦ ✐♥t❡❣r❛t❡ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥s✳ ❆s ✇❡ ❡①♣•❛✐♥❡❞ ✐♥ ❝❤❛♣t❡r ✶✱ ✐❢ ✇❡ ✜① ❛ ❦❡r♥❡• K V ❢♦r t❤❡ ✈❡❝t♦r ✜❡•❞s ✐♥ E ✭❣❡♥❡r✐❝❛••② ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦❢ s❝❛•❡ σ V ✮✱ ❛t ❡❛❝❤ t✐♠❡ st❡♣✱ ❛ ❞✐s♣•❛❝❡♠❡♥t ✜❡•❞ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♠♦♠❡♥t❛ ❛t ❡✈❡r② ♣♦✐♥t ♦❢ t❤❡ s♦✉r❝❡ s❤❛♣❡ ♥❡❡❞ t♦ ❜❡ ❝♦♠♣✉t❡❞ ❛t ❡❛❝❤ ♦❢ t❤❡ s♦✉r❝❡ s❤❛♣❡ ♣♦✐♥ts✳ ▲❡t✬s ❝♦♥s✐❞❡r ❢♦r ❡①❛♠♣•❡ t❤❡ ❢♦r✇❛r❞ ❡q✉❛t✐♦♥ ♦♥ q ✿ ❛t ❡❛❝❤ t✐♠❡ st❡♣✱ ♦♥❡ ❤❛s t♦ ❝♦♠♣✉t❡ t❤❡ ❞✐s♣•❛❝❡♠❡♥t ✜❡•❞ n j=1 K V (q t,i , q t,j )p t,j ❢♦r ❡❛❝❤ ♦❢ t❤❡ s♦✉r❝❡ ♣♦✐♥t ♣♦s✐t✐♦♥s q t,i ✳ ❆s ✇❡ s❡❡✱ t❤✐s ❤❛s ❛ n 2 ❝♦♠♣•❡①✐t② ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ t❤❡ s♦✉r❝❡ s❤❛♣❡ ✇❤✐❝❤ ❤❛s t♦ ❜❡ t❤❡♥ ♠✉•t✐♣•✐❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡ st❡♣s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ s❝❤❡♠❡✳ ✽✷ ❙♦✉r❝❡ ❛♥❞ t❛r❣❡t ❊st✐♠❛t❡❞ r❡s✐❞✉❛• h ❉❡❢♦r♠❡❞ s❤❛♣❡ ✇✐t❤ r❡s✐❞✉❛• ❋✐❣✉r❡ ✷✳✶✻✿ ❊①❛♠♣•❡ ♦❢ ♠❛t❝❤✐♥❣ ✇✐t❤ ❡st✐♠❛t✐♦♥ ♦❢ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❝♦•♦r❡❞ s✉r❢❛❝❡s✳ ❆ s❡❝♦♥❞ ❜♦tt•❡♥❡❝❦ ✐♥ t❡r♠s ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ✐♥t❡♥s✐t② ❝♦♠❡s ✇✐t❤ t❤❡ ❢❝✉rr❡♥t ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❛♥❞ ✐ts ❣r❛❞✐❡♥t✳ ❚❤✐s ✐s ❛❣❛✐♥ ❛♥ ❡✈❛•✉❛t✐♦♥ ♦❢ ❦❡r♥❡•s ✇❤✐❝❤ ✐s t❤✐s t✐♠❡ ❛ t❡♥s♦r ♣r♦❞✉❝t ❦❡r♥❡• K g ⊗ k f •✐✈✐♥❣ ♦♥ t❤❡ ❜✐❣❣❡r ♣r♦❞✉❝t s♣❛❝❡ E × M ✳ ❆s ✇❡ s❡❡ ✇✐t❤ ❡q✳✭✷✳✶✽✮✱ ❝♦♠♣✉t✐♥❣ t❤❡ ❞✐st❛♥❝❡ ✐s ♦♥❝❡ ❛❣❛✐♥ q✉❛❞r❛t✐❝ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❉✐r❛❝s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t✳ ❚❤✐s ♥✉♠❜❡r ♦❢ ❉✐r❛❝s ❝♦rr❡s♣♦♥❞s t②♣✐❝❛••② t♦ t❤❡ ♥✉♠❜❡r ♦❢ s❡❣♠❡♥ts ✐❢ ✇❡ ❛r❡ ✇♦r❦✐♥❣ ♦♥ ❝✉r✈❡s ♦r t❤❡ ♥✉♠❜❡r ♦❢ tr✐❛♥❣•❡s ❢♦r s✉r❢❛❝❡s✱ ✇❤✐❝❤ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r r♦✉❣❤•② ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ t❤❡ s❤❛♣❡s✳ ■♥ ❜♦t❤ ❝❛s❡s✱ ✇❡ s❡❡ t❤❛t t❤❡ r❡❝✉rr❡♥t ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❡ss❡♥t✐❛••② n ❦❡r♥❡• ✬❝♦♥✲ ✈♦•✉t✐♦♥s✬✱ t❤❛t ✇❡ ❝❛♥ ✇r✐t❡ ✐♥ ❣❡♥❡r❛• ✿ ∀i = 1, .., n, n j=1 K(a i , a j )α j ✭✷✳✹✼✮ ✇❤❡r❡ a i ❛♥❞ α i ❛r❡ ❡✐t❤❡r ❡•❡♠❡♥ts ♦❢ E ♦r E × M ✳ ❚❤✐s q✉❛❞r❛t✐❝ ❝♦♠♣•❡①✐t② ✐s ❛ ❝•❡❛r •✐♠✐t❛t✐♦♥ ♦❢ t❤❡ ❛♣♣r♦❛❝❤ ✇❤❡♥ ♦♥❡ ❤❛s t♦ ✇♦r❦ ✇✐t❤ ❤✐❣❤•②✲s❛♠♣•❡❞ s❤❛♣❡s✱ ✇❤✐❝❤ ✐s r❛t❤❡r ❢r❡q✉❡♥t ✐♥ ♠❡❞✐❝❛• ✐♠❛❣✐♥❣ ❞❛t❛s❡ts✳ ❆s ✇❡ ✇✐•• ❤✐❣❤•✐❣❤t ✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ ❜r✉t❡ ❢♦r❝❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❦❡r♥❡•s ✐♥ ▼❆❚▲❆❇ ♦♥ ❛ st❛♥❞❛r❞ P❈ ❜❡❝♦♠❡s ♥❡❛r•② ✐♥tr❛❝t❛❜•❡ ❢♦r s❤❛♣❡s ✇✐t❤ ♠♦r❡ t❤❛♥ ✷✵✵✵ ♣♦✐♥ts ✇❤❡r❡❛s ♠❛♥② ♦❢ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❡ ❞❡❛• ✇✐t❤ ✐♥ t❤✐s ♠❛♥✉s❝r✐♣t ✐♥✈♦•✈❡ s✉r❢❛❝❡s ✇✐t❤ ♠♦r❡ t❤❛♥ ✶✵✵✵✵ ♣♦✐♥ts✳ ❚❤✐s ✐s t❤❡ r❡❛s♦♥ ✇❤② s❡✈❡r❛• ❛•t❡r♥❛t✐✈❡ ♥✉♠❡r✐❝❛• ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ✐♥ t❤❡ ♣❛st ❛♥❞ ❛r❡ st✐•• ❡①♣•♦r❡❞ ❛t t❤❡ ♣r❡s❡♥t t✐♠❡✳ ❚❤♦s❡ ❝❛♥ ❝♦♥s✐st ❡✐t❤❡r ✐♥ ❝♦♠♣✉t✐♥❣ ❡q✳✭✷✳✹✼✮ ✉s✐♥❣ ❞✐✛❡r❡♥t ♥✉♠❡r✐❝❛• ♣•❛t❢♦r♠s t❤❛♥ ▼❛t•❛❜ t❤❛t ✇♦✉•❞ t❛❦❡ ♣r♦✜t ✐♥ ♣❛r❛••❡• ❝♦♠♣✉t✐♥❣ ❢♦r ✐♥st❛♥❝❡ ♦r ✐♥ ✜♥❞✐♥❣ ❛♣♣r♦①✐♠❛t❡ ❜✉t ❢❛st❡r ♥✉♠❡r✐❝❛• s❝❤❡♠❡s✳ ❲❡ ♣r❡s❡♥t ❛ ❢❡✇ ♣♦ss✐❜✐•✐t✐❡s ✿ ❛ ❝♦♠♣❛r❛t✐✈❡ t❛❜•❡ ♦❢ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥s ✇✐•• ❜❡ ❣✐✈❡♥ ❛♥❞ ❝♦♠♠❡♥t❡❞ ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✳ ✕ ❆ ✜rst str❛✐❣❤t❢♦r✇❛r❞ t❤✐♥❣ t❤❛t ✇❡ tr✐❡❞ ✐s t♦ ✐♠♣•❡♠❡♥t ❛•• ❦❡r♥❡• ❝♦♠♣✉t❛t✐♦♥s ❛s ❈✰✰ ♠❡① ✜•❡s t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ▼❆❚▲❆❇ ❜♦❞② ♣r♦❣r❛♠✳ ❚❤✐s ❛••♦✇s ❛♥ ❡❛s② ✉s❡ ♦❢ ♣❛r❛••❡• ❝♦♠♣✉t✐♥❣ •✐❜r❛r✐❡s ✐♥ ❈ t❤❛t ✐♠♣r♦✈❡ ❜② ❛ s✉❜st❛♥t✐❛• ❢❛❝t♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡✳ ❆♥♦t❤❡r ♣r♦♠✐s✐♥❣ ❛•t❡r♥❛t✐✈❡ ✐s ♣r♦✈✐❞❡❞ ❜② t❤❡ r❡❝❡♥t ❞❡✈❡•♦♣♠❡♥t ♦❢ ❣r❛♣❤✐❝s ❝❛r❞s✳ ❲❡ t❤✉s ❡①♣❡r✐♠❡♥t❡❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❡q✳✭✷✳✹✼✮ ❜❛s❡❞ ♦♥ ❛ ♠❡① ❈❯❉❆ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦♥ •P❯ ❛♥❞ ✐♥❝•✉❞❡❞ ✐t ❛•s♦ ✐♥ ♦✉r ❝♦❞❡✳ ✕ ❆t t❤❡ ♠♦♠❡♥t✱ ❛•• t❤❡s❡ ♠❡t❤♦❞s ❛r❡ ❡①❛❝t ♦r ♥❡❛r•② ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❦❡r♥❡• ✽✸ ❋✐❣✉r❡ ✷✳✶✼✿ ■••✉str❛t✐♦♥ ♦❢ t❤❡ ❣r✐❞✲♣❛rt✐t✐♦♥✐♥❣ str❛t❡❣② ✐♥ ✷❉✳ ❙♠❛•• ❜•❛❝❦ ❞♦ts ❛r❡ t❤❡ ♦r✐❣✐♥❛• ❣r✐❞ ♣♦✐♥ts✱ r❡❞ ❝r♦ss❡s ❝♦rr❡s♣♦♥❞ t♦ ❛•• ♣♦ss✐❜•❡ ❝❡♥t❡rs ♦❢ ❜•♦❝❦s✳ ❚❤❡② ❢♦r♠ ❛ r❡❣✉•❛r s✉r✲❣r✐❞ ♦❢ ❜✐❣❣❡r st❡♣ ✭✉s✉❛••② ♦❢ t❤❡ ♦r❞❡r 3σ✮✳ s✉♠s✳ ❖t❤❡r ♠❡t❤♦❞s ✐♥t❡♥❞ t♦ ❛♣♣r♦①✐♠❛t❡ ❡q✳✭✷✳✹✼✮✳ ❯s✉❛••②✱ s✉❝❤ ♠❡t❤♦❞s ❛r❡ ❞❡s✐❣♥❡❞ ❢♦r t❤❡ ❝❛s❡ ♦❢ r❛❞✐❛• s❝❛•❛r ❦❡r♥❡•s✳ ❲❡ ✇✐•• ❛ss✉♠❡✱ t♦ s✐♠♣•✐❢② t❤❡ ♣r❡s❡♥t❛t✐♦♥✱ t❤❛t ✇❡ ❛r❡ ✇♦r❦✐♥❣ ✇✐t❤ ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ✇✐t❤ ❛ ❝❡rt❛✐♥ s❝❛•❡ ♣❛r❛♠❡t❡r σ✳ ❆ ✜rst s✐♠♣•❡ ✐❞❡❛ ✐s t♦ ♥♦t✐❝❡ t❤❛t ✐❢ ♣♦✐♥ts ❛r❡ ❢❛rt❤❡r ❛✇❛② ❢r♦♠ ❡❛❝❤ ♦t❤❡r t❤❛♥ 3σ t❤❡♥ t❤❡ ✈❛•✉❡ ♦❢ t❤❡ ❦❡r♥❡• ✐s •♦✇❡r t❤❛♥ 10 -4 ✿ t❤✉s ❦❡r♥❡•s ♥❡❡❞ ♥♦t ❜❡ ❡✈❛•✉❛t❡❞ ❜❡t✇❡❡♥ ❛•• ♣❛✐rs ♦❢ ♣♦✐♥ts ❜✉t ♦♥•② ❢♦r ♣♦✐♥ts •②✐♥❣ ❛t ❛ s♠❛•• ❡♥♦✉❣❤ ❞✐st❛♥❝❡✳ ❚❤✐s s✉❣❣❡sts t❤❛t ✇❡ ❝❛♥ s❛✈❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❡①♣♦♥❡♥t✐❛•s ❡s♣❡❝✐❛••② ✇❤❡♥ σ ✐s s♠❛•• ❝♦♠♣❛r❡❞ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ♦❜•❡❝ts✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❛t✱ ♦♥❡ ❝♦✉•❞ ❛•s♦ ♣r❡✲❝♦♠♣✉t❡ ❛ t❛❜•❡ ♦❢ ✈❛•✉❡s ❢♦r t❤❡ •❛✉ss✐❛♥ ♦♥ ❛ ❞✐s❝r❡t✐③❡❞ ❣r✐❞ ♦❢ ❞✐st❛♥❝❡s ❛♥❞ t❤❡♥ ❛❝❝❡ss t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜♦① ✐♥st❡❛❞ ♦❢ r❡❝♦♠♣✉t✐♥❣ ✐t ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ •♦♦♣✳ ❚❤❡ s❡❛r❝❤ ❢♦r ♣♦✐♥ts ✐♥ ❛ t❤❡ ❞❡s✐r❡❞ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛ ❣✐✈❡♥ ♣♦s✐t✐♦♥ ❝♦✉•❞ ❜❡ ❛•s♦ ✐♠♣r♦✈❡❞ ❝♦♠♣✉t❛t✐♦♥❛••② ❜② ✉s✐♥❣ ❑❉✲tr❡❡ ❛•❣♦r✐t❤♠s✳ ✕ ■♥ t❤❡ s❛♠❡ ✈❡✐♥✱ ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ❛•s♦ ❞❡✈❡•♦♣❡❞ ❢♦r •❛✉s✲ s✐❛♥ ❦❡r♥❡•s✱ s✉❝❤ ❛s t❤❡ ❋❛st •❛✉ss ❚r❛♥s❢♦r♠ ❬✽✹(✱ ✇❤✐❝❤ ✇❛s ❛♣♣•✐❡❞ •❛t❡r ♦♥ t♦ ❝♦♠♣✉t❡ ❞✐st❛♥❝❡s ♦♥ ❝✉rr❡♥ts ✐♥ ❬✹✶(✳ ❚❤✐s ♠❡t❤♦❞ ❝♦♠❜✐♥❡s t❤❡ ✐❞❡❛ ♦❢ ❝•✉st❡r✐♥❣ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ♥❡❣•✐❣✐❜•❡ ❦❡r♥❡• ✈❛•✉❡s ✇✐t❤ t❤❡ ♣r✐♥❝✐♣•❡ ♦❢ ❢❛st ♠✉•t✐♣•❡ ♠❡t❤♦❞s ✭❋▼▼✮ ❢♦r ❛❝❝❡•❡r❛t✐♥❣ ❡①♣♦♥❡♥t✐❛•s✬ ❡✈❛•✉❛t✐♦♥s✱ •❡❛❞✐♥❣ t♦ ❛♥ ❛•❣♦r✐t❤♠ t❤❛t ❤❛s ❛ t❤❡♦r❡t✐❝❛• ❝♦♠♣•❡①✐t② ♦❢ O(np) ✇❤❡r❡ p ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝•✉st❡rs✳ (❡t t❤✐s ♥✉♠❜❡r ♦❢ ❝•✉st❡rs ✐s ❞✐r❡❝t•② ❝♦rr❡•❛t❡❞ t♦ t❤❡ s❝❛•❡ σ ❛♥❞ t♦ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡✳ ❙✐♥❝❡ p ✐s ♥♦t s❡t ✉♣ ❛✉t♦♠❛t✐❝❛••② ✐♥ t❤❡ ❛•❣♦r✐t❤♠ ♦❢ ❬✽✹(✱ ♦♥❡ ❞✐✣❝✉•t② ♦❢ t❤❡ ♠❡t❤♦❞ ✐s t♦ ❝❤♦♦s❡ ✐t ♣r♦♣❡r•② t♦ ❤❛✈❡ ❜♦t❤ ❛ ❣♦♦❞ ❛❝❝✉r❛❝② ✇✐t❤♦✉t s•♦✇✐♥❣ ❞♦✇♥ ❝♦♠♣✉t❛t✐♦♥s t♦♦ ♠✉❝❤✳ ■♥ ♣r❛❝t✐❝❡✱ ❛•t❤♦✉❣❤ ✐t •❡❛❞s t♦ ❢❛st ❝♦♠♣✉t❛t✐♦♥s ❡s♣❡❝✐❛••② ❢♦r ❜✐❣ σ ❛♥❞ ♥♦t t♦♦ ❤✐❣❤ ❞✐♠❡♥s✐♦♥✱ ✐t ♠❛❦❡s t❤❡ q✉❛•✐t② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❞✐✣❝✉•t t♦ ❡✈❛•✉❛t❡✳ ✕ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ ♥✉♠❡r✐❝❛• s❝❤❡♠❡ ✇❛s ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❬✷✽(✱ ✇❤✐❝❤ ❜❛s✐❝❛••② ♣r♦•❡❝ts ❛•• t❤❡ ♠♦♠❡♥t❛ α i •♦❝❛t❡❞ ❛t ♣♦✐♥ts q i ♦♥ ❛ ✜①❡❞ •✐♥❡❛r•② s♣❛❝❡❞ ❣r✐❞ ❛♥❞ t❤❡♥ ❡①♣r❡ss❡s ❡q✳✭✷✳✹✼✮ ❛s ❛ ❝♦♥✈♦•✉t✐♦♥ ♦♥ t❤✐s ❣r✐❞ ✇❤✐❝❤ ❝❛♥ ❜❡ ✽✹ n ✶✵✵ ✶✵✵✵ ✷✵✵✵ ✶✵✵✵✵ ✺✵✵✵✵ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ▼❆❚▲❆❇ ❜r✉t❡ ❢♦r❝❡ ❝♦♠♣✉t❛t✐♦♥ ✵✳✾✽s ✾✽✳✷s ✸✾✶✳✹s ∞ ∞ ②❡s ❈✰✰ ♣❛r❛••❡•✐③❡❞ ❝♦♠♣✉t❛t✐♦♥ ✵✳✵✷✺s ✵✳✹✷s ✶✳✻s ✹✵✳✽s ✶✵✸✵s ②❡s ❚r✉♥❝❛t✐♦♥ ❛♥❞ ♣r❡✲❝♦♠♣✉t❛t✐♦♥ ♦❢ •❛✉ss✐❛♥
♦ •r✐❞ ❝♦♥✈♦•✉t✐♦♥ ✵✳✶✶s ✵✳✶✶s ✵✳✶✶s ✵✳✶✻s ✵✳✹✸s ♥♦ ❚❛❜•❡ ✷✳✶✿ ❈♦♠♣✉t❛t✐♦♥❛• t✐♠❡ ❢♦r ✷✵ ❝♦♥✈♦•✉t✐♦♥s ✇✐t❤ σ = 0.3✳ ❝♦♠♣✉t❡❞ ❜② ❋❋❚ ❛♥❞ ✐♥✈❡rs❡ ❋❋❚✳ ❚❤❡ ♣r♦•❡❝t✐♦♥ st❡♣ ✐s •✐♥❡❛r ✐♥ n ✇❤✐•❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈♦•✉t✐♦♥ ✐♥ t❤❡ ❣r✐❞ ❞♦♠❛✐♥ ✐s ♦❢ ❝♦♠♣•❡①✐t② O(n g log n g ) ✇❤❡r❡ n g ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ t❤❡ ❣r✐❞✳ ◆♦✇✱ t❤❡ ❣r✐❞ ✐s ❝❤♦s❡♥ ❛s ❛ •✐♥❡❛r•② s♣❛❝❡❞ s❡t ♦❢ ♣♦✐♥ts ❞❡•✐♠✐t❡❞ ❜② ❛ r❡❝t❛♥❣✉•❛r ♣❛r❛••❡•❡♣✐♣❡❞ ❝♦♥t❛✐♥✐♥❣ ❛•• t❤❡ ♣♦✐♥ts ❛♥❞ ✇❤♦s❡ s❛♠♣•✐♥❣ ✐s ❝❤♦s❡♥ ❛s ❛ ❢r❛❝t✐♦♥ ♦❢ t❤❡ s❝❛•❡ ♦❢ t❤❡ •❛✉ss✐❛♥ σ✳ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❜❡ t❤❡♥ ❝♦♥tr♦••❡❞ ❜② t❤❡ ❣r✐❞ st❡♣✱ ❛s s❤♦✇♥ ✐♥ ❬✷✽( ❝❤❛♣t❡r ✷✳ ❆s ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ❣r✐❞✱ ✇❡ s❡❡ t❤❛t ✐t ❣r♦✇s ❛s s -D ✇❤❡r❡ s ✐s t❤❡ st❡♣ ♦❢ t❤❡ ❣r✐❞ ❛♥❞ D t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡✳ ❚❤❡ ✐♥t❡r❡st✐♥❣ t❤✐♥❣ t♦ ♦❜s❡r✈❡ ✐s t❤❛t✱ ✐❢ ✇❡ ❡①❝❡♣t t❤❡ ♣r♦•❡❝t✐♦♥ ♦♥ t❤❡ ❣r✐❞ ✭✇❤✐❝❤ ✐s ❛♥②❤♦✇ ♥♦t •✐♠✐t✐♥❣✮✱ t❤❡ ❝♦♠♣•❡①✐t② ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ❛♥②♠♦r❡ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ t❤❡ s❤❛♣❡s ❜✉t ♦♥•② ♦♥ t❤❡ ❣r✐❞ s✐③❡✱ ✇❤✐❝❤ ♠❛❦❡s ✐t ♣❛rt✐❝✉•❛r•② ❛❞❛♣t❡❞ ✇❤❡♥ n ✐s ✈❡r② •❛r❣❡✳ ❍♦✇❡✈❡r✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r✐❞ ✐ts❡•❢ ❣r♦✇s ❡①♣♦♥❡♥t✐❛••② ✇✐t❤ t❤❡ ❞✐♠❡♥s✐♦♥ ✿ ✐♥ ♣r❛❝t✐❝❡ ✐t r❡♠❛✐♥s ❛✛♦r❞❛❜•❡ ✐♥ ❞✐♠❡♥s✐♦♥ ✸ ♦r •❡ss ❜✉t ✐t ❜❡❝♦♠❡s r❛♣✐❞•② ✐♥❝r❡❞✐❜•② ❝♦♥s✉♠✐♥❣ ❜♦t❤ ✐♥ t✐♠❡ ❛♥❞ ✐♥ ♠❡♠♦r② ✐♥ ❞✐♠❡♥s✐♦♥ ✹ ❛♥❞ ♠♦r❡✳ ■♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❡ ❤❛✈❡ ❜❡❡♥ ❝♦♥s✐❞❡r✐♥❣ s♦ ❢❛r ✭✇✐t❤ E = R 3 ✮✱ t❤✐s ♠❡t❤♦❞ ✐s ✇❡••✲❛❞❛♣t❡❞ ❢♦r t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♣❛rt✱ ❜✉t ✇❡ s❡❡ t❤❛t ❢❝✉rr❡♥t ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ✐♥✈♦•✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❦❡r♥❡•s •✐✈✐♥❣ ♦♥ E × M = R 4 ✳ ■♥ t❤❛t ❝❛s❡✱ ✐♥tr♦❞✉❝✐♥❣ ✹❉ ❣r✐❞s t❤❛t ❝♦♥t❛✐♥ ❛•• t❤❡ ♣♦✐♥ts {(x i , f i )} ♣r♦❞✉❝❡s ❤✉❣❡ n g ✳ ❆s ✇❡ s❡❡✱ t❤✐s ✐s ❡ss❡♥t✐❛••② ❛ ♣r♦❜•❡♠ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ❜❡❝❛✉s❡ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s •✐✈❡s ❜❛s✐❝❛••② ♦♥ ❛ ✷✲ ❞✐♠❡♥s✐♦♥❛• ♠❛♥✐❢♦•❞ ✐♥ R 4 ✳ ❆ ♣♦ss✐❜•❡ s♦•✉t✐♦♥ t❤❛t ✇❡ ✐♠♣•❡♠❡♥t❡❞ t♦ ❝♦♥tr♦• t❤❡ s✐③❡ ♦❢ t❤❡ ❣r✐❞s ✐s t♦ ♠❛❦❡ ❜•♦❝❦ ❝♦♠♣✉t❛t✐♦♥s ❜❛s❡❞ ♦♥ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛• ❣r✐❞ ❛s ✐••✉str❛t❡❞ ✐♥ ✜❣✉r❡ ✷✳✶✼✱ ✇❤✐❝❤ r❡str✐❝ts t❤❡ ❝♦♥✈♦•✉t✐♦♥s t♦ ♦♥•② ♦✈❡r•❛♣♣✐♥❣ ❜•♦❝❦s t❤❛t ❛❝t✉❛••② ❝♦♥t❛✐♥ ♣♦✐♥ts ♦❢ t❤❡ s❤❛♣❡✳ (❡t✱ ❡✈❡♥ r❡❞✉❝❡❞✱ t❤❡ ❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥ ✐s st✐•• ♦❝❝✉rr✐♥❣ ✐♥ ❡❛❝❤ ❜•♦❝❦✱ ✇❤✐❝❤ ♠❛❦❡s t❤✐s ❛♣♣r♦❛❝❤ ❜❛r❡•② ❢❡❛s✐❜•❡ ❢♦r ✹❉ ❜✉t st✐•• ✉♥t❤✐♥❦❛❜•❡ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ ❈♦♠♣❛r❛t✐✈❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡s ❚♦ ❣✐✈❡ ❛ ♠♦r❡ ❝♦♥❝r❡t❡ ✈✐s✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞✐s❝✉ss✐♦♥✱ ✇❡ s❤♦✇ ❛ ❢❡✇ ❝♦♠♣✉t❛✲ t✐♦♥❛• t✐♠❡ ❢♦r t❤❡ ❞✐✛❡r❡♥t ♥✉♠❡r✐❝❛• ♠❡t❤♦❞s t❤❛t ❤❛✈❡ ❜❡❡♥ ❡✈♦❦❡❞ ♣r❡✈✐♦✉s•②✳ ❆•• ❝♦♠♣✉t❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣❡r❢♦r♠❡❞ ♦♥ ❛ st❛♥❞❛r❞ q✉❛❞ ❝♦r❡ •❛♣t♦♣ ✇✐t❤ ❛ ◆❱■❉■❆ ◆❱❙ ✺✶✵✵ ❣r❛♣❤✐❝s ❝❛r❞✳ ❲❡ ✜rst s❤♦✇ t❤❡ ❡•❛♣s❡❞ t✐♠❡ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✷✵ ❝♦♥✈♦•✉t✐♦♥s ❛s ✐♥ ❡q✳✭✷✳✹✼✮ ❢♦r ❛ ✸❉ •❛✉ss✐❛♥ ❦❡r♥❡• ❛♥❞ ✇✐t❤ ❛♥ ✐♥❝r❡❛s✐♥❣ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts n✳ ❚❤❡ ♣♦s✐t✐♦♥s q i ✇❡r❡ ♣✐❝❦❡❞ r❛♥❞♦♠•② ✐♥ t❤❡ ❝✉❜❡ [0, 1] 3 ✳ ■♥ t❛❜•❡ ✷✳✶✱ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✜rst ❞♦♥❡ ❢♦r ❛ ❜✐❣ ❦❡r♥❡• ✭σ = 0.3✮✳ ❇r✉t❡✲❢♦r❝❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥ ▼❆❚▲❆❇ ❜❡❝♦♠❡s ✉♥❢❡❛s✐❜•❡ ✇✐t❤ ♠♦r❡ t❤❛♥ ✷✵✵✵ ♣♦✐♥ts✳ ❲❡ ❛•s♦ s❡❡ t❤❛t tr✉♥✲ ❝❛t✐♦♥ ♦❢ ❡①♣♦♥❡♥t✐❛•s ✐s ❝•❡❛r•② ♥♦t ❡✣❝✐❡♥t ✐♥ t❤❛t ❝❛s❡ ✇❤❡r❡❛s ❢♦r ❛ s♠❛••❡r s❝❛•❡ σ = 0.1 ✭t❛❜•❡ ✷✳✷✮✱ ✇❡ ❝❛♥ ❣❛✐♥ ❛ ❢❛❝t♦r ✷ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ✐♥ ✽✺ n ✶✵✵ ✶✵✵✵ ✷✵✵✵ ✶✵✵✵✵ ✺✵✵✵✵ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ▼❆❚▲❆❇ ❜r✉t❡ ❢♦r❝❡ ❝♦♠♣✉t❛t✐♦♥ ✵✳✾✽s ✾✽✳✶s ✸✾✷s ∞ ∞ ②❡s ❈✰✰ ♣❛r❛••❡•✐③❡❞ ❝♦♠♣✉t❛t✐♦♥ ✵✳✵✸✸s ✵✳✹✷s ✶✳✻s ✹✶s ✶✵✸✾s ②❡s ❚r✉♥❝❛t✐♦♥ ❛♥❞ ♣r❡✲❝♦♠♣✉t❛t✐♦♥ ♦❢ •❛✉ss✐❛♥
♦ •r✐❞ ❝♦♥✈♦•✉t✐♦♥ ✶✳✷✵s ✶✳✷✹s ✶✳✷✵s ✶✳✸✷s ✶✳✼s ♥♦ ❚❛❜•❡ ✷✳✷✿ ❈♦♠♣✉t❛t✐♦♥❛• t✐♠❡ ❢♦r ✷✵ ❝♦♥✈♦•✉t✐♦♥s ✇✐t❤ σ = 0.1✳ n g ✸✷✵✵✵ ✻✹✵✵✵ ✷✻✵✵✵✵ ✶✵✵✵✵✵✵ ❚✐♠❡ ✵✳✶✸s ✵✳✸✹s ✵✳✽s ✹✳✼s |❡•❛t✐✈❡ ❡rr♦r 5% 2.9% 1% 0.4% ❚❛❜•❡ ✷✳✸✿ Pr❡❝✐s✐♦♥ ✈s t✐♠❡ ✇✐t❤ t❤❡ ❣r✐❞ ❛♣♣r♦❛❝❤ ✭✇✐t❤ σ = 0.2✮ ❈✰✰✳ ◆♦t❡ t❤❡ ✐♠♣♦rt❛♥t ❛❝❝❡•❡r❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② •P❯ ❝♦♠♣✉t❛t✐♦♥ ✐♥ ❜♦t❤ ❝❛s❡s t❤❛t ♦✉t♣❡r❢♦r♠s t❤❡ ♣r❡✈✐♦✉s ♠❡t❤♦❞s ❛s s♦♦♥ ❛s n ✐s ❜✐❣❣❡r t❤❛♥ ✶✵✵✵✳ ■t ✐s ♣❛rt✐❝✲ ✉•❛r•② ♦♣t✐♠❛• ✐♥ t❤❡ r❛♥❣❡ ✶✵✵✵✲✶✵✵✵✵ ♣♦✐♥ts✳ ❲✐t❤ •❛r❣❡r n✱ s♦♠❡ ❛♣♣r♦①✐♠❛t✐✈❡ ♠❡t❤♦❞s ♠❛② ❜❡❝♦♠❡ ❢❛st❡r✳ ❚❤❡ ❋•❚✱ ❢♦r ✐♥st❛♥❝❡✱ ✐s q✉✐t❡ ❣♦♦❞ ❢♦r •❛r❣❡ s❝❛•❡ ❦❡r♥❡•s s✐♥❝❡✱ t♦ ♦❜t❛✐♥ ❛ ❣✐✈❡♥ ♣r❡❝✐s✐♦♥✱ t❤❡ r❡q✉✐r❡❞ ♥✉♠❜❡r p ♦❢ ❝•✉st❡rs ❞❡❝r❡❛s❡s ✇✐t❤ σ✳ ❋♦r s♠❛••❡r ❦❡r♥❡•s✱ t❤❡ ❣❛✐♥ ✐s ♥♦t ❛s ❝•❡❛r✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs ✭p ✐♥ ♣❛rt✐❝✉•❛r✮ ✐s ❛•s♦ ❛ ♥♦t tr✐✈✐❛• st❡♣ ✇✐t❤ ❋•❚✳ ■♥ t❤❡ ❡①♣❡r✐♠❡♥ts ✇❡ s❤♦✇ ❤❡r❡✱ ✇❡ ♠❛❞❡ s❡✈❡r❛• tr✐❡s t♦ ❛❞❛♣t t❤❡ ♥✉♠❜❡r ♦❢ ❝•✉st❡rs ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ r❡•❛t✐✈❡ ♣r❡❝✐s✐♦♥ ❛r♦✉♥❞ 1% ❢♦r ❡❛❝❤ ❝❛s❡✳ ❚❤❡ ❣r✐❞ ❝♦♥✈♦•✉t✐♦♥ ♠❡t❤♦❞ ✐s ❛•s♦ ✈❡r② ❡✣❝✐❡♥t✳ ■♥ t❤♦s❡ ❡①❛♠♣•❡s✱ ✇❡ ✉s❡❞ ❣r✐❞s s❛♠♣•❡❞ ❛t σ/5 ✇❤✐❝❤ ❣✐✈❡s n g ≈ 32000 ✐♥ t❛❜•❡ ✷✳✶ ❛♥❞ n g ≈ 260000 ✐♥ t❛❜•❡ ✷✳✷✳ ❆s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ❛♥❞ ❝♦♥✜r♠❡❞ ❜② ♥✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡ ✐s t❤❡♥ ❛•♠♦st ✐♥❞❡♣❡♥❞❡♥t ♦❢ n✱ ✇❤✐❝❤ ✐s ♣❛rt✐❝✉•❛r•② ✐♥t❡r❡st✐♥❣ ❢♦r ❤✐❣❤•② s❛♠♣•❡❞ s❤❛♣❡s ✭t②♣✐❝❛••② n ≥ 10000✮✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ tr✉❡ r❡s✉•t✳ ■♥ t❛❜•❡ ✷✳✸✱ ✇❡ s❤♦✇ ❜♦t❤ t✐♠❡ ❛♥❞ ♣r❡❝✐s✐♦♥ ❢♦r ❛ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦♥✈♦•✉t✐♦♥s ✇✐t❤ σ = 0.2 ✉s✐♥❣ ✜♥❡r ❛♥❞ ✜♥❡r ❣r✐❞s✳ •❡tt✐♥❣ ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ r❡q✉✐r❡s ♠♦r❡ ❛❝❝✉r❛t❡ ❣r✐❞s ❛♥❞ •❡❛❞s t♦ •♦♥❣❡r ❝♦♠♣✉t❛t✐♦♥s✳ ❚❤✐s ✐s ❜❛s✐❝❛••② ✇❤② s✉❝❤ ❛ ♠❡t❤♦❞ ✐s ♥♦t ✇❡••✲✜tt❡❞ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥ t❤❛♥ ✸✳ ■♥ ✹❉ ❢♦r ✐♥st❛♥❝❡✱ ✇✐t❤ σ = 0.2✱ r❡q✉✐r✐♥❣ ❛ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ 1% ♥❡❝❡ss✐t❛t❡s t❤❡ ✉s❡ ❛♥❞ st♦r❛❣❡ ♦❢ ❣r✐❞s ✇✐t❤ ❛t •❡❛st n g = 5 × 10 6 ♣♦✐♥ts✱ ✇✐t❤ t❤❡ •✐♠✐t❛t✐♦♥s t❤❛t t❤✐s ✐♥❞✉❝❡s ✐♥ t✐♠❡ ❛♥❞ ♠❡♠♦r②✳ ❙♦ ✇❤✐❝❤ ♦♥❡ ✐s t❤❡ ❜❡st ❛♥❞ ❤♦✇ ❢❛r ❝❛♥ ✇❡ ❣♦ ❄ ■t ✐s st✐•• ❤❛r❞✱ ❛t t❤✐s ♣♦✐♥t✱ t♦ ❣✐✈❡ ❛ ❞❡✜♥✐t❡ ❛♥s✇❡r t♦ t❤❡ ♥✉♠❡r✐❝❛• ✐ss✉❡✳ ❚❤❡ ♣♦s✐t✐✈❡ ❛s♣❡❝t ✐s t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ❛r❡ s❡✈❡r❛• ♣♦ss✐❜•❡ ♥✉♠❡r✐❝❛• s❝❤❡♠❡s ❛✈❛✐•❛❜•❡ ❛♥❞ st✐•• s♦♠❡ ♥❡✇ ♦♥❡s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❋♦❝✉s✐♥❣ ♦♥•② ♦♥ t❤❡ ♦♥❡s ✇❡ ❤❛✈❡ ❡①♣❡r✐♠❡♥t❡❞ ♦♥✱ t❤❡ ♣r❡✈✐♦✉s ❞✐s❝✉ss✐♦♥ t❡♥❞s t♦ s❤♦✇ t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♥✉♠❡r✐❝❛• ♠❡t❤♦❞ s❤♦✉•❞ ❜❡ ♣r♦❜❛❜•② t❛✐•♦r❡❞ t♦ t❤❡ s✐t✉❛t✐♦♥✳ ❲♦r❦✐♥❣ ♦♥ ❝♦♠✲ ♣✉t❡rs ❡q✉✐♣♣❡❞ ✇✐t❤ ◆❱■❉■❆ ❣r❛♣❤✐❝s ❝❛r❞s✱ •P❯ ❝♦♠♣✉t❛t✐♦♥ ✐s ♦❜✈✐♦✉s•② ❛ ✈❡r② s♦•✐❞ ❝❛♥❞✐❞❛t❡ t❤❛t ✐s ❛❞❛♣t❡❞ t♦ ♠♦st ❝❛s❡s s✐♥❝❡ ✐t ♣r♦✈✐❞❡ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❦❡r♥❡•s ✇✐t❤ •✐♠✐t❡❞ ♠❡♠♦r② ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ✈❡r② ❝♦♠♣❡t✐t✐✈❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡s✱ ♥♦t ♠❡♥t✐♦♥✐♥❣ t❤❛t t❤❡ t✐♠❡s s❤♦✇♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ✇❡r❡ ♥♦t ♦❜✲ ✽✻ t❛✐♥❡❞ ✇✐t❤ t❤❡ ♠♦st r❡❝❡♥t ❛♥❞ ♣♦✇❡r❢✉• ❣r❛♣❤✐❝s ❝❛r❞✳ (❡t✱ t❤✐s ✐s st✐•• ❛ q✉❛❞r❛t✐❝ s❝❤❡♠❡ ✇❤✐❝❤ ❜❡❝♦♠❡s ✐♥❡✈✐t❛❜•② •✐♠✐t❡❞ ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ❡①❝❡❡❞ 10000✳ ❋♦r s✉❝❤ ❤✐❣❤ r❡s♦•✉t✐♦♥ s❤❛♣❡s✱ t❤❡ ❣r✐❞ str❛t❡❣② ♦✛❡rs ❛ ❣♦♦❞ ❛•t❡r♥❛t✐✈❡ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠s✬ ♣❛rt ✭✐♥✈♦•✈✐♥❣ t❤❡ ❦❡r♥❡• K V ✇❤✐❝❤ ✐s ❛•✇❛②s ✐♥ ✸❉ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✮ ❜❡❝❛✉s❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡ ✐s t❤❡♥ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r✐❞ ❛♥❞ ♥♦t t❤❡ s❛♠♣•✐♥❣ ♦❢ t❤❡ s❤❛♣❡✳ ❍♦✇❡✈❡r✱ ✐t ❞♦❡s ♥♦t ✇♦r❦ ❛s ✇❡•• ✇✐t❤ t❤❡ ❢❝✉rr❡♥t ❞✐st❛♥❝❡ ❝♦♠♣✉t❛t✐♦♥ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥ ✐♥❝r❡❛s❡ ❛♥❞ t❤❡ r❡s✉•t✐♥❣ ❣r✐❞s✬ s✐③❡s✳ ❆ s❛t✐s❢②✐♥❣ s♦•✉t✐♦♥ ✐♥ t❤♦s❡ ❝❛s❡s ✐s t♦ ✉s❡ ❣r✐❞ ❝♦♠♣✉t❛t✐♦♥s ❢♦r ❞✐✛❡♦♠♦r♣❤✐s♠s ✭✇❤✐❝❤ ✐s t❤❡ ♣❛rt t❤❛t ✐♥✈♦•✈❡s ✐♥ ♣r❛❝t✐❝❡ t❤❡ •❛r❣❡r ♥✉♠❜❡r ♦❢ ❝♦♥✈♦•✉t✐♦♥s✮ ❛♥❞ •P❯ ❢♦r t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t t❡r♠✳ ▼♦r❡ r❡❝❡♥t ✈❡rs✐♦♥s ♦❢ t❤❡ ❋•❚✱ ❛s ✐♥ ❬✻✻✱ ✼✸( ❢♦r ✐♥st❛♥❝❡✱ ❝♦✉•❞ ❜❡ ❛•s♦ ❛♥ ✐♥t❡r❡st✐♥❣ tr❛❝❦ ❜✉t ✇❡r❡ ♥♦t ✐♥✈❡st✐❣❛t❡❞ ❤❡r❡✳ ❚❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ✇❡ ❤❛✈❡ ♠❛❞❡ ✐♥t❡❣r❛t❡s ❛•• t❤❡ ❞✐✛❡r❡♥t ♠❡t❤♦❞s t❤❛t ✇❡r❡ ♠❡♥t✐♦♥❡❞✱ ❛••♦✇✐♥❣ t❤❡r❡❢♦r❡ ❛ r❡•❛t✐✈❡ ❛❞❛♣t❛❜✐•✐t② t♦ ♠❛♥② ❝❛s❡s✳ ❚♦ ❣✐✈❡ ❛♥ ♦✈❡r❛•• ✐❞❡❛✱ ❛t t❤❡ ♣r❡s❡♥t st❛t❡✱ ❛ ❝♦♠♣•❡t❡ r❡❣✐str❛t✐♦♥ ♦♥ ❛ ❝♦♠♣✉t❛t✐♦♥❛• s❡r✈❡r ✭✶✵✵ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣s✮ ❜❡t✇❡❡♥ s✉r❢❛❝❡s ✇✐t❤ ✶✵✵✵ ♣♦✐♥ts t❛❦❡s ❛ ❢❡✇ ♠✐♥✉t❡s ❛♥❞ •❡ss t❤❛♥ ❤❛•❢ ❛♥ ❤♦✉r ✇✐t❤ ✶✵✵✵✵ ♣♦✐♥ts ✉s✐♥❣ t❤❡ ♠♦st ❛❞❛♣t❡❞ ♠❡t❤♦❞✳ ❚❤❡ ❛•❣♦r✐t❤♠ ✐s ❛•s♦ ❛❜•❡ t♦ r✉♥ ✐♥ ❞❡❝❡♥t t✐♠❡ ❢♦r s❤❛♣❡s ✇✐t❤ ✉♣ t♦ ✺✵✵✵✵ ♣♦✐♥ts✳ ✷✳✹ ❈♦♥❝•✉s✐♦♥ ❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛ ✇❛② t♦ ❣❡♥❡r❛•✐③❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❝✉rr❡♥ts ❢♦r t❤❡ ♣✉r♣♦s❡ ♦❢ ✐♥t❡❣r❛t✐♥❣ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✐♥t♦ ❛ ❝♦❤❡r❡♥t ❛♥❞ r♦❜✉st r❡♣r❡s❡♥t❛t✐♦♥✳ ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♣r♦✈✐❞❡ ❛ ❢r❛♠❡✇♦r❦ t♦ ♠♦❞❡• ❣❡♦♠❡tr✐❝❛••②✲s✉♣♣♦rt❡❞ s✐❣♥❛•s ♦❢ ♥❡❛r•② ❛♥② t②♣❡ ❛♥❞ r❡❣✉•❛r✐t② ✇❤✐•❡ ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❝✉rr❡♥ts t♦ ♠♦❞❡• t❤❡ ❣❡♦♠❡tr②✳ ❲❡ ❤❛✈❡ s❤♦✇♥ t❤❛t |❑❍❙ ♠❡tr✐❝s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✐♥ s♣❛❝❡s ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ♣r♦✈✐❞✐♥❣ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t❤❛t ❛••♦✇ t❤❡ s✐♠✉•t❛♥❡♦✉s ❝♦♠♣❛r✐s♦♥ ♦❢ ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛• ✇✐t❤♦✉t r❡q✉✐r✐♥❣ ❛♥② ♣r❡•✐♠✐♥❛r② ❡①❛❝t ♠❛t❝❤✐♥❣ ❜❡t✇❡❡♥ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rts✳ ❚❤✐s ❝•❛ss ♦❢ ♥♦r♠s ❛•s♦ s❤❛r❡s ✐♠♣♦rt❛♥t ❝♦♥tr♦• ♣r♦♣❡rt✐❡s ❜♦t❤ ✇✐t❤ r❡s♣❡❝t t♦ ❞❡❢♦r♠❛t✐♦♥s ❛♥❞ s✐❣♥❛• tr❛♥s✲ ❢♦r♠❛t✐♦♥s✱ ❛s ✇❡•• ❛s r♦❜✉st♥❡ss t♦ s♠❛•• ❞✐s❝♦♥♥❡❝t✐♦♥s ❛♥❞ ❝❤❛♥❣❡s ♦❢ t♦♣♦•♦❣②✳ ❚❤❡ r❡s✉•t✐♥❣ ❍✐•❜❡rt str✉❝t✉r❡ ♦♥ ❢❝✉rr❡♥ts ♦♣❡♥s t❤❡ ✇❛② t♦ ❛ ✈❡r② ✇✐❞❡ ❝•❛ss ♦❢ ❛♣♣•✐❝❛t✐♦♥s✳ ❲❡ ❞❡t❛✐•❡❞ t✇♦ ♣♦ss✐❜•❡ ♥✉♠❡r✐❝❛• ❛♣♣•✐❝❛t✐♦♥s t❤❛t ❛r❡ ♦❢ ✉s✉❛• ✐♥✲ t❡r❡st ✐♥ t❤❡ ✜❡•❞ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ✿ ❛ ♠❛t❝❤✐♥❣ ♣✉rs✉✐t s❝❤❡♠❡ t♦ ❛❞❞r❡ss ❢❝✉rr❡♥t ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❛✈❡r❛❣✐♥❣ ❛♥❞ ❛♥ ❛❞❛♣t❛t✐♦♥ ♦❢ ▲❉❉▼▼ ❛•❣♦r✐t❤♠ ❢♦r ❞✐❢✲ ❢❡♦♠♦r♣❤✐❝ r❡❣✐str❛t✐♦♥ ♦❢ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❆ ♥❛t✉r❛• ❛♥❞ ❡①♣❡❝t❡❞ ❡①t❡♥s✐♦♥ ♦❢ r❡❣✐str❛t✐♦♥ ✐s t❤❡ ♣r♦❜•❡♠ ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❛ ❣r♦✉♣ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ✇❤✐❝❤ ❜r✐♥❣s s♣❡❝✐❛• ♣r♦❜•❡♠❛t✐❝s ❛♥❞ ✇✐•• ❜❡ t❤❡ s✉❜•❡❝t ♦❢ ❝❤❛♣t❡r ✹✳ ❋r♦♠ ❛♥✲ ♦t❤❡r ❛♥❣•❡✱ t❤❡ q✉❡st✐♦♥ ♦❢ ♦r✐❡♥t❛t✐♦♥ ♦❢ s❤❛♣❡s t❤❛t ✐s ❢✉♥❞❛♠❡♥t❛• ✇✐t❤ ❝✉rr❡♥ts ✐s ❛•s♦ tr❛♥s❢❡r❡❞ t♦ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✿ t❤✐s ✐ss✉❡ ✐s ♣r❡❝✐s❡•② t❤❡ s✉❜•❡❝t ♦❢ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ✽✼ ✽✽ ❈❤❛♣t❡r ✸ ◆♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❛♥❞ ✈❛r✐❢♦•❞s ❈♦♥t❡♥ts ✸✳✶ ❖r✐❡♥t❛t✐♦♥ ❛♥❞ •✐♠✐t❛t✐♦♥s ♦❢ ❝✉rr❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✸✳✶✳✶ ❖r✐❡♥t❛t✐♦♥ ❛♠❜✐❣✉✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✸✳✶✳✷ ❙❤❛♣❡ ❛♥♥✐❤✐•❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✸✳✶✳✸ ◆♦♥✲♦r✐❡♥t❡❞ r❡♣r❡s❡♥t❛t✐♦♥s ✐♥ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ✳ ✾✸ ✸✳✷ ❱❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✷✳✶ ❉❡✜♥✐t✐♦♥s ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✸✳✷✳✷ ◆♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❛s ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✸✳✸ ❑❡r♥❡• ♠❡tr✐❝s ♦♥ ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✸✳✸✳✶ ❑❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✸✳✸✳✷ ❑❡r♥❡•s ♦♥ ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✸✳✹ Pr♦♣❡rt✐❡s ♦❢ |❑❍❙ ♥♦r♠s ♦♥ ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✸✳✹✳✶ ❉❡♣❡♥❞❡♥❝② ♦♥ t❤❡ s❝❛•❡s ♦❢ t❤❡ ❦❡r♥❡•s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✸✳✹✳✷ ❈♦♥tr♦• ♦❢ t❤❡ ✈♦•✉♠❡ ❜② W ′ ✲♥♦r♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✸✳✹✳✸ ▼❡tr✐❝ ✈❛r✐❛t✐♦♥ ❢♦r♠✉•❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✸✳✺ |❡❣✐str❛t✐♦♥ ♦❢ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✸✳✺✳✶ ❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✸✳✺✳✷ ❉✐s❝r❡t❡ ❡①♣r❡ss✐♦♥s ❢♦r ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✻ ✸✳✺✳✸ ◆✉♠❡r✐❝❛• ❝♦♠♣•❡①✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✸✳✺✳✹ ❙✐♠✉•❛t✐♦♥s ❛♥❞ r❡s✉•ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✸✳✻ ❉✐s❝✉ss✐♦♥✱ ❡①t❡♥s✐♦♥s ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ✸✳✻✳✶ ❙❝❛•❡ ❛♥❞ ♥♦✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷ ✸✳✻✳✷ ❋✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✸✳✻✳✸ ❖r✐❡♥t❡❞ ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼ ✸✳✻✳✹ ❈♦♥❝•✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼ |❡s✉•ts ♦❢ t❤✐s ❝❤❛♣t❡r ✇❡r❡ ♣❛rt✐❛••② ♣✉❜•✐s❤❡❞ ✐♥ ❬✷✷(✳ ✽✾ ❋✐❣✉r❡ ✸✳✶✿ ❖r✐❡♥t❛t✐♦♥ ✐ss✉❡s ♦♥ ♥♦♥✲♦r✐❡♥t❛❜•❡ s✉r❢❛❝❡s✳ ✸✳✶ ❖r✐❡♥t❛t✐♦♥ ❛♥❞ •✐♠✐t❛t✐♦♥s ♦❢ ❝✉rr❡♥ts ❆s ✇❡ ♠❡♥t✐♦♥❡❞ s❡✈❡r❛• t✐♠❡s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs✱ ❝✉rr❡♥ts ❛r❡ ✐♥tr✐♥s✐❝❛••② ♠♦❞❡••✐♥❣ ♦r✐❡♥t❡❞ ♦❜•❡❝ts✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♣r♦❝❡ss ♦❢ r❡♣r❡s❡♥t✐♥❣ ❛ r❡❝t✐✜❛❜•❡ s✉❜s❡t X ❜② ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❝✉rr❡♥t C X ❛s ✐♥ ❝❤❛♣t❡r ✶ r❡q✉✐r❡s ❛♥ ♦r✐❡♥t❛t✐♦♥ ♦♥ X✳ ■❢ X ✐s t②♣✐❝❛••② ❛ ❝♦♥♥❡❝t❡❞ s✉❜♠❛♥✐❢♦•❞✱ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜•❡ ♦r✐❡♥t❛t✐♦♥s ❢♦r X✳ ❉❡♥♦t✐♥❣ X ❛♥❞ X t❤❡ t✇♦ r❡s✉•t✐♥❣ ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞s✱ ✇❡ ❤❛✈❡ ❜② ❞❡✜♥✐t✐♦♥ ✿ C X = -C X ❲❤❡♥ ✉s✐♥❣ ❦❡r♥❡• ♠❡tr✐❝s ♦♥ ❝✉rr❡♥ts✱ t❤❡ ♦r✐❡♥t❛t✐♦♥ ✐s ❞✐✛✉s❡❞ ✐♥ s♣❛❝❡ •❡❛❞✐♥❣ t❤✉s t♦ ♣♦t❡♥t✐❛• ❝❛♥❝❡••❛t✐♦♥s ♦❢ s♦♠❡ ♣❛rts ♦❢ s❤❛♣❡s✳ ❚②♣✐❝❛••②✱ ❢♦r ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦❢ ✇✐❞t❤ σ✱ δ ξ x + δ -ξ y W ′ = 2|ξ| 2 1 -e -|x-y| 2 /σ 2 ✭✸✳✶✮ ✇❤✐❝❤ ❣❡ts s♠❛•• ✇❤❡♥❡✈❡r |x -y| ✐s s♠❛•• ❝♦♠♣❛r❡❞ t♦ σ✳ ■♥ s❡✈❡r❛• s✐t✉❛t✐♦♥s✱ t❤✐s ♦r✐❡♥t❛t✐♦♥ ❞❡♣❡♥❞❡♥❝② ❛♥❞ ❝❛♥❝❡••❛t✐♦♥ ❡✛❡❝t ✐s ❛♥ ✐♠♣♦rt❛♥t ❞r❛✇❜❛❝❦ ♦❢ ❝✉rr❡♥ts ❛s ✜rst ♣♦✐♥t❡❞ ♦✉t ✐♥ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ❬✽✱ ✻✼( ❛♥❞ •❛t❡r ♦♥ ✐♥ ❬✷✽✱ ✹✶( ❢♦r t❤❡ ✉s❡ ♦❢ ❝✉rr❡♥ts ✐♥ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥s✬ s❡tt✐♥❣s✳ ■♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ •✐st t❤❡ ❞✐✛❡r❡♥t •✐♠✐t❛t✐♦♥s t❤❛t ♦r✐❡♥t❛t✐♦♥ ✐♥❞✉❝❡s ✐♥ ♦✉r ❝♦♥t❡①t✳ ✸✳✶✳✶ ❖r✐❡♥t❛t✐♦♥ ❛♠❜✐❣✉✐t② ❉✐✣❝✉•t✐❡s ❝❛♥ ✜rst ♦❝❝✉r ✇✐t❤ ♥♦♥✲♦r✐❡♥t❛❜•❡ s❤❛♣❡s ♦♥ ✇❤✐❝❤ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❞✐✛❡r✲ ❡♥t✐❛• ❢♦r♠s •✐❦❡ ✐♥ ❡q✉❛t✐♦♥ ✶✳✶✵ ✐s ♥♦t ✇❡••✲❞❡✜♥❡❞✳ ■❢ ✇❡ ❝♦♥s✐❞❡r ❢♦r ✐♥st❛♥❝❡ ❛ ▼ö❜✐✉s ❜❛♥❞ ❛s ✐♥ ✜❣✉r❡ ✸✳✶✱ ♦♥❡ ❝♦✉•❞ st✐•• ✐♠❛❣✐♥❡ t❤❛t ✇❡ ❝✉t t❤❡ ▼ö❜✐✉s ❜❛♥❞ ❛t s♦♠❡ •♦❝❛t✐♦♥ t♦ ♦❜t❛✐♥ ❛ ✉s✉❛• ♦r✐❡♥t❛❜•❡ ❜❛♥❞ t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❝✉r✲ r❡♥t✳ (❡t✱ ❢♦r t❤❡ ❦❡r♥❡• ♠❡tr✐❝✱ ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♦♣♣♦s✐t❡ ♦r✐❡♥t❛t✐♦♥s ✇✐t❤✐♥ ❛ ❝❡rt❛✐♥ ❞♦♠❛✐♥ ✐♥ s♣❛❝❡ ❛♥❞ ❡q✳✭✸✳✶✮✱ ✇❡ ❛❝t✉❛••② ❝❛♥❝❡• ❛ ♥♦♥✲③❡r♦ s✉r❢❛❝❡ ♣❛rt ♦❢ t❤❡ s❤❛♣❡ ❛r♦✉♥❞ t❤❡ ❝✉tt✐♥❣ ❝✉r✈❡ t❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ t②♣✐❝❛• ✇✐❞t❤ ♦❢ t❤❡ ❦❡r♥❡•✳ ❚❤✉s t❤❡r❡ ✐s ♥♦ ❝❛♥♦♥✐❝❛• ❛♥❞ s❛t✐s❢②✐♥❣ ✇❛② t♦ r❡♣r❡s❡♥t ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ✇✐t❤ ❝✉rr❡♥ts✳ ❚❤✐s ✜rst ❞r❛✇❜❛❝❦ r❡♠❛✐♥s ❛♥❡❝❞♦t❛• t❤♦✉❣❤✱ s✐♥❝❡ ♠♦st s❤❛♣❡s ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❛r❡ ♦r✐❡♥t❛❜•❡✳ ❍♦✇❡✈❡r✱ ❡✈❡♥ ✐❢ ♦r✐❡♥t❛❜•❡✱ ♦r✐❡♥t❛t✐♥❣ s❤❛♣❡s ❝♦♥s✐st❡♥t•② ✾✵ ❋✐❣✉r❡ ✸✳✷✿ ❆♥ ❡①❛♠♣•❡ ♦❢ ✇❤✐t❡ ♠❛tt❡r ✜❜❡r ❜✉♥❞•❡ ❡st✐♠❛t❡❞ ❢r♦♠ ❉✐✛✉s✐♦♥ ❚❡♥s♦r ■♠❛❣✐♥❣ ✭❉❚■✮ ✐••✉str❛t✐♥❣ t❤❡ ♣♦t❡♥t✐❛• ❞✐✣❝✉•t② ♦❢ ❝♦♥s✐st❡♥t ♦r✐❡♥t❛t✐♦♥ ♦❢ ❛•• ❞✐✛❡r❡♥t ✜❜❡rs✳ ❝❛♥ ❜❡ ❡✐t❤❡r ❛ ❞✐✣❝✉•t ♦r ❡✈❡♥ ✐••✲♣♦s❡❞ ♣r♦❜•❡♠ ✐♥ ❝❡rt❛✐♥ ❞❛t❛s❡ts✳ ❚❤✐s ✐s ♥♦t❛❜•② t❤❡ ❝❛s❡ ❢♦r ✜❜❡r ❜✉♥❞•❡s ✐♥ t❤❡ ✷❉ ♦r ✸❉ s♣❛❝❡ ❝♦♥s✐st✐♥❣ ♦❢ ♠❛♥② ❞✐✛❡r❡♥t ❛♥❞ ♣♦ss✐❜•② ❞✐s❝♦♥♥❡❝t❡❞ ♣✐❡❝❡s ♦❢ ❝✉r✈❡s✳ ■♥❞❡❡❞✱ ✐❢ ❛ ❣✐✈❡♥ s❤❛♣❡ ❤❛s N ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✱ t❤❡r❡ ❛r❡ 2 N ❞✐✛❡r❡♥t ♣♦ss✐❜•❡ ♦r✐❡♥t❛t✐♦♥s✳ ❯s✐♥❣ ❝✉rr❡♥ts ✇✐t❤ s✉❝❤ ♦❜•❡❝ts r❡q✉✐r❡s ❛ ✇❛② t♦ ♣r♦♣❛❣❛t❡ ♦r✐❡♥t❛t✐♦♥ ❢r♦♠ ♦♥❡ ♣❛rt ♦❢ t❤❡ s❤❛♣❡ t♦ ❛•• t❤❡ r❡st✳ ❚❡❝❤♥✐❝❛••②✱ t❤✐s •❡❛❞s t♦ ❛❞❞✐t✐♦♥❛• ♣r❡✲♣r♦❝❡ss✐♥❣ ✇❤✐❝❤ ❝❛♥ ❜❡ ♣❛rt✐❝✉•❛r•② tr✐❝❦② ❢♦r ❤✐❣❤•② ❞✐s❝♦♥♥❡❝t❡❞ ❜✉♥❞•❡ ♦❢ ❝✉r✈❡s ✇✐t❤ ♠❛♥② ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s •✐❦❡ t❤❡ ♦♥❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✇❤✐t❡ ♠❛tt❡r ✜❜❡rs ❡st✐♠❛t❡❞ ❢r♦♠ ❉❚■ ✐♥ ❜r❛✐♥ ✐♠❛❣✐♥❣✱ ✇❡ s❤♦✇ t❤❡ ❡①❛♠♣•❡ ♦❢ ♦♥❡ s✉❜•❡❝t ✐♥ ✜❣✉r❡ ✸✳✷✳ ❚❤✐s ❡①❛♠♣•❡ ❛•s♦ ❡♠♣❤❛s✐③❡s ❛♥♦t❤❡r ❞✐✣❝✉•t② ❛♣♣❡❛r✐♥❣ ✇✐t❤ s❡ts ♦❢ ❝✉r✈❡s ❝r♦ss✐♥❣ ❡❛❝❤ ♦t❤❡r ❢♦r ✇❤✐❝❤ t❤❡ ✈❡r② ♥♦t✐♦♥ ♦❢ ✬❝♦♥s✐st❡♥t✬ ♦r✐❡♥t❛t✐♦♥ ♠❛② ❜❡❝♦♠❡ ♠❡❛♥✐♥❣•❡ss✳ ■♥ s✉❝❤ ❝❛s❡s✱ t❤❡ ♦r✐❡♥t❛t✐♦♥ ✐s ❝•❡❛r•② ✐rr❡•❡✈❛♥t ❛♥❞ ♦♥❡ ✇♦✉•❞ •✐❦❡ t♦ tr❡❛t ♦❜•❡❝ts ❛s s❡ts ♦❢ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s✳ ✸✳✶✳✷ ❙❤❛♣❡ ❛♥♥✐❤✐•❛t✐♦♥ ❖r✐❡♥t❛t✐♦♥ ♠❛② ❛•s♦ r❡♣r❡s❡♥t ❛♥ ♦❜st❛❝•❡ ❡✈❡♥ ✐♥ t❤❡ s✐♠♣•❡st ❝❛s❡ ♦❢ ✉s✉❛• ♦r✐❡♥t❡❞ ❛♥❞ ❝♦♥♥❡❝t❡❞ s✉❜♠❛♥✐❢♦•❞s ❜❡❝❛✉s❡ str✉❝t✉r❡s •✐❦❡ s❤❛r♣ s♣✐♥❡s ♦r t❛✐•s ♥❛t✉r❛••② •❡❛❞✱ ✇❤❡♥ r❡♣r❡s❡♥t❡❞ ✐♥ ❛ ❦❡r♥❡• ❍✐•❜❡rt s♣❛❝❡ ♦❢ ❝✉rr❡♥ts✱ t♦ ❛♥♥✐❤✐•❛t✐♦♥ ♦❢ s♦♠❡ ♣❛rts ♦❢ t❤❡ s❤❛♣❡s✳ ❚❤✐s ❝♦♠❡s ❛❣❛✐♥ ❢r♦♠ ❡q✉❛t✐♦♥ ✭✸✳✶✮✱ ♦r✱ t♦ st❛t❡ ✐t ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛• ✇❛②✱ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t✱ ✇❤✐•❡ t❤❡ W ′ ✲♥♦r♠ ♦❢ ❛ s❡t ✐s ❛•✇❛②s ❞♦♠✐♥❛t❡❞ ❜② ✐ts ❍❛✉s❞♦r✛ ♠❡❛s✉r❡ ❛s ✇❡ s❛✇ ✐♥ s❡❝t✐♦♥ ✶✳✸✳✹ ❡q✳✶✳✶✼✱ t❤❡r❡ ❡①✐sts s❡ts ♦❢ ❣✐✈❡♥ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡ ❜✉t ✇✐t❤ ❛r❜✐tr❛r✐•② s♠❛•• W ′ ✲♥♦r♠✳ ❚❤❡ ♣•❛♥❡ ❝✉r✈❡ ♦❢ ✜❣✉r❡ ✸✳✸ ✐s ❛♥ ❡①❛♠♣•❡✳ ■♥❞❡❡❞✱ t❤❡ 1✲❍❛✉s❞♦r✛ ♠❡❛s✉r❡ ❜❡✐♥❣ t❤❡ •❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ✿ H 1 (γ) = 2(1 + ǫ) ≥ 2✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ γ ✐s ♣❛r❛♠❡tr✐③❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛• I✱ t❤❡♥ ✇❡ ❤❛✈❡ ✿ C γ 2 W ′ = I×I γ ′ (s) T K(γ(s), γ(t))γ ′ (t)dsdt ■❢ ✇❡ ❛ss✉♠❡✱ t♦ s✐♠♣•✐❢② ❝♦♠♣✉t❛t✐♦♥s✱ t❤❛t ❦❡r♥❡• K ✐s ❛ tr❛♥s•❛t✐♦♥ ✐♥✈❛r✐❛♥t ❦❡r♥❡• ♦❢ t❤❡ ❢♦r♠ K(x, y) = k(x -y) ■❞ R 2 ✇✐t❤ ▲✐♣s❝❤✐t③ r❡❣✉•❛r✐t②✱ t❤❡♥✱ ❜② ❡①♣❛♥❞✐♥❣ t❤❡ ✐♥t❡❣r❛•s ❛s t❤❡ s✉♠ ♦❢ t❤❡ ❢♦✉r ♣✐❡❝❡s ♦❢ ❝✉r✈❡s γ v 1 ✱ γ v 2 ✱ γ h 1 ✱ γ h 2 ✱ ✇❡ ❝❛♥ ❡✈❡♥t✉❛••② ✾✶ -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε γ v 1 γ v 2 γ h 2 γ h 1 ❋✐❣✉r❡ ✸✳✸✿ ❈♦✉♥t❡r✲❡①❛♠♣•❡ t♦ t❤❡ r❡❝✐♣r♦❝❛• ❞♦♠✐♥❛t✐♦♥✳ ❋♦r ❛•• ǫ✱ H 1 (γ) = 2(1 + ǫ) ≥ 2 ✇❤❡r❡❛s C γ W ′ -→ ǫ→0 0 s✐♠♣•✐❢② t❤❡ r❡s✉•t t❤❛♥❦s t♦ t❤❡ tr❛♥s•❛t✐♦♥✲✐♥✈❛r✐❛♥❝❡✱ ✇❤✐❝❤ ❣✐✈❡s ✿ C γ 2 W ′ = 2( C γv 1 2 W ′ -C γv 1 , C γv 2 W ′ ) + 2( C γ h 1 2 W ′ -C γ h 1 , C γ h 2 W ′ ) . ❙✐♥❝❡ C γ h 1 2 W ′ = C γ h 2 2 W ′ = [0,1]×[0,1] k(γ(s) -γ(t))ǫ 2 dsdt✱ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t❤❛t C γ h 1 2 W ′ -C γ h 1 , C γ h 2 W ′ = O(ǫ 2 )✳ ▼♦r❡♦✈❡r✱ C γv 1 2 W ′ -C γv 1 , C γv 2 W ′ = [0,1]×[0,1] [k(γ(s) -γ(t)) -k(γ(s) -γ(t) -(ǫ, 0))] dsdt . ❙✐♥❝❡ t❤❡ ❦❡r♥❡• k ✐s ❛ss✉♠❡❞ t♦ ❜❡ ▲✐♣s❝❤✐t③✱ [k(γ(s) -γ(t)) -k(γ(s) -γ(t) -(ǫ, 0))] ≤ ❈t❡ ǫ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❡✈❡♥t✉❛••② ♣r♦✈❡❞ t❤❛t C γ W ′ = O( √ ǫ) ✇❤❡r❡❛s ❢♦r ❛•• ǫ✱ H 1 (γ) ≥ 2✳ ■❢ t❤✐s ❛♥♥✐❤✐•❛t✐♦♥ ❝❛♥ ❜❡ ❜❡♥❡✜❝✐❛• ✐♥ ❝❡rt❛✐♥ s✐t✉❛t✐♦♥s ❜❡❝❛✉s❡ ✐t ✐s ✇❛② t♦ s♠♦♦t❤ ♥♦✐s❡✱ ✐t ❛•s♦ ♠❡❛♥s t❤❛t s♦♠❡ ♠❡❛♥✐♥❣❢✉• str✉❝t✉r❡s ♦❢ s❤❛♣❡s✱ ♠♦❞❡•❡❞ ❛s ❡•❡♠❡♥ts ♦❢ ❛ |❑❍❙ ♦❢ ❝✉rr❡♥ts✱ ✇✐•• ✈❛♥✐s❤ ✐♥ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✳ ❲❡ ✇✐•• ❞❡t❛✐•✱ ✐♥ ❛ ❝♦♠✐♥❣ s❡❝t✐♦♥✱ ✇❤❛t ✐♠♣♦rt❛♥t •✐♠✐t❛t✐♦♥s t❤✐s ❝❛♥ ✐♥❞✉❝❡ ✐♥ ♠❛t❝❤✐♥❣ ❝❡rt❛✐♥ ❡①❛♠♣•❡s ♦❢ ❝✉r✈❡s ♦r s✉r❢❛❝❡s ❤❛✈✐♥❣ s✉❝❤ ❦✐♥❞ ♦❢ str✉❝t✉r❡s✳ ❆♥♦t❤❡r ❢✉♥❞❛♠❡♥t❛• ❝♦r♦••❛r② t♦ t❤❛t ❝❛♥ ❜❡ s✉♠♠❡❞ ✉♣ ❜② t❤❡ ♠♦tt♦ ✬❝✉rr❡♥ts ❞♦♥✬t s❡❡ ❛♥②t❤✐♥❣ ❛t •❛r❣❡ s❝❛•❡s✬✳ ■♥❞❡❡❞✱ ✐❢ ✇❡ ❝♦♥s✐❞❡r ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ t❤❡ s♣❛❝❡ E ❛♥❞ ♠❛❦❡ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ s❝❛•❡ t❡♥❞ t♦ ∞✱ ✐✳❡ k e (x, y) = Id E ✱ t❤❡♥ ❢♦r ❛♥② d✲❞✐♠❡♥s✐♦♥❛• s✉❜♠❛♥✐❢♦•❞ ♦❢ X✱ t❤❡ ♥♦r♠ ♦❢ t❤❡ ❝✉rr❡♥t C X ❜❡❝♦♠❡s ✿ C X 2 W ′ = X×X ξ(x), ξ(y) = X ξ(x), X ξ(x) ◆♦✇✱ ❢r♦♠ ❙t♦❦❡s✬ ❢♦r♠✉•❛✱ ♦♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t X ξ(x) ✐s ❛ t❡r♠ t❤❛t ♦♥•② ❞❡♣❡♥❞s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ X✳ ❚❤❡r❡❢♦r❡✱ ❛t •❛r❣❡ s❝❛•❡✱ |❑❍❙ ♥♦r♠s ♦♥ ❝✉rr❡♥ts ❛r❡ ♦♥•② s❡♥s✐t✐✈❡ t♦ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ ♦❜•❡❝ts✱ ❛♥❞ ❛•• s❤❛♣❡s ✇✐t❤ ♥♦ ❜♦✉♥❞❛r② ✈❛♥✐s❤ ✐♥ W ′ ✳ ✾✷ ❋✐❣✉r❡ ✸✳✹✿ ◆♦r♠❛• ❜✉♥❞•❡s ♦❢ ❝✉r✈❡s ✇✐t❤ ❜♦✉♥❞❛r② ❜♦t❤ ✐♥ ✷❉ ❛♥❞ ✸❉✳ ✸✳✶✳✸ ◆♦♥✲♦r✐❡♥t❡❞ r❡♣r❡s❡♥t❛t✐♦♥s ✐♥ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ❚❤❡ ✐ss✉❡ ♦❢ ♦r✐❡♥t❛t✐♦♥ ✐s ♥♦t ♥❡✇ ✐♥ t❤❡ ✜❡•❞ ♦❢ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ❛♥❞ ❤❛s ❞r❛✇♥ ❛tt❡♥t✐♦♥ s✐♥❝❡ ✐ts ❡❛r•② ❜❡❣✐♥♥✐♥❣s✳ ❆❢t❡r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝✉rr❡♥ts✱ s❡✈❡r❛• ❝♦♥❝❡♣ts ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ❛❝❝♦✉♥t ❢♦r ♥♦♥✲♦r✐❡♥t❛❜✐•✐t②✳ ❚❤❡ ❝•♦s❡st t♦ t❤❡ ✐❞❡❛ ♦❢ ❝✉rr❡♥ts ✐s t♦ ❝♦♥s✐❞❡r ✢❛t ❝❤❛✐♥s ❛♥❞ ❝✉rr❡♥ts ♠♦❞✉•♦ 2 ❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✸✼( s❡❝t✐♦♥ ✹✳✷✳✷✻ ❛♥❞ s✉♠♠❡❞ ✉♣ ✐♥ ❬✻✼(✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ✐s t♦ ❞❡✜♥❡ ❛ q✉♦t✐❡♥t s♣❛❝❡ ❜② ✐❞❡♥t✐❢②✐♥❣ r❡❝t✐✜❛❜•❡ ❝✉rr❡♥ts t❤❛t ❞✐✛❡rs ❜② 2.Q ✇❤❡r❡ Q ✐s s♦♠❡ r❡❝t✐✜❛❜•❡ ❝✉rr❡♥t✳ ■♥ t❤❛t q✉♦t✐❡♥t s♣❛❝❡✱ ✇❡ ❡❛s✐•② s❡❡ t❤❛t C X = -C X ❢♦r ❛♥② r❡❝t✐✜❛❜•❡ s✉❜s❡t X✱ ❛♥❞ t❤✉s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ✐♥s❡♥s✐t✐✈❡ t♦ ♦r✐❡♥t❛t✐♦♥✳ ❆•t❤♦✉❣❤ s❡✈❡r❛• ❝♦♠♣❛❝t♥❡ss✱ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ❡①✐st❡♥❝❡ r❡s✉•ts ✇❡r❡ s❤♦✇♥ ❢♦r t❤❡s❡ ♦❜•❡❝ts✱ t❤❡② ❛r❡ ❡ss❡♥t✐❛••② t❤❡♦r❡t✐❝❛• ❛t t❤❡ ♣r❡s❡♥t t✐♠❡ ❛♥❞ ❞♦ ♥♦t s❡❡♠ t♦ ❛❞❛♣t ❡❛s✐•② t♦ ♦✉r t②♣❡ ♦❢ ❛♣♣•✐❝❛t✐♦♥s✳ ❈❤r♦♥♦•♦❣✐❝❛••②✱ t❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ t❤❛t ✇❛s ♣r♦♣♦s❡❞ ✐s t❤❡ ♦♥❡ ♦❢ ✈❛r✐❢♦•❞s✱ ❛s ♣✐♦♥❡❡r❡❞ ❜② ❋✳ ❆•♠❣r❡♥ ✐♥ ❬✽(✱ •❛t❡r ❞❡✈❡•♦♣❡❞ ❜② ❲✳ ❆••❛r❞ ✐♥ ❬✹( ❛♥❞ ✉s❡❞ ✐♥ ✈❛r✐♦✉s ❛♣♣•✐❝❛t✐♦♥s ✐♥ ❝❛•❝✉•✉s ♦❢ ✈❛r✐❛t✐♦♥s ❬✶✼✱ ✺✺(✳ ❲❡ s❤❛•• ❢♦❝✉s ♦♥ t❤✐s ♣❛rt✐❝✉•❛r ♦♥❡ ❛♥❞ ♣r❡s❡♥t ✐t ✐♥ ❞❡t❛✐• ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❆s ✇❡ s❤❛•• s❡❡ ✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ t❤❡ ❡ss❡♥t✐❛• st❡♣ ✐s t♦ ❞❡✜♥❡ ❛ ♠♦❞❡• ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡s✱ ✇❤✐❝❤ ✐s t❤❡ r♦•❡ ♦❢ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ❋✐♥❛••②✱ ✇❡ ❝♦✉•❞ ❛•s♦ ❡✈♦❦❡ t❤❡ ♠♦r❡ r❡❝❡♥t ♥♦t✐♦♥ ♦❢ ♥♦r♠❛• ❝②❝•❡ ♣r❡s❡♥t❡❞ ✐♥ ❬✸✾(✱ ✇❤✐❝❤ ✇❛s ❡ss❡♥t✐❛••② ✉s❡❞ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❣❡♦♠❡tr② ❛s ❛ ✇❛② t♦ ❞❡✜♥❡ ❛♥❞ ❝♦♠♣✉t❡ ❝✉r✈❛t✉r❡ t❡♥s♦rs ❢♦r ❞✐s❝r❡t❡ ♣♦•②❤❡❞r❛• s✉r❢❛❝❡s ❛♥❞ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ t❤❡ ❝♦♥t✐♥✉♦✉s ✈❡rs✐♦♥s ❬✷✺(✳ ❚♦ ♣r❡s❡♥t ✐t ❜r✐❡✢②✱ t❤❡ ✐❞❡❛ ♦❢ ♥♦r♠❛• ❝②❝•❡s ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ✉♥✐t ♥♦r♠❛• ❜✉♥❞•❡ ♦❢ ❛ s✉❜♠❛♥✐❢♦•❞ X ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s N U (X) = {(x, n(x))/ x ∈ X, n(x) ∈ (T x X) ⊥ , n(x) = 1} ❛♥❞ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s✉❜♠❛♥✐❢♦•❞ ♦❢ ❞✐♠❡♥s✐♦♥ n -1 ✐♥ t❤❡ 2n✲❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡ E × E✳ ❚❤❡ ✐♠♣♦rt❛♥t ♣♦✐♥t ✐s t❤❛t N U (X) ✐s ❛•✇❛②s ♦r✐❡♥t❛❜•❡ ✭❡✈❡♥ ✐❢ X ✐s ♥♦t✮ ❛♥❞ ❝❛♥ ❜❡ ❝❛♥♦♥✐❝❛••② ♦r✐❡♥t❡❞✳ ■t ❝❛♥ ❜❡ r♦✉❣❤•② ✐❞❡♥t✐✜❡❞ t♦ ❛ t✉❜✉•❛r ❡♥✈❡•♦♣❡ ❛r♦✉♥❞ X ❛s ♦♥ t❤❡ ❝✉r✈❡ ❡①❛♠♣•❡s ♦❢ ✜❣✉r❡ ✸✳✹✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ❝❛♥ ❜❡ ❣❡♥❡r❛•✐③❡❞ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ X ✐s ❛ ♠❛♥✐❢♦•❞ ✇✐t❤ ❜♦✉♥❞❛r✐❡s ♦r ❢♦r ♣♦•②❤❡❞r♦♥s ❜② ❛❞❞✐♥❣ ♥♦r♠❛• ❝♦♥❡s ✿ ✇❡ r❡❢❡r t♦ ❬✷✺( ❢♦r ♠♦r❡ ❞❡t❛✐•s✳ ◆♦✇ t❤❡ ♥♦r♠❛• ❝②❝•❡ ❛ss♦❝✐❛t❡❞ t♦ X ✐s s✐♠♣•② t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ❛ ❝✉rr❡♥t ✐♥ E × E ♦❢ t❤❡ ♥♦r♠❛• ❜✉♥❞•❡ N U (X)✳ ■♥ ♦✉r ❝♦♥t❡①t ♦❢ ✇♦r❦✱ t❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ❞✐✛❡r s✐❣♥✐✜❝❛♥t•② ❢r♦♠ ❝✉rr❡♥ts ✐♥ t❤❡ t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇ ✿ ❦❡r♥❡•s ❝♦✉•❞ ❜❡ ❞❡✜♥❡❞ ❛s ✇❡•• ❛♥❞ ♣r♦✈✐❞❡ ♠❡tr✐❝s ❜❡t✇❡❡♥ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❛s ✇✐t❤ ✉s✉❛• ❝✉rr❡♥ts ✇❤✐❝❤ ❝♦✉•❞ ❜❡ ✉s❡❞ t♦ ❞r✐✈❡ ♠❛t❝❤✐♥❣ ❛•❣♦r✐t❤♠s✳ ❚❤✐s ✐s ♣r❡s❡♥t•② st✉❞✐❡❞ ❜② ❏✳ ••❛✉♥ès ❛♥❞ ❤❛s ✾✸ ❜❡❡♥ ✐♠♣•❡♠❡♥t❡❞ ✐♥ t❤❡ ♣•❛♥❛r ❝✉r✈❡ ❝❛s❡✳ ❊✈❡♥ t❤♦✉❣❤ t❤✐s ♠♦❞❡• ✐s r✐❝❤ ❛♥❞ ✐s ♥♦t ✐♥tr❛❝t❛❜•❡ ❢♦r ❛❝t✉❛• ❝♦♠♣✉t❛t✐♦♥s✱ ✐t •❡❛❞s t♦ ❛♥ ✐♠♣♦rt❛♥t ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡s✱ ✜rst ❜❡❝❛✉s❡ ✇❡ ❛r❡ •♦♦❦✐♥❣ ❛t ❝✉rr❡♥ts ✐♥ t❤❡ ❜✐❣❣❡r s♣❛❝❡ E × E ❜✉t ❛•s♦ ❜❡❝❛✉s❡ t❤❡ ❛❝t✉❛• ♠♦❞❡•❡❞ ♦❜•❡❝t ✐s ♥♦ ♠♦r❡ X ❜✉t ✐ts ✉♥✐t ♥♦r♠❛• ❜✉♥❞•❡ N U (X) t❤❛t ❝❛♥ ❜❡ ❛ s✉❜♠❛♥✐❢♦•❞ ♦❢ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥ ✭❡✳❣ ✐♥ t❤❡ ✸❉ ❝✉r✈❡ ❝❛s❡✮✳ ❚❤✉s✱ ✐t s❡❡♠s ✉♥•✐❦❡•②✱ ❛t t❤❡ ♣r❡s❡♥t t✐♠❡✱ t❤❛t ♥♦r♠❛• ❝②❝•❡ r❡♣r❡s❡♥t❛t✐♦♥s ❝♦✉•❞ ♣r♦✈✐❞❡ ♠❡tr✐❝s ✇✐t❤ ♥✉♠❡r✐❝❛• ❝♦♠♣•❡①✐t✐❡s ❝♦♠♣❛r❛❜•❡ t♦ ❝✉rr❡♥ts✳ ✸✳✷ ❱❛r✐❢♦•❞s ❲❡ ♥♦✇ ❢♦❝✉s ❡①❝•✉s✐✈❡•② ♦♥ ✈❛r✐❢♦•❞s✳ ❲❡ s❤❛•• ❡①♣•❛✐♥ ✐♥ ✇❤❛t r❡s♣❡❝t ✈❛r✐❢♦•❞s ♦✛❡r ✐♥ ♦✉r s❡♥s❡ ❛ ♥❛t✉r❛• ❛♥❞ ❝♦♥✈❡♥✐❡♥t s❡tt✐♥❣ t♦ ❡♠❜❡❞ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❜♦t❤ ❢r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• ❛♥❞ ♥✉♠❡r✐❝❛• ❛s♣❡❝t✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r❡s❡♥t ✈❛r✐❢♦•❞s ✜rst ❛s ❛ t❤❡♦r❡t✐❝❛• ♦❜•❡❝t✱ ❢♦••♦✇✐♥❣ t❤❡ ♦r✐❣✐♥❛• ✇♦r❦ ♦❢ ❆•♠❣r❡♥ ✐♥ ❬✽( ❛♥❞ •❛t❡r ♦❢ ❆••❛r❞ ✐♥ ❬✹(✳ ❲❡ t❤❡♥ ❡①♣•❛✐♥ ✐♥ ❞❡t❛✐•✱ ✐♥ ❛ s♣✐r✐t ✈❡r② s✐♠✐•❛r t♦ ❝✉rr❡♥ts✱ t❤❡ ❢✉♥❞❛♠❡♥t❛• r❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ✈❛r✐❢♦•❞s ❛♥❞ ✉♥♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s❡ts ❛♥❞ s❤♦✇ ❤♦✇ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ❡✣❝✐❡♥t•② tr❛♥s❝r✐❜❡❞ ✐♥ ❛ ♣r❛❝t✐❝❛• ✇❛②✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❦❡r♥❡• ♠❡tr✐❝s ♦♥ ✈❛r✐❢♦•❞s ✐s ♣♦st♣♦♥❡❞ t♦ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ✸✳✷✳✶ ❉❡✜♥✐t✐♦♥s ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ❱❛r✐❢♦•❞s ❤❛✈❡ ❜❡❡♥ ✜rst ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ❛s ❛ ✇❛② t♦ ❛❞❞r❡ss P•❛t❡❛✉✬s ♣r♦❜•❡♠ ♦❢ ✜♥❞✐♥❣ •❡❛st ❛r❡❛ s✉r❢❛❝❡s ✇✐t❤ ❛ ♣r❡s❝r✐❜❡❞ ❜♦✉♥❞❛r②✳ ❚❤❡s❡ ❞❡✈❡•♦♣♠❡♥ts ❝•❡❛r•② ❞♦ ♥♦t ❡♥t❡r ✐♥ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ♠❛♥✉s❝r✐♣t✳ ❖✉r ♣✉r♣♦s❡ ❤❡r❡ ✐s r❛t❤❡r t♦ ❝♦♥♥❡❝t ❝♦♥✈❡♥✐❡♥t•② ✈❛r✐❢♦•❞s t♦ t❤❡ ❢r❛♠❡✇♦r❦ ❛♥❞ •❛♥❣✉❛❣❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠②✳ ❚❤❡r❡❢♦r❡✱ ✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ ✇❡ ✇✐•• ❢♦❝✉s ♠❛✐♥•② ♦♥ ❞❡✜♥✐t✐♦♥s ♦❢ s✉❝❤ ♦❜•❡❝ts ❛♥❞ ❡①♣•❛✐♥ ✐♥ ✇❤❛t r❡s♣❡❝t t❤❡s❡ ❞❡✜♥✐t✐♦♥s ❛r❡ ❝♦♠✲ ♣✉t❛t✐♦♥❛••② r❡•❡✈❛♥t✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ❜❡❤✐♥❞ ✈❛r✐❢♦•❞s ✐s t♦ r❡♣r❡s❡♥t ❛♥② r❡❝t✐✜❛❜•❡ s❡t ✭♦r✐❡♥t❛❜•❡ ♦r ♥♦t✮ ❛s ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ ✉♥♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡s s♣r❡❛❞ ✐♥ t❤❡ ❡♠❜❡❞❞✐♥❣ s♣❛❝❡ E✳ ❆s ✇❡ s❤❛•• s❡❡✱ ✈❛r✐❢♦•❞s ❡♥❝♦♠♣❛ss ♥♦t ♦♥•② ♠❛♥✐❢♦•❞s ❛♥❞ r❡❝t✐✜❛❜•❡ s❡ts ❜✉t ♠♦r❡ ❣❡♥❡r❛••② s❡ts ♦❢ ❞✐r❡❝t✐♦♥s ✐♥ t❤❡ s♣❛❝❡✳ ▲❡t✬s t❛❦❡ ❛❣❛✐♥ t❤❡ ❝♦♥✈❡♥t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs✱ E ❜❡✐♥❣ ❛ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ n ❛♥❞ d ❛♥ ✐♥t❡❣❡r ✇✐t❤ 0 ≤ d ≤ n✳ ❲❤❛t ✇❡ ✜rst ♥❡❡❞ ✐s ❛ ✇❛② t♦ r❡♣r❡s❡♥t t❛♥❣❡♥t s♣❛❝❡s ♦❢ ❞✐♠❡♥s✐♦♥ d ✐♥ t❤❡ s♣❛❝❡ E✳ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❚❤❡ ♣r♦♣❡r s♣❛❝❡ ✐♥ t❤❛t r❡❣❛r❞ ✐s ♣r❡❝✐s❡•② t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❞❡✜♥❡❞ ❛s ✿ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✶✳ ❚❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ♦❢ ❞✐♠❡♥s✐♦♥ d ✐♥ E✱ ❞❡♥♦t❡❞ G d (E)✱ ✐s t❤❡ s❡t ♦❢ ❛•• d✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ♦❢ E✳ ■t ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ t♦ t❤❡ q✉♦t✐❡♥t s♣❛❝❡ ♦❢ ❛•• ❢❛♠✐•✐❡s ♦❢ d ✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦rs ♦❢ E ❜② t❤❡ ❡q✉✐✈❛•❡♥❝❡ r❡•❛t✐♦♥ ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ ❢❛♠✐•✐❡s t❤❛t ❣❡♥❡r❛t❡ t❤❡ s❛♠❡ s✉❜s♣❛❝❡✳ ❆s ❛ q✉♦t✐❡♥t s♣❛❝❡✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t G d (E) ✐♥❤❡r✐ts ❛ str✉❝t✉r❡ ♦❢ ❝♦♠♣❛❝t |✐❡✲ ♠❛♥♥✐❛♥ ♠❛♥✐❢♦•❞ ♦❢ ❞✐♠❡♥s✐♦♥ d(n-d) ✭❝❢ ❬✶✻( ❝❤❛♣ ✹ ❛♥❞ ❬✽✸(✮✳ ■t ✐s ❛•s♦ ❛ ❝•❛ss✐❝❛• r❡s✉•t t❤❛t G d (E) ✐s ❛ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ •▲(E) ❛♥❞ t❤❡ s✉❜❣r♦✉♣ ♦❢ ❞✐r❡❝t ✐s♦♠❡tr✐❡s ❙❖(E)✳ ✾✹ |❡♠❛r❦ ✸✳✷✳✶✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ❛♣♣•✐❝❛t✐♦♥ G d (E) → G n-d (E), V → V ⊥ ✱ ♦♥❡ s❡❡s ❡❛s✐•② t❤❛t G d (E) ❛♥❞ G n-d (E) ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ ❡❛❝❤ ♦t❤❡r✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡s ✇❤❡r❡ d = 1 ♦r d = n -1✱ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ✐s ♥♦t❤✐♥❣ ❡•s❡ t❤❛♥ t❤❡ r❡❛• ♣r♦•❡❝t✐✈❡ s♣❛❝❡ ♦❢ E✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❝❛♥ ❜❡ ❛•s♦ ❞❡s❝r✐❜❡❞ ✐♥ ❛ •❡ss ❛❜str❛❝t ✇❛② ❜② ❣✐✈✐♥❣ ❡①♣•✐❝✐t •♦❝❛• ❝❤❛rts✳ ■❢ V ✐s ❛♥ ❡•❡♠❡♥t ♦❢ G d (E)✱ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r U V = {W ∈ G d (E) | W ∩ V ⊥ = {0}}✱ ✇❤✐❝❤ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ V ✐♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ▲❡♠♠❛ ✸✳✷✳✶✳ ❊✈❡r② ❡•❡♠❡♥t W ∈ U V ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s W = {v + l(v), v ∈ V } ❢♦r ❛ ❝❡rt❛✐♥ •✐♥❡❛r ❢✉♥❝t✐♦♥ l ∈ L(V, V ⊥ ) ❛♥❞ l ✐s ✉♥✐q✉❡•② ❞❡t❡r♠✐♥❡❞ ❜② W ✳ Pr♦♦❢✳ ■❢ W ∈ U V ✱ ❢♦r ❞✐♠❡♥s✐♦♥❛•✐t② r❡❛s♦♥✱ ✇❡ ❤❛✈❡ W ⊕ V ⊥ = E ❛♥❞ ❞❡♥♦t✐♥❣ h t❤❡ •✐♥❡❛r ♣r♦•❡❝t♦r ♦♥ V ⊥ ❛ss♦❝✐❛t❡❞ t♦ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❡ ❞❡❞✉❝❡ t❤❛t W = {u -h(u) / u ∈ E}✳ ◆♦✇ ✇r✐t✐♥❣ ❢♦r ❛•• u ∈ E, u = v + v ⊥ ✇✐t❤ v ∈ V ❛♥❞ v ⊥ ∈ V ⊥ ✱ ✇❡ ❤❛✈❡ u -h(u) = v + v ⊥ -h(v) -h(v ⊥ ) = v -h(v) s✐♥❝❡ h(v ⊥ ) = v ⊥ ✳ ❚❤✉s W = {v + l(v) / v ∈ V } ✇✐t❤ l = (-h) |V ❛♥❞ l ✐s ✐♥❞❡❡❞ ✉♥✐q✉❡•② ❞❡t❡r♠✐♥❡❞ ❜② W ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ ❛ ❜✐•❡❝t✐✈❡ ♠❛♣ ψ V : U V → L(V, V ⊥ ), W → l✳ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✶✳ ❚❤❡ s❡t ♦❢ ❝❤❛rts (U V , ψ V ) ❞❡✜♥❡s ❛♥ ❛t•❛s ♦♥ t❤❡ ♠❛♥✐❢♦•❞ G d (E) ❛♥❞✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ❛ ♥❛t✉r❛• ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T V G d (E) ❛♥❞ L(V, V ⊥ )✳ Pr♦♦❢✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✻✽( s❡❝t✐♦♥ ✷✳✷ ❢♦r t❤❡ ❝♦♠♣•❡t❡ ♣r♦♦❢ ♦❢ t❤❛t✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ •r❛ss♠❛♥♥✐❛♥ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s✉❜♠❛♥✐❢♦•❞ ♦❢ L(E) ❜② ✐❞❡♥✲ t✐❢②✐♥❣ ❛♥② V ∈ G d (E) ✇✐t❤ t❤❡ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥ p V ♦♥ V ✇❤✐❝❤ ❣✐✈❡s t❤❡ ❡♠❜❡❞❞✐♥❣ ✿ G d (E) -→ L(E) V -→ p V ✭✸✳✷✮ ❲❡ s❤❛•• ♠❛❦❡ ✉s❡ ♦❢ t❤❛t ✇❤❡♥ ❞❡✜♥✐♥❣ ❦❡r♥❡•s ♦♥ G d (E) ✭s❡❝t✐♦♥ ✸✳✸✮✳ ■t ❣✐✈❡s ❛•s♦ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛② t♦ ✐❞❡♥t✐❢② t❛♥❣❡♥t s♣❛❝❡s ❛♥❞ ❝♦♠♣✉t❡ ✈❛r✐❛t✐♦♥s✳ ■♥❞❡❡❞✱ ✐❢ V t ❢♦r t ∈] -ǫ, ǫ[ ✐s ❛ s♠♦♦t❤ ❝✉r✈❡ ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ t❤❡♥ p Vt ✐s ❛ s♠♦♦t❤ ❝✉r✈❡ ♦♥ L(E)✳ ▲❡t✬s ❞❡♥♦t❡ δp V . = d dt t=0 p Vt ✳ ❚❤❡♥✱ ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤❡ r❡•❛t✐♦♥s p Vt • p Vt = p Vt ❛♥❞ p Vt = p * Vt ✱ ✇❡ ✜♥❞ t❤❛t ✿ δp V = δ p V • p V + (δ p V • p V ) * ▼♦r❡♦✈❡r✱ ❢♦r ❛•• v, v ′ ∈ V ✱ ✇❡ ❤❛✈❡ ✿ δp V (v), v ′ = δp V (v) + δp * V (v), v ′ = δp V (v), v ′ + v, δp V (v ′ ) ❆♥❞ t❤✉s ❢♦r ❛•• v, v ′ ∈ V ✱ v, δp V (v ′ ) = 0✳ ❲❡ ❞❡❞✉❝❡ t❤❛t δp V = p V ⊥ • δp V ❛♥❞ ✿ δp V = p V ⊥ • δ p V • p V + (p V ⊥ • δ p V • p V ) * . ✾✺ ■t r❡s✉•ts t❤❛t ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ❛♥② ❡•❡♠❡♥t δp V ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T V G d (E) ✇✐t❤ t❤❡ ❛♣♣•✐❝❛t✐♦♥ p V ⊥ • δ p V • p V = (p V ⊥ • δ p V ) |V ∈ L(V, V ⊥ ) . ▼♦r❡♦✈❡r✱ ✐❢ A(t) ✐s ❛ ❝✉r✈❡ ✐♥ GL(E) ✇✐t❤ A(0) = ■❞ ❛♥❞ A(t).V ❞❡♥♦t❡s t❤❡ ❛❝t✐♦♥ ♦❢ GL(E) ♦♥ G d (E)✱ ✐t ✐s ❛ str❛✐❣❤t❢♦r✇❛r❞ ✈❡r✐✜❝❛t✐♦♥ t❤❛t✱ ✉♥❞❡r t❤✐s ✐❞❡♥t✐✜❝❛t✐♦♥ ✿ d dt t=0 A(t).V = p ⊥ V • A ′ (0) |V . ✭✸✳✸✮ ❋✐♥❛••②✱ •❡t✬s ♠❡♥t✐♦♥ t❤❛t t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❝❛♥ ❜❡ ❛•s♦ •✐♥❦❡❞ t♦ t❤❡ s♣❛❝❡ ♦❢ d✲✈❡❝t♦rs ✉s❡❞ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝✉rr❡♥ts✳ ❚❤✐s ✐s ❢♦r♠✉•❛t❡❞ ❜② t❤❡ P•ü❝❦❡r ❡♠❜❡❞❞✐♥❣ ♣r♦♣❡rt② ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✷✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❛♣♣•✐❝❛t✐♦♥ ✐s ❛♥ ❡♠❜❡❞❞✐♥❣ ✿ i P : G d (E) -→ P Λ d E vect(v 1 , ..., v d ) -→ [v 1 ∧ ... ∧ v d ] ✇❤❡r❡ P Λ d E ✐s t❤❡ r❡❛• ♣r♦•❡❝t✐✈❡ s♣❛❝❡ ✭✐✳❡ t❤❡ s❡t ♦❢ ❛•• •✐♥❡s✮ ♦❢ Λ d E✳ ■t ✐s ❡✈❡♥ ❛ ❤♦♠❡♦♠♦r♣❤✐s♠ ✐♥ t❤❡ ❝❛s❡s d = 1 ♦r d = n -1✳ ❆s ❛ r❡s✉•t✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛♥ ✉♥♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• t❛♥❣❡♥t s♣❛❝❡ ❛s ❛♥ ❡•❡♠❡♥t ♦❢ P Λ d E ✱ ✇❤✐❝❤ ✐s ❤♦✇❡✈❡r ❛ s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ n d -1✱ ♠✉❝❤ ❜✐❣❣❡r ✐♥ ❣❡♥❡r❛• t❤❛♥ G d (E)✳ Pr✐♥❝✐♣•❡ ❛♥❣•❡s ❆ ❝♦♠♠♦♥ ❛♥❞ ✉s❡❢✉• ♥♦t✐♦♥ ❝•♦s❡•② r❡•❛t❡❞ t♦ •r❛ss♠❛♥♥✐❛♥ ✐s t❤❡ ♦♥❡ ♦❢ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s ❜❡t✇❡❡♥ s✉❜s♣❛❝❡s✳ ❲❡ ✈❡r② ❜r✐❡✢② r❡❝❛•• t❤❡ ❢❡✇ ♣r♦♣❡rt✐❡s t❤❛t s❤❛•• ❜❡ ✐♥t❡r❡st✐♥❣ ❢♦r ✉s ✐♥ t❤❡ ❢♦••♦✇✐♥❣✳ ■❢ V, W ∈ G d (E) ❛r❡ t✇♦ d✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ♦❢ E✱ t❤❡ d ♣r✐♥❝✐♣•❡ ❛♥❣•❡s θ 1 , .., θ d ❜❡t✇❡❡♥ t❤❡♠ ❛r❡ ❞❡✜♥❡❞ r❡❝✉rs✐✈❡•② ❜② t❤❡ r❡•❛t✐♦♥s ✿ cos(θ k ) = max v k ∈V max w k ∈W v k |v k | , w k |w k | with ∀i ∈ {1, .., k -1}, v k ⊥v i , w k ⊥w i ❆s ✇❡ s❡❡✱ t❤❡ ✜rst ♣r✐♥❝✐♣•❡ ❛♥❣•❡ ✐s t❤❡ s♠❛••❡st ❛♥❣•❡ ❜❡t✇❡❡♥ ❛•• ♣❛✐rs ♦❢ ✉♥✐t ✈❡❝t♦rs ✐♥ V ❛♥❞ W ✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t✇♦ •✐♥❡s✱ ✐t ✐s ❡①❛❝t•② t❤❡ ♥♦♥✲♦r✐❡♥t❡❞ ❛♥❣•❡ ❜❡t✇❡❡♥ t❤❡♠✳ ❚❤❡ ♥❡①t θ i ✬s ❛r❡ ❞❡✜♥❡❞ ✐♥ ❛ s✐♠✐•❛r ✇❛② ❜② r❡str✐❝t✐♥❣ t♦ ♦rt❤♦❣♦♥❛• ✈❡❝t♦rs✳ ❚❤❡ ❛♥❣•❡s ❛r❡ ❛•• ✈❛♥✐s❤✐♥❣ ✐❢ ❛♥❞ ♦♥•② ✐❢ V = W ❛♥❞ ✇❡ ❡❛s✐•② s❡❡ t❤❛t t❤❡r❡ ❛r❡ ❛t ♠♦st min(d, n -d) ♥♦♥✲③❡r♦ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s✳ ❚❤❡ ❝❡♥tr❛• r❡s✉•t t❤❛t ✇❡ ❜♦rr♦✇ t♦ ❬✽✸( s❤♦✇s t❤❛t t❤❡ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s ❝♦♠♣•❡t❡•② ❞❡t❡r♠✐♥❡ t❤❡ r❡•❛t✐✈❡ ♣♦s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ s✉❜s♣❛❝❡s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡r❡ ❡①✐sts ♦rt❤♦♥♦r♠❛• ❢r❛♠❡s (v 1 , .., v d ) ❛♥❞ (w 1 , .., w d ) ♦❢ V ❛♥❞ W s✉❝❤ t❤❛t cos(θ i ) = v i , w i ❛♥❞ s✉❝❤ t❤❛t t❤❡ ❞✐✛❡r❡♥t ❛♥❣•❡ 2✲♣•❛♥❡s ❙♣❛♥{v i , w i } ❛r❡ ♠✉t✉❛••② ♦rt❤♦❣♦♥❛• t♦ ❡❛❝❤ ♦t❤❡r✳ ❇❡ ❝❛r❡❢✉• t❤❛t ❙♣❛♥{v i , w i } ✐s ❛ ❞❡❣❡♥❡r❛t❡ 2✲♣•❛♥❡ ✇❤❡♥❡✈❡r θ i = 0✳ ❉❡✜♥✐t✐♦♥ ♦❢ ✈❛r✐❢♦•❞s ❋♦••♦✇✐♥❣ ❝•♦s❡•② ❬✹(✱ ✈❛r✐❢♦•❞s ❛r❡ ♥♦✇ ❞❡✜♥❡❞ ♣r❡❝✐s❡•② ❛s ❞✐str✐❜✉t✐♦♥s ♦❢ ✉♥♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡s ✐♥ E ✿ ✾✻ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✷✳ ❆ d✲❞✐♠❡♥s✐♦♥❛• ✈❛r✐❢♦•❞ ♦♥ E ✐s ❛ ❇♦r❡• ✜♥✐t❡ ♠❡❛s✉r❡ ✭♦r ❞✐s✲ tr✐❜✉t✐♦♥✮ ♦♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ E × G d (E)✱ ✐✳❡ ❛♥ ❡•❡♠❡♥t ♦❢ C 0 (E × G d (E)) ′ ✳ ◆♦t❡ t❤❛t t❤✐s ❞✐✛❡rs ❢r♦♠ t❤❡ ♦r✐❣✐♥❛• ❞❡✜♥✐t✐♦♥ ♦❢ ✈❛r✐❢♦•❞s ❣✐✈❡♥ ❜② ❆•♠❣r❡♥ ❜✉t ❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❛s ❡①♣•❛✐♥❡❞ ✐♥ t❤❡ ♣r❡❢❛❝❡ ♦❢ ❬✽(✳ ❚❤❡ r❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ♠❡❛s✉r❡ ❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♣♦✐♥t ♦❢ ✈✐❡✇ ✐s✱ ❛s ✉s✉❛•✱ ❣✐✈❡♥ ❜② |✐❡s③ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ✿ ∀ω ∈ C 0 (E × G d (E)), µ(ω) = E×G d (E) ω(x, V )dµ(x, V ) ■♥ ♣❛rt✐❝✉•❛r✱ ❉✐r❛❝s ✐♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s ❛r❡ ♦❢ t❤❡ ❢♦r♠ δ (x,V ) ✇✐t❤ x ∈ E ❛♥❞ V ∈ G d (E) ❛♥❞ ❛❝t ♦♥ ❢✉♥❝t✐♦♥s ♦❢ C 0 (E × G d (E)) ❜② t❤❡ r❡•❛t✐♦♥ ✿ ∀ω ∈ C 0 (E × G d (E)), δ (x,V ) (ω) = ω(x, V ) ■♥ t❤✐s ❝♦♥t❡①t✱ ❛ ❉✐r❛❝ ❝♦♥s✐sts ✐♥ t❤❡ ❞❛t❛ ♦❢ ❛ ♣♦s✐t✐♦♥ x ✐♥ t❤❡ s♣❛❝❡ ❛tt❛❝❤❡❞ t♦ ❛ d✲❞✐♠❡♥s✐♦♥❛• ✭♥♦♥✲♦r✐❡♥t❡❞✮ s✉❜s♣❛❝❡ V t❤❛t ✇✐•• ♣•❛② t❤❡ r♦•❡ ♦❢ t❛♥❣❡♥t s♣❛❝❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ r❡❝t✐✜❛❜•❡ s❡ts✳ ◆♦t❡ t❤❛t ✈❛r✐❢♦•❞s ❞✐✛❡r ❢r♦♠ t❤❡ ✐❞❡❛ ♦❢ ♠❡❛s✉r❡s ❛s ❝♦♥s✐❞❡r❡❞ ✐♥ ❬✹✶(✱ ✇❤✐❝❤ ❛r❡ s✐♠♣•② ❡•❡♠❡♥ts ♦❢ t❤❡ ❞✉❛• ♦❢ C 0 (E)✳ ❙♦♠❡❤♦✇✱ ✈❛r✐❢♦•❞s ❡♥r✐❝❤ t❤❡ ♠❡❛s✉r❡ r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ ❛ ♥♦t✐♦♥ ♦❢ •♦❝❛• ❞✐r❡❝t✐♦♥s✳ ❲❡ ❛•s♦ ❞❡✜♥❡ t❤❡ ❛❞❞✐t✐♦♥❛• ♥♦t✐♦♥ ♦❢ ♣♦s✐t✐✈❡ ✈❛r✐❢♦•❞ t❤❛t s❤❛•• ❜❡ ✉s❡❢✉• ❢♦r t❤❡ ❢♦••♦✇✐♥❣ ✿ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✸✳ ❆ ✈❛r✐❢♦•❞ µ ✐s s❛✐❞ t♦ ❜❡ ♣♦s✐t✐✈❡ ✐❢ ✐t ✐s ✐♥ ❛❞❞✐t✐♦♥ ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ ♦♥ E ×G d (E) ♦r ❡q✉✐✈❛•❡♥t•② ✐❢ ❢♦r ❛•• ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ω ∈ C 0 (E ×G d (E))✱ µ(ω) ≥ 0✳ ❋♦r ❛ ♣♦s✐t✐✈❡ ✈❛r✐❢♦•❞ µ✱ ✇❡ ✇✐•• ❛•s♦ ❝❛•• t❤❡ s✉♣♣♦rt ♦❢ µ ❛♥❞ ❞❡♥♦t❡ ❙✉♣♣(µ) t❤❡ s♠❛••❡st ❝•♦s❡❞ s✉❜s❡t C ⊂ E × G d (E) s✉❝❤ t❤❛t µ((E × G d (E))\C) = 0 . ✸✳✷✳✷ ◆♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❛s ✈❛r✐❢♦•❞s |❡❝t✐✜❛❜•❡ ✈❛r✐❢♦•❞s ■♥ s❡❝t✐♦♥ ✶✳✸✳✹✱ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s❡ts ♦❢ ❞✐♠❡♥s✐♦♥ d ❛r❡ ❝❛♥♦♥✲ ✐❝❛••② r❡♣r❡s❡♥t❡❞ ❛s d✲❝✉rr❡♥ts✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s✱ t❤❡ r✐❣❤t ♥♦t✐♦♥ ✐s ♣r❡❝✐s❡•② t❤❡ ♦♥❡ ♦❢ ✈❛r✐❢♦•❞s✳ ■♥❞❡❡❞✱ •❡t X ❜❡ ❛ ♥♦♥✲♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s❡t ♦❢ E ♦❢ ❞✐♠❡♥s✐♦♥ d✳ ❚♦ X✱ ♦♥❡ ❝❛♥ ❛ss♦❝✐❛t❡ ❛ ✈❛r✐❢♦•❞ µ X ✱ ✇❤✐❝❤✱ ✐♥ t❤❡ ♠❡❛s✉r❡ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐s ❣✐✈❡♥ ❜② ✿ µ X (A) = H d ({x ∈ X | (x, T x X) ∈ A}) ❢♦r ❛•• ❇♦r❡• s✉❜s❡t A ⊂ E × G d (E)✳ ◆♦✇✱ s❡❡♥ ❛s ❛ ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ ♦♥ C 0 (E × G d (E))✱ µ X ✇r✐t❡s✱ ❢♦r ❛•• ω ∈ C 0 (E × G d (E)) ✿ µ X (ω) = E×G d (E) ω(x, V )dµ X (x, V ) ❛♥❞ s✐♥❝❡ ❛ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ❤❛s ❛ t❛♥❣❡♥t s♣❛❝❡ ❛♣♣•✐❝❛t✐♦♥ x → T x X ❞❡✜♥❡❞ ❢♦r H d ✲❛•♠♦st ❛•• x ∈ X ✇❤✐❝❤ ✐s ❛•s♦ H d ♠❡❛s✉r❛❜•❡ ✭❝❢ s❡❝t✐♦♥ ✶✳✶✼✮✱ ✇❡ ❝❛♥ ✇r✐t❡ ✿ µ X (ω) = X ω(x, T x X)dH d (x) . ✭✸✳✹✮ ✾✼ ❲❡ ✇✐•• ❝❛•• s✉❝❤ s♣❡❝✐❛• ✈❛r✐❢♦•❞s r❡❝t✐✜❛❜•❡ ✈❛r✐❢♦•❞s ✭♥♦t❡ t❤❛t ✇❡ ❞♦ ♥♦t ❝♦♥s✐❞❡r✱ ✐♥ ♦✉r ❞❡✜♥✐t✐♦♥✱ ♠✉•t✐♣•✐❝✐t✐❡s ♦r ❞❡♥s✐t② ❢✉♥❝t✐♦♥s ❛s ✐♥ ❬✹(✮✳ ❚❤✐s •❛st ✐♥t❡❣r❛• ❝❛♥ ❜❡ ❛•s♦ ✇r✐tt❡♥ ✇✐t❤ ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐❢ X ✐s ♠♦r❡ s✐♠♣•② ❛ s♠♦♦t❤ ❝♦♠♣❛❝t s✉❜♠❛♥✐❢♦•❞ ♦❢ E ❣✐✈❡♥ ❜② ❛ ♣❛r❛♠❡tr✐③❛t✐♦♥ γ : U → E ✇✐t❤ U ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ R d ✳ ❚❤❡♥ ❢♦r ❛•• ω ∈ C 0 (E × G d (E)) µ X (ω) = U ω(γ(u), T γ(u) X)|γ ′ (u)|du ✭✸✳✺✮ ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥ γ ′ (u) = Λ d i=1 ∂γ/∂u i ∈ Λ d (E)✱ |γ ′ (u)| r❡♣r❡s❡♥t✐♥❣ t❤❡ •♦❝❛• d✲ ✈♦•✉♠❡ ❡•❡♠❡♥t✳ ◆♦t❡ t❤❛t t❤❡ ✐♥t❡❣r❛• ♦❢ ❡q✳✭✸✳✺✮ ✐s✱ ❛s ❡①♣❡❝t❡❞✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛♥② r❡♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ X✱ ♣♦s✐t✐✈❡•② ♦r ♥❡❣❛t✐✈❡•② ♦r✐❡♥t❡❞✳ P♦•②❣♦♥❛• ♠❡s❤❡s ◆♦✇✱ ✐♥ ❞✐s❝r❡t❡ ❣❡♦♠❡tr②✱ s❤❛♣❡s ❛r❡ ❣✐✈❡♥ ❛s ♣♦•②❣♦♥❛• s❡ts ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ❛r❡ ❛•s♦ ❡♥❝♦♠♣❛ss❡❞ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ t❤❡r❡❢♦r❡ r❡♣r❡s❡♥t❛❜•❡ ❛s ✈❛r✐❢♦•❞s✳ ■♥ t❤❡ s❛♠❡ s♣✐r✐t ❛s ✇✐t❤ ❝✉rr❡♥ts ❛♥❞ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ❛♥② ♠❡s❤ s❡t ❝❛♥ ❜❡ ❝♦❞❡❞ ❛s ❛ ✜♥✐t❡ s✉♠ ♦❢ ❞✐r❛❝ ✈❛r✐❢♦•❞s ♦❢ t❤❡ ❢♦r♠ n i=1 m i .δ (p i ,V i ) ✳ ❆❣❛✐♥✱ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝❡••s ♦❢ t❤❡ ♠❡s❤✱ p i ∈ E ❛r❡ t❤❡ ❝❡♥t❡rs ♦❢ ❡❛❝❤ ❝❡••✱ V i ∈ G d (E) t❤❡ ✉♥♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡ t♦ t❤❡ s❤❛♣❡ ❛t ♣♦✐♥t p i ❛♥❞ m i ∈ R * + t❤❡ ✈♦•✉♠❡ ♦❢ t❤❡ i✲t❤ ❝❡••✳ ▼♦r❡ s♣❡❝✐✜❝❛••②✱ •❡t✬s ❡①❛♠✐♥❡ t❤❡ t✇♦ ♠♦st ✉s✉❛• ❡①❛♠♣•❡s ✐♥ ♣r❛❝t✐❝❡✳ ❋✐rst✱ ❝♦♥s✐❞❡r ❛ ❝✉r✈❡ X ❣✐✈❡♥ ❛s ❛ s❡t ♦❢ ♣♦✐♥ts {x k } k=1..N ❛♥❞ ❛ s❡t ♦❢ ❡❞❣❡s (f 1 i , f 2 i ) ∈ {1, .., N } 2 ❢♦r i = 1, .., n✳ ❚❤❡♥✱ ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ❝♦♥✈❡♥t✐♦♥s ✿                p i = x f 1 i + x f 2 i 2 V i = (x f 1 i x f 2 i ) m i = ----→ x f 1 i x f 2 i . ✭✸✳✻✮ ▲❡t ✉s ♣r❡❝✐s❡ t❤❛t ❤❡r❡ (x f 1 i x f 2 i ) ❞❡♥♦t❡s t❤❡ •✐♥❡ ✐♥ G 1 (R 2 ) t❤❛t ✐s s♣❛♥♥❡❞ ❜② t❤❡ ✈❡❝t♦r ----→ x f 1 i x f 2 i ✳ ◆♦t❡ ❛•s♦ t❤❛t ❛•• t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s r❡♠❛✐♥ ✉♥❝❤❛♥❣❡❞ ✐❢ ❢♦r ❛♥② i ✇❡ t❛❦❡ (f 2 i , f 1 i ) ✐♥st❡❛❞ ♦❢ (f 1 i , f 2 i ) ❛s ❛ ❢❛❝❡✱ ✇❤✐❝❤ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ✐❞❡❛ ♦❢ ♥♦♥✲♦r✐❡♥t❛t✐♦♥✳ ■♥ ❛ s✐♠✐•❛r ✇❛②✱ ✐❢ X ✐s ♥♦✇ ❢♦r ✐♥st❛♥❝❡ ❛ tr✐❛♥❣✉•❛t❡❞ s✉r❢❛❝❡ ✐♥ E = R 3 ❣✐✈❡♥ ❜② ❛ s❡t ♦❢ ♣♦✐♥ts {x k } k=1..N ✇✐t❤ ❛ s❡t ♦❢ n tr✐❛♥❣•❡s (f 1 i , f 2 i , f 3 i ) ∈ {1, .., N } 3 ❢♦r i = 1, .., n t❤❡♥ ✇❡ ❣❡t ✿                    p i = x f 1 i + x f 2 i + x f 3 i 3 V i = ----→ x f 1 i x f 2 i , ----→ x f 1 i x f 3 i m i = 1 2 . ----→ x f 1 i x f 2 i ∧ ----→ x f 1 i x f 3 i . ✭✸✳✼✮ ❍❡r❡ ----→ x f 1 i x f 2 i , ----→ x f 1 i x f 3 i ❞❡♥♦t❡s t❤❡ 2-❞✐♠❡♥s✐♦♥❛• s♣❛❝❡ ❣❡♥❡r❛t❡❞ ❜② ----→ x f 1 i x f 2 i ❛♥❞ ----→ x f 1 i x f 3 i ✳ ❆❣❛✐♥✱ ❛•• t❤❡s❡ ❡q✉❛t✐♦♥s ❛r❡ ♥♦t ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ♦r✐❡♥t❛t✐♦♥ ❣✐✈❡♥ t♦ ❡❛❝❤ tr✐❛♥❣•❡✳ ✾✽ ❆❝t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s ❆ •❛st ✐♠♣♦rt❛♥t ♣♦✐♥t ❢♦r t❤❡ ❢♦••♦✇✐♥❣ ✐s t♦ ❜❡ ❛❜•❡ t♦ ❡①♣r❡ss t❤❡ tr❛♥s♣♦rt ♦❢ ✈❛r✐❢♦•❞s ❜② ❞✐✛❡♦♠♦r♣❤✐s♠ ✐♥ ❛ ✇❛② t❤❛t ✐s ❝♦♠♣❛t✐❜•❡ ✇✐t❤ t❤❡ tr❛♥s♣♦rt ♦❢ s❤❛♣❡s✳ ▲❡t✬s ✜① ❛ ✈❛r✐❢♦•❞ µ ∈ C 0 (E × G d (E)) ′ ❛♥❞ φ ∈ ❉✐✛(E)✱ ✇❡ ❞❡✜♥❡ t❤❡ tr❛♥s♣♦rt φ * µ ♦❢ µ ❜② φ ❜② ♣✉••✲❜❛❝❦ ❛♥❞ ♣✉s❤✲❢♦r✇❛r❞ ♦♣❡r❛t✐♦♥s ✿ ∀ω ∈ C 0 (E × G d (E)), (φ * µ) (ω) = µ (φ * ω) ✭✸✳✽✮ ✇❤❡r❡ φ * ω ✐s t❤❡ ♣✉••✲❜❛❝❦ ♦❢ ω ❜② φ✳ ❋♦r ❛♥② x ∈ E ❛♥❞ V ∈ G d (E) ❛ d✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ✇✐t❤ ♦rt❤♦♥♦r♠❛• ❜❛s✐s (u 1 , ..., u d )✱ φ * ω ✐s ❞❡✜♥❡❞ ❜② t❤❡ r❡•❛t✐♦♥ ✿ (φ * ω) (x, V ) = |d x φ(u 1 ) ∧ ... ∧ d x φ(u d )| ω(φ(x), d x φ.V ) . ✭✸✳✾✮ ■♥ t❤❡ •❛st ❡q✉❛t✐♦♥✱ d x φ.V ❞❡♥♦t❡s t❤❡ ❡•❡♠❡♥t ♦❢ G d (E) t❤❛t ✐s t❤❡ ✐♠❛❣❡ ♦❢ V ❜② d x φ✱ t❤❡ t❡r♠ |d x φ(u 1 ) ∧ ... ∧ d x φ(u d )| ✐s t❤❡ d✲❞✐♠❡♥s✐♦♥❛• ❏❛❝♦❜✐❛♥ ♦❢ φ ♦♥ t❤❡ s✉❜s♣❛❝❡ V ❛♥❞ r❡♣r❡s❡♥ts t❤❡ •♦❝❛• ❝❤❛♥❣❡ ♦❢ d✲❞✐♠❡♥s✐♦♥❛• ✈♦•✉♠❡ ♦❢ t❤❡ tr❛♥s✲ ❢♦r♠❛t✐♦♥✳ ❲✐t❤ ❡q✉❛t✐♦♥s ✭✸✳✽✮ ❛♥❞ ✭✸✳✾✮✱ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝♦♠♣✉t❡ t❤❡ tr❛♥s♣♦rt ♦❢ ❛ ❞✐r❛❝ δ (x,V ) ✿ φ * δ (x,V ) = |d x φ(u 1 ) ∧ ... ∧ d x φ(u d )| .δ (φ(x),dxφ.V ) ✭✸✳✶✵✮ ✐❢ (u 1 , ..., u d ) ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ V ✳ ◆♦✇✱ ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ❞❡✜♥✐t✐♦♥s✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ✈❛r✐❢♦•❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ r❡❝t✐✜❛❜•❡ s❡t ❝♦♠♠✉t❡s ✇✐t❤ t❤❡ tr❛♥s♣♦rt ❜② ❞✐✛❡♦♠♦r♣❤✐s♠ ✐♥ t❤❡ s❡♥s❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✸✳ ■❢ X ✐s ❛ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ❛♥❞ φ ∈ ❉✐✛(E)✱ t❤❡♥ ✿ φ * µ X = µ φ(X) . Pr♦♦❢✳ ■❢ X ✐s ❛ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E✱ ✇✐t❤ ❡q✳✭✸✳✹✮✱ ✇❡ ❤❛✈❡ ❢♦r ❛•• ω ✿ µ X (ω) = X ω(x, T x X)dH d (x) ❛♥❞ t❤✉s φ * µ X (ω) = µ X (φ * ω) = X (φ * ω)(x, T x X)dH d (x) = X ω(φ(x), d x φ(T x X))|d x φ.T x X|dH d (x) ✇❤❡r❡ |d x φ.T x X| ✐s t❤❡ d✲❞✐♠❡♥s✐♦♥❛• ❏❛❝♦❜✐❛♥ ♦❢ φ ❛•♦♥❣ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T x X✱ ❛s ♣r❡✈✐♦✉s•②✳ ❋♦r ❛•♠♦st ❛•• x ∈ X✱ d x φ(T x X) ✐s ❛ t❛♥❣❡♥t s♣❛❝❡ ♦❢ φ(X) ❛♥❞ ❜② t❤❡ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ❢♦r r❡❝t✐✜❛❜•❡ s✉❜s❡ts ✭❝♦r♦••❛r② ✸✳✷✳✷✵ ✐♥ ❬✸✼(✮✱ ✇❡ ♦❜t❛✐♥ ✿ φ * µ X (ω) = φ(X) ω(y, T y φ(X))dH d (y) . ❚❤✉s✱ ❢♦r ❛♥② ω✱ φ * µ X (ω) = µ φ(X) (ω) ❛♥❞ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ ♣r♦♣♦s✐t✐♦♥ ✸✳✷✳✸✳ ✾✾ ❚♦ ❝♦♥❝•✉❞❡ t❤✐s s❡❝t✐♦♥✱ ✈❛r✐❢♦•❞s ♦✛❡r ❛ ♣♦ss✐❜•❡ ❢r❛♠❡✇♦r❦ t♦ r❡♣r❡s❡♥t ♥♦♥✲ ♦r✐❡♥t❡❞ s❤❛♣❡s ❜♦t❤ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉♦✉s s❡tt✐♥❣ ❛♥❞ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛• ❝❛s❡s ♦❢ ♠❡s❤❡❞ ❝✉r✈❡s✱ s✉r❢❛❝❡s✳✳✳ ❯♣ t♦ t❤✐s ♣♦✐♥t✱ ✇❡ ❤❛✈❡ s❤♦✇♥ ❤♦✇ t♦ r❡♣r❡s❡♥t ❝♦♠✲ ♣✉t❛t✐♦♥❛••② ❛ ♠❡s❤❡❞ s❤❛♣❡ ❛s ❛ ✜♥✐t❡ s❡t ♦❢ ❞✐r❛❝s✱ ❡❛❝❤ ♦❢ t❤❡♠ ❝❛rr②✐♥❣ ❛ •♦❝❛• ✐♥❢♦r♠❛t✐♦♥ ♦❢ ♣♦s✐t✐♦♥✱ ✉♥♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡ ❛♥❞ •♦❝❛• ✈♦•✉♠❡ ❛♥❞ ❞❡r✐✈❡ t❤❡ ❡q✉❛t✐♦♥s ♦❢ ✈❛r✐❢♦•❞s✬ tr❛♥s♣♦rt ❜② ❞❡❢♦r♠❛t✐♦♥✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ✐ss✉❡ t♦ ❡①❛♠✐♥❡ ✐s t❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s✱ ✇❤✐❝❤ ✇❡ ❛❞❞r❡ss ✐♥ t❤❡ ❢♦••♦✇✐♥❣✳ ✸✳✸ ❑❡r♥❡• ♠❡tr✐❝s ♦♥ ✈❛r✐❢♦•❞s ❚❤❡ ✉s❡ ♦❢ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡•s ✐s ❛ ❢✉♥❞❛♠❡♥t❛• st❡♣ ✇❤❡♥ ✇♦r❦✐♥❣ ✇✐t❤ ❝✉rr❡♥ts ❜❡❝❛✉s❡ ✐t ♣r♦✈✐❞❡s r❡❣✉•❛r✐③❡❞ ♠❡tr✐❝s ❢♦r t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ s❤❛♣❡s ✇❤✐❝❤ ❤❛s t❤❡ ❛❞❞✐t✐♦♥❛• ❛❞✈❛♥t❛❣❡ ♦❢ ❤❛✈✐♥❣ ❛♥ ✉♥❞❡r•②✐♥❣ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡✳ ❲❡ ❤❛✈❡ ❛•s♦ ♠❡♥t✐♦♥❡❞ s❡✈❡r❛• t✐♠❡s t❤❛t ❦❡r♥❡•s ❛r❡ ♣❛rt✐❝✉•❛r•② ✇❡••✲✜tt❡❞ t♦ ❝♦♠♣✉t❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❞✐s❝r❡t✐③❡❞ s❤❛♣❡s ❜❡❝❛✉s❡ ♦❢ t❤❡ s✐♠♣•❡ ❡①♣r❡ss✐♦♥ ♦❢ ❞♦t ♣r♦❞✉❝t ❜❡t✇❡❡♥ t✇♦ ❞✐r❛❝s✳ ❆•• t❤❡s❡ r❡❛s♦♥s ♠♦t✐✈❛t❡ ❛ s✐♠✐•❛r ❛♣♣r♦❛❝❤ ✐♥ t❤❡ tr❡❛t♠❡♥t ♦❢ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s t❤r♦✉❣❤ t❤❡ ✈❛r✐❢♦•❞ s❡tt✐♥❣ t❤❛t ❤❛s ❜❡❡♥ ♣r❡s❡♥t❡❞✳ ■t✬s ✇♦rt❤ ♠❡♥t✐♦♥✐♥❣ t❤❛t ♦t❤❡r ❛♣♣r♦❛❝❤❡s ❝♦✉•❞ ❜❡ ♣♦ss✐❜•❡✱ ❜② ✇♦r❦✐♥❣ ✇✐t❤ ♠♦r❡ ❣❡♥❡r❛• |✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝s ♦♥ ❞✐str✐❜✉t✐♦♥ ♦❢ t❛♥❣❡♥t s♣❛❝❡s✱ ❣❡♥❡r❛•✐③✐♥❣ ❢♦r ✐♥st❛♥❝❡ ✇❤❛t ✐s ❞♦♥❡ ❢♦r t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ✐♥ ❬✶(✳ ❲❡ ❛r❣✉❡ ❤♦✇❡✈❡r t❤❛t |❑❍❙ ♠❡tr✐❝s ❛r❡ ✈❡r② ❝♦♥✈❡♥✐❡♥t ❢r♦♠ ❛ ❝♦♠♣✉t❛t✐♦♥❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❡s♣❡❝✐❛••② ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s t♦ s❤❛♣❡ ♠❛t❝❤✐♥❣ ❛♥❞ ❛♥❛•②s✐s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦♣♦s❡ ❛ ❣❡♥❡r✐❝ ❛♥❞ s✐♠♣•❡ ✇❛② t♦ ❜✉✐•❞ r❡•❡✈❛♥t r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡•s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s✱ ❜② ❜❛s✐❝❛••② ♠❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ♦❢ ❦❡r♥❡•s ♦♥ E ❛♥❞ ♦♥ G d (E)✳ ✸✳✸✳✶ ❑❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❚❤❡ ✜rst st❡♣ ✐s t♦ ✇♦r❦ ♦♥ t❤❡ •r❛ss♠❛♥♥✐❛♥✳ ❙✐♥❝❡ ✐t ✐s ❛ r❛t❤❡r ❛❜str❛❝t ♠❛♥✐❢♦•❞✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❦❡r♥❡•s ✐s ♥♦t ❛ tr✐✈✐❛• ♣r♦❜•❡♠ ❛♥❞ ❤❛s ❜❡❡♥ t❤❡ s✉❜•❡❝t ♦❢ s❡✈❡r❛• r❡❝❡♥t ✇♦r❦s r❡•❛t❡❞ t♦ ♠❛❝❤✐♥❡ •❡❛r♥✐♥❣✱ ❛s ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✹✻✱ ✽✷(✳ ❲❡ ♣r♦♣♦s❡ ❛♥ ❛•t❡r♥❛t✐✈❡ ✇❛② t❤❛t ❡①♣•♦✐ts t❤❡ ❞✐✛❡r❡♥t ❡♠❜❡❞❞✐♥❣s ♦❢ G d (E) ✐♥t♦ ♦t❤❡r s♣❛❝❡s ✐♥ ✇❤✐❝❤ ❦❡r♥❡•s ❛r❡ ♠♦r❡ ❡❛s✐•② ❞❡✜♥❡❞✳ ❲❡ ✇✐•• ❡ss❡♥t✐❛••② ❝♦♥s✐❞❡r t✇♦ ♣♦ss✐❜✐•✐t✐❡s ✿ ❛ ❝♦♥str✉❝t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ✇✐t❤ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t♦rs ❛♥❞ ❛ s❡❝♦♥❞ ♦♥❡ ✉s✐♥❣ t❤❡ P•ü❝❦❡r ❡♠❜❡❞❞✐♥❣✳ ❊♠❜❡❞❞✐♥❣ ✐♥ L(E) ❆s ❡①♣•❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✷✱ ♦♥❡ ❝❛♥ ✐❞❡♥t✐❢② ❛♥② V ∈ G d (E) ✇✐t❤ t❤❡ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥ p V ∈ L(E)✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ♦♥❡ ❝❛♥ ✐♥❞✉❝❡ str❛✐❣❤t❢♦r✇❛r❞•② ❛ ❦❡r♥❡• ♦♥ G d (E) ❜② r❡str✐❝t✐♦♥ ♦❢ ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡• ❞❡✜♥❡❞ ♦♥ L(E)✳ ❙✐♥❝❡ L(E) ✐s ❛ ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐✣❝✉•t✐❡s ✐♥ ❞❡✜♥✐♥❣ s✉❝❤ ❦❡r♥❡•s✳ ❚②♣✐❝❛• ❝❤♦✐❝❡s ❛r❡ ❣✐✈❡♥ ❜② ✿ k P (V, W ) = p V , p W k ✭✸✳✶✶✮ ✇✐t❤ k ∈ N * ❛♥❞ ., . t❤❡ ✉s✉❛• ❋r♦❜❡♥✐✉s ♠❡tr✐❝ ✐♥ L(E)✳ ▼♦r❡ ❣❡♥❡r❛••②✱ ❛♥② ❢✉♥❝t✐♦♥ P ( p V , p W )✱ P ❜❡✐♥❣ ❛ ♣♦•②♥♦♠✐❛• ✇✐t❤ ♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥ts✱ ✐s ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ L(E)✳ ❲❡ s❤❛•• ❝❛•• t❤❡s❡ ♣♦•②♥♦♠✐❛• ❦❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❛♥❞ •✐♥❡❛r ❦❡r♥❡• t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ ❢♦r k = 1 ✐♥ ❡q✳✭✸✳✶✶✮✱ ✐✳❡ ✿ k L (V, W ) = p V , p W ✭✸✳✶✷✮ ✶✵✵ ❖t❤❡r ✐♠♣♦rt❛♥t ♣♦ss✐❜✐•✐t✐❡s ❛r❡ t❤❡ ❦❡r♥❡•s ✐♥❞✉❝❡❞ ❜② •❛✉ss✐❛♥s ✐♥ L(E)✱ ♥❛♠❡•② ✿ k G (V, W ) = e - |p V -p W | 2 σ 2 ✭✸✳✶✸✮ ❙✉❝❤ ❦❡r♥❡•s ❛••♦✇ t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ s✉❜s♣❛❝❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❝❡rt❛✐♥ s❝❛•❡ ❣✐✈❡♥ ❜② ♣❛r❛♠❡t❡r σ✳ ▼❛♥② ♦t❤❡r ❝♦✉•❞ ❜❡ ❞❡✜♥❡❞ ✇✐t❤ t❤✐s ♠❡t❤♦❞✳ ■t✬s ❛•s♦ ♣♦ss✐❜•❡ t♦ ❡①♣r❡ss t❤❡s❡ ❦❡r♥❡•s ✉s✐♥❣ t❤❡ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s ✐♥tr♦❞✉❝❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✷✳✶✳ ■❢ ✇❡ ❞❡♥♦t❡ (v 1 , .., v d ) ❛♥❞ (w 1 , .., w d ) t❤❡ ♦rt❤♦♥♦r♠❛• ❢r❛♠❡s ♦❢ V ❛♥❞ W s✉❝❤ t❤❛t cos(θ i ) = v i , w i ❛♥❞ ❙♣❛♥(v i , w i )⊥❙♣❛♥(v j , w j ) ❢♦r j = i t❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ ✿ p V , p W = d i=1 v * i ⊗ v i , d j=1 w * j ⊗ w j = d i,j=1 v i , w j 2 = d i=1 cos 2 (θ i ) ■t r❡s✉•ts t❤❛t t❤❡ ❦❡r♥❡• ♦❢ ❡q✳✭✸✳✶✶✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ✿ k P (V, W ) = d i=1 cos 2 (θ i ) k ✭✸✳✶✹✮ ❚❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• ❛•s♦ ❤❛s t❤❡ ❛•t❡r♥❛t✐✈❡ ❡①♣r❡ss✐♦♥ ✿ k G (V, W ) = e -2 σ 2 . d i=1 (1-cos 2 (θ i )) = d i=1 e -2 σ 2 sin 2 (θ i ) ✭✸✳✶✺✮ ❚❤✐s ❢♦r♠✉•❛t✐♦♥ ✇✐t❤ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s ♦✛❡rs✱ ✐♥ ♦✉r s❡♥s❡✱ ❛ ♥✐❝❡•② ✐♥t❡r♣r❡t❛❜•❡ ✇❛② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ s✉❝❤ ❦❡r♥❡•s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s✉❜s♣❛❝❡s✬ r❡•❛t✐✈❡ ♣♦s✐t✐♦♥✳ ■♥ ♣❛rt✐❝✉•❛r✱ ♦♥❡ ❝❛♥ ❜❡tt❡r ✉♥❞❡rst❛♥❞ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ✇✐❞t❤ σ ❢♦r t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡•✳ ❚❤❡ ❣❡♥❡r❛• ❝♦♥str✉❝t✐♦♥ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ t❤✉s ♣r♦✈✐❞❡s ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ✐♥❞✉❝❡❞ ❦❡r♥❡•s ♦♥ G d (E) t❤❛t ❛r❡ ❡✛❡❝t✐✈❡•② ❝♦♠♣✉t❛❜•❡ ✭❡✐t❤❡r ❜② t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ♣r♦•❡❝t✐♦♥ ♠❛tr✐① ♦r ❜② ❝♦♠♣✉t✐♥❣ t❤❡ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s✮✳ ❊♠❜❡❞❞✐♥❣ ✐♥ P (Λ d E) ❋♦••♦✇✐♥❣ ❛ s✐♠✐•❛r ♣❛t❤✱ ❦❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥✐❛♥ ❝♦✉•❞ ❜❡ ❛•s♦ ❞❡✜♥❡❞ ❜❛s❡❞ ♦♥ t❤❡ P•ü❝❦❡r ❡♠❜❡❞❞✐♥❣ ♠❡♥t✐♦♥❡❞ ✐♥ ✸✳✷✳✶✱ t❤❛t ❡♠❜❡❞s G d (E) ✐♥t♦ t❤❡ ♣r♦•❡❝t✐✈❡ s♣❛❝❡ P (Λ d E)✳ ❚❤❡ P•ü❝❦❡r ❛♣♣•✐❝❛t✐♦♥ ❛ss♦❝✐❛t❡s t♦ ❛♥② V = ❙♣❛♥{v 1 , .., v d } t❤❡ •✐♥❡ ✐♥ t❤❡ ❡①t❡r✐♦r ♣r♦❞✉❝t ❣❡♥❡r❛t❡❞ ❜② ξ V = v 1 ∧ ... ∧ v d ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡❞ ❜② [ξ V ]✳ ❚❤❡ ✐❞❡❛ ✇♦✉•❞ ❜❡ ❛❣❛✐♥ t♦ ✐♥❞✉❝❡ ❛ ❦❡r♥❡• ♦♥ G d (E) ❢r♦♠ ❛ ❦❡r♥❡• ♦♥ P (Λ d E)✳ ❚❤✐s ✐s ❤♦✇❡✈❡r ♥♦t ❛s tr✐✈✐❛• ❛s ♣r❡✈✐♦✉s•② s✐♥❝❡ P (Λ d E) ✐s st✐•• ♥♦t ❛ ✈❡❝t♦r s♣❛❝❡✳ ❇✉t ✇❡ ❝❛♥ ❜✉✐•❞ ❦❡r♥❡•s ♦♥ t❤✐s ♣r♦•❡❝t✐✈❡ s♣❛❝❡ ❜❛s❡❞ ♦♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳✶✳ ▲❡t k 0 ❜❡ ❛ r❡❛• ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ Λ d E s✉❝❤ t❤❛t ❢♦r ❛•• ξ, ξ ′ ∈ Λ d E✱ k 0 (-ξ, -ξ ′ ) = k 0 (ξ, ξ ′ )✳ ❚❤❡♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥ ✿ k([ξ], [ξ ′ ]) = 1 2 k 0 ξ |ξ| , ξ ′ |ξ ′ | + k 0 ξ |ξ| , - ξ ′ |ξ ′ | ✭✸✳✶✻✮ ✶✵✶ ❞❡✜♥❡s ❛ r❡❛• ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ P (Λ d E)✳ Pr♦♦❢✳ ❚❤❡ ✜rst t❤✐♥❣ t♦ ♥♦t✐❝❡ ✐s t❤❛t t❤❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ ❡q✳✭✸✳✶✻✮ ✐s ✇❡••✲❞❡✜♥❡❞ ♦♥ P (Λ d E)✱ s✐♥❝❡ ✐t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ r❡♣r❡s❡♥t❛♥ts ξ ❛♥❞ ξ ′ ✐♥ t❤❡ ❡q✉✐✈❛•❡♥❝❡ ❝•❛ss❡s [ξ], [ξ ′ ]✳ ❚❤❡ ♥❡①t ♣♦✐♥t ✐s t❤❛t t❤❡ ❛♣♣•✐❝❛t✐♦♥ k 0 ( ξ |ξ| , ξ ′ |ξ ′ | ) ✐s ❜② r❡str✐❝t✐♦♥ ❛ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ ♦❢ t❤❡ ✈❡❝t♦r s♣❛❝❡ Λ d E✳ ▲❡t✬s ❝❛•• H t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ |❑❍❙✳ H ✐s ❛ ❍✐•❜❡rt s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ s♣❤❡r❡ ❛♥❞ ✇❡ r❡♠✐♥❞ t❤❛t✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ |❑❍❙✱ k 0 ξ |ξ| , ξ ′ |ξ ′ | = k 0 ξ ′ |ξ ′ | , ξ |ξ| = k 0 ξ |ξ| , . , k 0 ξ ′ |ξ ′ | , . H . ◆♦✇ ❢♦r ❛♥② p ∈ N * ❛♥❞ ❛♥② ❢❛♠✐•② (ξ i ) i=1,..,p ∈ Λ d E ❛♥❞ (α i ) i=1,..,p ∈ R✱ ✇❡ ❤❛✈❡ ✿ p i,j=1 k([ξ i ], [ξ j ])α i α j = 1 2 p i,j=1 k 0 ξ i |ξ i | , ξ j |ξ j | + k 0 ξ i |ξ i | , - ξ j |ξ j | α i α j = 1 4 p i,j=1 k 0 ξ i |ξ i | , ξ j |ξ j | + k 0 - ξ i |ξ i | , ξ j |ξ j | + k 0 ξ i |ξ i | , - ξ j |ξ j | + k 0 - ξ i |ξ i | , - ξ j |ξ j | α i α j = 1 4 p i,j=1 k 0 ξ i |ξ i | , . + k 0 - ξ i |ξ i | , . , k 0 ξ j |ξ j | , . + k 0 - ξ j |ξ j | , . H α i α j = 1 4 p i=1 α i k 0 ξ i |ξ i | , . + k 0 - ξ i |ξ i | , . , α j   p j=1 k 0 ξ j |ξ j | , . + k 0 - ξ j |ξ j | , .   H = p i=1 α i k 0 ξ i |ξ i | , . + k 0 - ξ i |ξ i | , . 2 H . ❚❤❡ •❛st ❡①♣r❡ss✐♦♥ ✐s t❤✉s ♣♦s✐t✐✈❡✳ •✐✈❡♥ ❛ ❦❡r♥❡• ❛s ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✸✳✶✱ ✇❡ ♦❜t❛✐♥ ❜② ✐♥❞✉❝t✐♦♥ ❛ ❦❡r♥❡• ♦♥ t❤❡ •r❛ss✲ ♠❛♥♥ ♠❛♥✐❢♦•❞✳ ❚❤✐s ❣✐✈❡s ❛❣❛✐♥ ❛ •♦t ♦❢ ♣♦ss✐❜✐•✐t✐❡s✳ ❆♠♦♥❣ t❤❡♠✱ ♦♥❡ ❝♦✉•❞ t❛❦❡ k 0 (ξ, ξ ′ ) = ξ, ξ ′ 2k ❢♦r k ∈ N * ✱ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❤②♣♦t❤❡s❡s ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❛♥❞ ❣❡t ❛ ❦❡r♥❡• ♦♥ G d (E) t❤❛t ✇r✐t❡s ✿ k CB (V, W ) = ξ V |ξ V | , ξ W |ξ W | 2k ✭✸✳✶✼✮ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ ♣♦ss✐❜✐•✐t② ✐s t♦ t❛❦❡ k 0 (ξ, ξ ′ ) = e -|ξ/|ξ|-ξ ′ /|ξ ′ || 2 σ 2 + e -|ξ/|ξ|+ξ ′ /|ξ ′ || 2 σ 2 ❛♥❞ ✇❡ ♦❜t❛✐♥ ✿ k G2 (V, W ) = e -2 σ 2 1- ξ V |ξ V | , ξ W |ξ W | + e -2 σ 2 1+ ξ V |ξ V | , ξ W |ξ W | ✭✸✳✶✽✮ ❆❣❛✐♥✱ t❤❡s❡ ❡①♣r❡ss✐♦♥s ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♣r✐♥❝✐♣•❡ ❛♥❣•❡s ❜❡✲ t✇❡❡♥ t❤❡ s✉❜s♣❛❝❡s✳ ■♥tr♦❞✉❝✐♥❣✱ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥✱ t❤❡ ♦rt❤♦♥♦r✲ ♠❛• ❜❛s❡s v 1 , ..., v d ❛♥❞ w 1 , ..., w d ♦❢ V ❛♥❞ W s✉❝❤ t❤❛t cos(θ i ) = v i , w i ❛♥❞ ❙♣❛♥(v i , w i )⊥❙♣❛♥(v j , w j ) ❢♦r j = i✱ ✇❡ ❤❛✈❡ ✿ ∧ d i=1 v i , ∧ d j=1 w j = det( v i , w j ) i,j = d i=1 cos(θ i ) ✶✵✷ ❛♥❞ ✐t r❡s✉•ts t❤❛t ✿ k CB (V, W ) = d i=1 cos 2k (θ i ) ✭✸✳✶✾✮ ❛♥❞ k G2 (V, W ) = e -2 σ 2 (1-d i=1 cos(θ i )) + e -2 σ 2 (1+ d i=1 cos(θ i )) ✭✸✳✷✵✮ ◆♦t❡ t❤❛t t❤❡ ❦❡r♥❡• k CB ♦❢ ❡q✳✭✸✳✶✾✮ ❣✐✈❡s✱ ❢♦r k = 1✱ t❤❡ ❝•❛ss✐❝❛• ❈❛✉❝❤②✲❇✐♥❡t ❦❡r♥❡• ♦♥ t❤❡ •r❛ss♠❛♥♥✐❛♥✱ st✉❞✐❡❞ ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✹✻(✳ ❲❤✐❝❤ ❦❡r♥❡•s t♦ ❝❤♦♦s❡ ❄ ❆s ✇❡ s❡❡✱ ❞❡s♣✐t❡ ♥♦t ❤❛✈✐♥❣ ❞❡✜♥❡❞ ❦❡r♥❡•s ❞✐r❡❝t•② ♦♥ G d (E) ❛s ❛♥ ❛❜str❛❝t ♠❛♥✐❢♦•❞✱ t❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s ✇❡ ♣r❡s❡♥t❡❞ ♣r♦✈✐❞❡ ❛ ✈❡r② ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♣♦ss✐❜✐•✐t✐❡s t❤❛t ❞❡♣❡♥❞ ♦❜✈✐♦✉s•② ♦♥ t❤❡ ❝❤♦s❡♥ ❡♠❜❡❞❞✐♥❣ ♦❢ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ❋r♦♠ ❛ ♣✉r❡•② t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐t✬s ♥♦t ❡❛s② t♦ ❛♥s✇❡r ②❡t ♦♥ ✇❤✐❝❤ ♦♥❡ ♦❢ t❤❡s❡ ❡♠❜❡❞❞✐♥❣s s❤♦✉•❞ ❜❡ ❢❛✈♦r❡❞✳ ❋♦r ❤✐❣❤ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❝♦❞✐♠❡♥s✐♦♥✱ ✇❡ ❝❛♥ st❛t❡ t❤❛t t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s♣❛❝❡ P (Λ d E) ❣r♦✇s ♠✉❝❤ ❢❛st❡r t❤❛♥ L(E) ✇✐t❤ d ❛♥❞ n ❛♥❞ s♦ t❤❡ •r❛ss♠❛♥♥✐❛♥ r❡♣✲ r❡s❡♥t ❛ ✈❡r② s♠❛•• ❞✐♠❡♥s✐♦♥❛• s✉❜♠❛♥✐❢♦•❞ ✐♥ t❤✐s ❜✐❣ s♣❛❝❡✳ ❆ s✐❞❡ ❡✛❡❝t ♣♦ss✐❜•② r❡s✉•t✐♥❣ ❢r♦♠ ✐t✱ ✐❢ ✇❡ •♦♦❦ ❛t t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❦❡r♥❡• ♦❢ ❡q✳✭✸✳✶✾✮✱ ✐s t❤❛t t✇♦ s✉❜s♣❛❝❡s V ❛♥❞ W ❛r❡ ♦rt❤♦❣♦♥❛• ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡tr✐❝ ❛s s♦♦♥ ❛s ♦♥•② ♦♥❡ ♦❢ t❤❡✐r ♣r✐♥❝✐♣•❡ ❛♥❣•❡ ✐s ❛ r✐❣❤t ❛♥❣•❡✳ ❲❡ ❝♦✉•❞ ❛r❣✉❡ t❤❛t✱ ✐♥ s✉❝❤ s✐t✉❛t✐♦♥s✱ t❤✐s ❦❡r♥❡• ✐s s♦♠❡❤♦✇ t♦♦ ✬❝♦♥❝❡♥tr❛t❡❞✬ ❛♥❞ ❞❡❝❛②s t♦♦ ❢❛st✱ ✐❢ ✇❡ ❝♦♠♣❛r❡ ✐t ❢♦r ✐♥st❛♥❝❡ t♦ t❤❡ ❦❡r♥❡• k P ♦❢ ❡q✳✭✸✳✶✹✮ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❡♠❜❡❞❞✐♥❣ ✐♥t♦ L(E)✳ ❚❤✐s ❦✐♥❞ ♦❢ ❜❡❤❛✈✐♦r ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✇❛s ♦❜s❡r✈❡❞ ✐♥ ❝•❛ss✐✜❝❛t✐♦♥s✬ ❡①♣❡r✐♠❡♥ts ❜❛s❡❞ ♦♥ s✉❝❤ ❦❡r♥❡•s ✐♥ ❬✹✻(✳ ❚❤❡r❡ ✐s ❛ s❡❝♦♥❞ ❛r❣✉♠❡♥t ♣•❡❛❞✐♥❣ r❛t❤❡r ✐♥ ❢❛✈♦r ♦❢ t❤❡ ✜rst ❛♣♣r♦❛❝❤✱ t❤❛t s❤❛•• ❛♣♣❡❛r ♠♦r❡ ❝•❡❛r•② ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥s ✇❤❡♥ ❡①❛♠✐♥✐♥❣ t❤❡ C 0 ✲✉♥✐✈❡rs❛•✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ❦❡r♥❡•s✳ ■♥❞❡❡❞✱ t❤❡ ❞❡✜♥✐t✐♦♥ ❜② q✉♦t✐❡♥t ✇✐t❤ t❤❡ ♣r♦•❡❝t✐✈❡ s♣❛❝❡ ♠❛❦❡s s✉❝❤ ♣r♦♣❡rt✐❡s✱ ❛t •❡❛st ❢r♦♠ ♦✉r ❡①♣❡r✐❡♥❝❡✱ s✐❣♥✐✜❝❛♥t•② ♠♦r❡ ❞✐✣❝✉•t t♦ ❞❡❛• ✇✐t❤✳ ■♥ ❛ ♠♦r❡ ♣r❛❝t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡ ❝❛s❡s ✇❡ s❤❛•• ❜❡ •♦♦❦✐♥❣ ❛t ✐♥ t❤✐s ♠❛♥✉s❝r✐♣t ❛r❡ ❢♦❝✉s❡❞ ♦♥ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✐♥ R 2 ♦r R 3 ❛♥❞ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞s ✐♥✈♦•✈❡❞ ❛r❡ ❡ss❡♥t✐❛••② G 1 (R 2 ) ❛♥❞ G 1 (R 3 ) = G 2 (R 3 ) ✭✇✐t❤ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ❢♦r♠✉•❛t❡❞ ✐♥ t❤❡ r❡♠❛r❦ ♦❢ ✸✳✷✳✶✮✳ ❚❤✉s t❤❡r❡ ✐s ♦♥•② ♦♥❡ ♣r✐♥❝✐♣•❡ ❛♥❣•❡ t♦ ❝♦♥s✐❞❡r ✐♥ ❛•• t❤❡s❡ ❝❛s❡s ❛♥❞ t❤❡ ❦❡r♥❡•s k P ❛♥❞ k CB ❢♦r ✐♥st❛♥❝❡ ❛r❡ ✐❞❡♥t✐❝❛•✳ ■♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ✉s❡ t❤❡ ❦❡r♥❡•s k L ❛♥❞ k G ✳ ✸✳✸✳✷ ❑❡r♥❡•s ♦♥ ✈❛r✐❢♦•❞s ❚❡♥s♦r ♣r♦❞✉❝ts✬ ❝♦♥str✉❝t✐♦♥ ❚❤❡ ✐ss✉❡ ♦❢ ❦❡r♥❡•s ♦♥ G d (E) ❜❡✐♥❣ ❛❞❞r❡ss❡❞✱ ✇❡ ♥♦✇ ♠♦✈❡ t♦ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s✳ ❙✐♥❝❡ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡✱ ❛ ♥❛t✉r❛• ✇❛② t♦ ❜✉✐•❞ ❛♣♣r♦♣r✐❛t❡ ❦❡r♥❡•s ✐s t❤❡ t❡♥s♦r ♣r♦❞✉❝t tr✐❝❦✳ ❚❤❡ r❡s✉•t ♦❢ •❡♠♠❛ ✷✳✷✳✶ ✐♥ ❝❤❛♣t❡r ✷ s❤♦✇s t❤❛t t❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ❛ r❡❛•✲✈❛•✉❡❞ ❦❡r♥❡• ♦♥ ❛ s❡t A ❛♥❞ r❡❛•✲✈❛•✉❡❞ ❦❡r♥❡• ♦♥ ❛ s❡t B ❣✐✈❡s ❛ r❡❛•✲✈❛•✉❡❞ ♣♦s✐t✐✈❡ ❦❡r♥❡• ♦♥ A×B✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s✱ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳✷✳ ❆ss✉♠❡ t❤❛t ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ♣♦s✐t✐✈❡ r❡❛• ❦❡r♥❡• k e ♦♥ t❤❡ s♣❛❝❡ E s✉❝❤ t❤❛t k e ✐s ❝♦♥t✐♥✉♦✉s✱ ❜♦✉♥❞❡❞ ❛♥❞ ❢♦r ❛•• x ∈ E✱ t❤❡ ❢✉♥❝t✐♦♥ k e (x, .) ✈❛♥✐s❤❡s ❛t ✐♥✜♥✐t②✳ ❆ss✉♠❡ t❤❛t ❛ s❡❝♦♥❞ ❦❡r♥❡• k t ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ♠❛♥✐❢♦•❞ G d (E) ❛♥❞ ✶✵✸ ✐s ❛•s♦ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡♥ t❤❡ |❑❍❙ W ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣♦s✐t✐✈❡ ❦❡r♥❡• k e ⊗ k t ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ t❤❡ s♣❛❝❡ C 0 (E × G d (E))✳ Pr♦♦❢✳ ❇② ❞❡✜♥✐t✐♦♥ ✿ k (x, V ), (y, Ṽ ) = k e (x, y) k t (V, Ṽ ) ✭✸✳✷✶✮ ❛♥❞ s♦✱ t❤❛♥❦s t♦ t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❦❡r♥❡•s✱ k((x, V ), .) ✐s ❝♦♥t✐♥✉♦✉s ♦♥ E × G d (E) ❛♥❞ ❜❡•♦♥❣s t♦ C 0 (E × G d (E))✳ ❚❤❡ ✈❡❝t♦r s♣❛❝❡ W 0 ❣❡♥❡r❛t❡❞ ❜② t❤❡s❡ ❢✉♥❝t✐♦♥s ✐s t❤✉s ✐♥❝•✉❞❡❞ ✐♥ C 0 (E × G d (E))✳ ▼♦r❡♦✈❡r✱ ✐❢ ω ∈ W 0 ✿ ω(x, V ) = δ (x,V ) (ω) = k((x, V ), .), ω W ❲✐t❤ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t② ✿ |ω(x, V )| ≤ k((x, V ), .) W . ω W ✳ ■♥ ❛❞❞✐t✐♦♥✱ k((x, V ), .) W = k((x, V ), (x, V )) ❛♥❞ ❜♦t❤ ❦❡r♥❡•s k e ❛♥❞ k t ❛r❡ ❜♦✉♥❞❡❞ s♦ t❤❛t k ✐s ❛•s♦ ❜♦✉♥❞❡❞✳ ❲❡ ❝♦♥❝•✉❞❡ t❤❛t |ω| ∞ ≤ |k| ∞ . ω W ✳ ❚❤✉s ❈❛✉❝❤② s❡q✉❡♥❝❡s ✐♥ W 0 ❢♦r t❤❡ W ✲♥♦r♠ ❛r❡ ❛•s♦ ❈❛✉❝❤② s❡q✉❡♥❝❡ ❢♦r t❤❡ ✐♥✜♥✐t❡ ♥♦r♠✳ ■t r❡s✉•ts t❤❛t ❛•• t❤❡✐r •✐♠✐ts ❜❡•♦♥❣ t♦ C 0 (E × G d (E)) ❛♥❞ t❤❡r❡❢♦r❡ W ✐s ✐♥❝•✉❞❡❞ ✐♥ C 0 (E × G d (E))✳ ❚❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t② t❤❡♥ ❤♦•❞s ❢♦r ❛•• ω ∈ W ✱ ✇❤✐❝❤ s❤♦✇s t❤❛t t❤❡ ✐♥❝•✉s✐♦♥ ❡♠❜❡❞❞✐♥❣ ı : W ֒→ C 0 (E × G d (E)) ✐s ✐♥❞❡❡❞ ❝♦♥t✐♥✉♦✉s✳ ❈♦♥s❡q✉❡♥t•②✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ i * ♦❢ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s C 0 (E × G d (E)) ′ ✐♥t♦ t❤❡ ❞✉❛• ♦❢ W ✳ ❏✉st ❛s ❢♦r ❝✉rr❡♥ts✱ ✐❢ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ✐s♦♠❡tr② K W : W ′ → W ❞❡✜♥❡❞ ❜② K W µ, ω W = µ(ω) ❢♦r ❛•• µ ∈ W ′ ❛♥❞ ω ∈ W ✳ ❇② t❤❡ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ♣r♦♣❡rt②✱ ✇❡ ❦♥♦✇ t❤❛t K W δ (x,V ) = k((x, V ), .)✳ ❚❤❡♥ ❢♦r ❛•• x 1 , x 2 ∈ E ❛♥❞ V 1 , V 2 ∈ G d (E)✱ δ (x 1 ,V 1 ) , δ (x 2 ,V 2 ) W ′ = K W δ (x 1 ,V 1 ) , K W δ (x 2 ,V 2 ) W = K W δ (x 2 ,V 2 ) (x 1 , V 1 ) ❛♥❞ t❤✉s ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ❜❡t✇❡❡♥ t✇♦ ❉✐r❛❝s ✿ δ (x 1 ,V 1 ) , δ (x 2 ,V 2 ) W ′ = k e (x 1 , x 2 ) k t (V 1 , V 2 ) . ✭✸✳✷✷✮ ❋♦r ❛♥② ✈❛r✐❢♦•❞s µ ❛♥❞ µ ′ ✐♥ C 0 (E × G d (E)) ′ ✱ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ❜❡t✇❡❡♥ t❤❡♠ ✇r✐t❡s ✿ µ, µ ′ W ′ = K W µ, K W µ ′ W = µ(K W µ ′ ) = E×G d (E) K W µ ′ (x, V )dµ(x, V ) = E×G d (E) K W µ ′ , K W δ (x,V ) W dµ(x, V ) = E×G d (E) E×G d (E) K W δ (x,V ) (x ′ , V ′ )dµ ′ (x ′ , V ′ ) dµ(x, V ) ❛♥❞ ❣✐✈❡s ✜♥❛••② ✿ µ, µ ′ W ′ = (E×G d (E)) 2 k e (x, x ′ ) k t (V, V ′ )dµ(x, V )dµ ′ (x ′ , V ′ ) . ✭✸✳✷✸✮ ✶✵✹ ■♥ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡ ♦❢ r❡❝t✐✜❛❜•❡ ✈❛r✐❢♦•❞s µ X ❛♥❞ µ Y ✱ ✇❡ ❡✈❡♥ ♦❜t❛✐♥ t❤❛♥❦s t♦ ❡q✳✭✸✳✷✸✮ ❛♥❞ ✭✸✳✹✮ ✿ µ X , µ Y W ′ = X Y k e (x, y) k t (T x X, T y Y )dH d (x)dH d (y) . ✭✸✳✷✹✮ ❲❡ ❤❛✈❡ t❤❡r❡❢♦r❡ ❛ ❣❡♥❡r✐❝ ✇❛② t♦ ❜✉✐•❞ ❦❡r♥❡•s ❢♦r ✈❛r✐❢♦•❞s ✇❤✐❝❤ ❛r❡ s❡♣❛r❛❜•❡ s✐♥❝❡ s✉❝❤ ❦❡r♥❡•s ❛r❡ t❡♥s♦r ♣r♦❞✉❝ts ♦❢ ❛ ❦❡r♥❡• ♦♥ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡ E ✇✐t❤ ❛ ❦❡r♥❡• ♦♥ t❤❡ s❡t ♦❢ ❛•• t❛♥❣❡♥t s♣❛❝❡s G d (E)✳ ❇✉✐•❞✐♥❣ ❦❡r♥❡•s ♦♥ t❤❡ ❡✉❝•✐❞❡❛♥ s♣❛❝❡ E r❛✐s❡s ♥♦ ❞✐✣❝✉•t✐❡s ✭❡①❛♠♣•❡s ✇❡r❡ ❛•r❡❛❞② ❣✐✈❡♥ ✐♥ ❝❤❛♣t❡r ✶✮ ❛♥❞ ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ✇❛② t♦ ❜✉✐•❞ ❦❡r♥❡•s ♦♥ G d (E)✳ ❉✉❛• ❡♠❜❡❞❞✐♥❣ ❛♥❞ C 0 ✲✉♥✐✈❡rs❛•✐t② ❍♦✇❡✈❡r✱ ❛t t❤✐s st❛❣❡✱ t❤❡ ♠❡tr✐❝ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s r❡♠❛✐♥s ♦♥•② ❛ ♣s❡✉❞♦✲ ❞✐st❛♥❝❡ ❜❡❝❛✉s❡ ✐t ✐s st✐•• ✉♥❝•❡❛r ✇❤❡t❤❡r t❤❡ ❞✉❛• ❛♣♣•✐❝❛t✐♦♥ i * : C 0 (E×G d (E)) ′ → W ′ ✐s ❛•✇❛②s ❛♥ ❡♠❜❡❞❞✐♥❣✳ ❆s ✇❡ ❛•r❡❛❞② ❡✈♦❦❡❞ ✐♥ ❝❤❛♣t❡r ✷ s❡❝t✐♦♥ ✷✳✷✳✷✱ ✐♥ ❣❡♥✲ ❡r❛•✱ t❤✐s ✐s ❛❝t✉❛••② ♥♦t t❤❡ ❝❛s❡ ✿ ❛•t❤♦✉❣❤ W ֒→ C 0 (E × G d (E)) ✐s ❛♥ ❡♠❜❡❞❞✐♥❣✱ t❤❡ ❞✉❛• ❛♣♣•✐❝❛t✐♦♥ ♥❡❡❞ ♥♦t ❜❡ ✐♥•❡❝t✐✈❡✳ ❋♦r ✐♥st❛♥❝❡✱ ❝❤♦♦s✐♥❣ t❤❡ •✐♥❡❛r ❦❡r♥❡• k L ♦❢ ❡q✳✭✸✳✶✷✮ ♦♥ t❤❡ •r❛ss♠❛♥♥✐❛♥ ♠❛❦❡s i * ♥♦t ✐♥•❡❝t✐✈❡✳ ■♥❞❡❡❞✱ •❡t✬s ♣•❛❝❡ ♦✉r✲ s❡•✈❡s ✐♥ t❤❡ ❝❛s❡ E = R 2 ✇✐t❤ ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s (e 1 , e ⊥ 1 ) ❛♥❞ (e θ , e ⊥ θ ) t❤❡ r♦t❛t❡❞ ✈❡❝t♦rs ❜② ❛ ❝❡rt❛✐♥ ❛♥❣•❡ θ✳ ▲❡t x ❜❡ ❛ ✜①❡❞ ♣♦✐♥t ✐♥ E✳ ❋♦r ❛♥② ❦❡r♥❡• k e ⊗ k L ✱ ✇✐t❤ k e ❛ ❦❡r♥❡• ♦♥ E✱ ✇❡ ❤❛✈❡ ✿ (δ (x,e 1 ) + δ (x,e ⊥ 1 ) ) -(δ (x,e θ ) + δ (x,e ⊥ θ ) ) 2 W ′ = (δ (x,e 1 ) + δ (x,e ⊥ 1 ) ) 2 W ′ + (δ (x,e θ ) + δ (x,e ⊥ θ ) ) 2 W ′ -2 (δ (x,e 1 ) + δ (x,e ⊥ 1 ) ), (δ (x,e θ ) + δ (x,e ⊥ θ ) ) W ′ = k e (x, x) 4 -2 p e 1 , p e θ -2 p e ⊥ 1 , p e θ -2 p e 1 , p e ⊥ θ -2 p e ⊥ 1 , p e ⊥ θ = k e (x, x) 4 -2 cos 2 (θ) -2 sin 2 (θ) -2 sin 2 (θ) -2 cos 2 (θ) = 0 ❆♥ t❤❡r❡❢♦r❡ i * (δ (x,e 1 ) + δ (x,e ⊥ 1 ) ) = i * (δ (x,e θ ) + δ (x,e ⊥ θ ) ) ❢♦r ❛•• θ✳ ❚❤❡ ❢❛❝t t❤❛t i * ✐s ❛♥ ❡♠❜❡❞❞✐♥❣ ✐s ❝❛••❡❞ t❤❡ C 0 ✲✉♥✐✈❡rs❛•✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ❦❡r♥❡• ❛♥❞✱ ❛s ♣r♦✈❡❞ ✐♥ ❬✷✵(✱ ✐t ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ ♣r♦♣❡rt② ♦❢ W ❜❡✐♥❣ ❞❡♥s❡ ✐♥ C 0 (E × G d (E))✳ ■♥ t❤❡ ♣r❡✈✐♦✉s ❝♦✉♥t❡r❡①❛♠♣•❡✱ t❤❡ •❛❝❦ ♦❢ ✐♥•❡❝t✐✈✐t② ❜❛s✐❝❛••② ❝♦♠❡s ❢r♦♠ t❤❡ ♣r❡s❡♥❝❡ ♦❢ s❡✈❡r❛• ❞✐r❡❝t✐♦♥s ♦❢ G d (E) •♦❝❛t❡❞ ❛t ♦♥❡ s✐♥❣•❡ ♣♦✐♥t x✳ ❲❤❡♥ r❡♣r❡s❡♥t✐♥❣ s✉❜♠❛♥✐❢♦•❞s ♦r r❡✉♥✐♦♥ ♦❢ s✉❜♠❛♥✐❢♦•❞s✱ ✐t ✐s ♥♦t s♦♠❡t❤✐♥❣ t❤❛t ❝♦✉•❞ ❤❛♣♣❡♥ ❡①❝❡♣t ♦♥ ✐s♦•❛t❡❞ ♣♦✐♥ts✳ ■♥ ❢❛❝t✱ ✇❡ ❝❛♥ s❤♦✇ r❡s✉•t ♦❢ ✬✇❡❛❦ ✐♥•❡❝t✐✈✐t②✬ t❤❛t ❛♣♣•✐❡s ❢♦r ❛ ✈❡r② ✇✐❞❡ r❛♥❣❡ ♦❢ ❦❡r♥❡•s✳ ■t ✐s st❛t❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳✸✳ ▲❡t k = k e ⊗ k t ❜❡ ❛ ❦❡r♥❡• ❛s ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✸✳✷✳ ❆ss✉♠❡ t❤❛t ❦❡r♥❡• k e ✐s C 0 ✲✉♥✐✈❡rs❛• ❛♥❞ t❤❛t t❤❡ ❦❡r♥❡• k t ✐s s✉❝❤ t❤❛t k t (V, V ) > 0 ❢♦r ❛•• V ∈ G d (E)✳ ▲❡t X = N i=1 X i ❛♥❞ Y = M j=1 Y j ❜❡ t✇♦ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❝♦♠♣❛❝t d✲❞✐♠❡♥s✐♦♥❛• s✉❜♠❛♥✐❢♦•❞s ♦❢ E✳ ■❢ µ X -µ Y W ′ = 0 t❤❡♥ X = Y ✳ Pr♦♦❢✳ ❲❡ ✇✐•• ❞❡♥♦t❡ ❜② W e ❛♥❞ W t t❤❡ |❑❍❙ ❛ss♦❝✐❛t❡❞ t♦ ❦❡r♥❡•s k e ❛♥❞ k t ✳ ▲❡t✬s st❛rt ❜② t❤❡ ❝❛s❡ ✇❤❡r❡ X ✐s ❛ s✐♥❣•❡ s✉❜♠❛♥✐❢♦•❞ ♦❢ E✳ ■❢ µ X -µ Y W ′ = 0 ❛♥❞ X = Y ✱ ♦♥❡ ❝❛♥ ✜♥❞ x 0 ∈ X\Y ❛♥❞ t❤❡r❡ ❡①✐sts r > 0 s✉❝❤ t❤❛t B(x 0 , r) ∩ Y = ∅✳ ▲❡t a ∈ C 0 (E, R) ❜❡ ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t t❤❡ s✉♣♣♦rt ♦❢ a ✐s ✐♥❝•✉❞❡❞ ✶✵✺ ✐♥ B(x 0 , r)✳ ▲❡t ω t = k t (T x 0 X, .) ∈ W t ✳ ❋♦r ❛•• ❢✉♥❝t✐♦♥s ω e ∈ W e ✱ ω e ⊗ ω t ∈ W ❛♥❞ t❤✉s (µ X -µ Y )(ω e ⊗ ω t ) = 0✱ ✐✳❡ X ω e (x)ω t (T x X)dH d (x) - Y ω e (y)ω t (T y Y )dH d (y) = 0 ◆♦✇✱ s✐♥❝❡ W e ✐s ❞❡♥s❡ ✐♥ C 0 (E, R)✱ ✇❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ✉♥✐❢♦r♠•② a ❜② ❢✉♥❝t✐♦♥s ✐♥ W e ✳ ❇② ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ ♣r❡✈✐♦✉s ✐♥t❡❣r❛• ❛♥❞ t❤❡ ❢❛❝t t❤❛t a = 0 ✐♥ Y ✱ ✇❡ ❣❡t ✿ X a(x)ω t (T x X)dH d (x) = 0 ❚❤✐s ❤♦•❞s ❢♦r ❛•• ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s a s✉♣♣♦rt❡❞ ✐♥ B(x 0 , r)✱ ✇❤✐❝❤ ✐s ❝•❡❛r•② ✐♠♣♦ss✐❜•❡ s✐♥❝❡ x → ω t (T x X) ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ω t (T x 0 X) = k t (T x 0 X, T x 0 X) = 0 . ■❢ X ✐s ♥♦✇ ❛ ✜♥✐t❡ r❡✉♥✐♦♥ ♦❢ ❝♦♠♣❛❝t s✉❜♠❛♥✐❢♦•❞s ❛♥❞ X = Y t❤❡♥✱ ❛s ♣r❡✈✐♦✉s•②✱ ✇❡ ❝❛♥ ✜♥❞ x 0 ∈ N i=1 Xi ✇✐t❤ B(x 0 , r 0 ) ∩ Y = ∅✳ B(x 0 , r 0 ) ∩ X ✐s ✐ts❡•❢ ❛ ♥♦♥✲ ❡♠♣t② ✜♥✐t❡ r❡✉♥✐♦♥ ♦❢ s✉❜♠❛♥✐❢♦•❞s✳ ■♥ ♦r❞❡r t♦ ❣❡t ❛ s✐♠✐•❛r ♣r♦♦❢ ❛s ❢♦r t❤❡ ♦♥❡ s✉❜♠❛♥✐❢♦•❞ ❝❛s❡✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❡ ❢♦••♦✇✐♥❣ ✿ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t x ∈ B(x 0 , r 0 ) ∩ X✱ ρ > 0 ✇✐t❤ B(x, ρ) ⊂ B(x 0 , r 0 ) ❛♥❞ j ∈ {1, .., N } s✉❝❤ t❤❛t ❢♦r ❛•• ❢✉♥❝t✐♦♥ ω s✉♣♣♦rt❡❞ ✐♥ B(x, ρ) × G d (E)✱ µ X (ω) = µ X j (ω)✳ ❚❤✐s ❝❛♥ ❜❡ ♣r♦✈❡❞ r❡❝✉rs✐✈❡•②✳ ❆ss✉♠✐♥❣ t❤❡ r❡s✉•t ❢♦r N -1✱ ✇❡ ❝❛♥ ✜♥❞ x 1 ✱ r 1 ✱j 1 ✇✐t❤ B(x 1 , r 1 ) ⊂ B(x 0 , r 0 ) ❛♥❞ µ N -1 i=1 X i = µ X j 1 ❢♦r ❢✉♥❝t✐♦♥s s✉♣♣♦rt❡❞ ✐♥ B(x 1 , r 1 )✳ ◆♦✇✱ t❤❡r❡ ❛r❡ t✇♦ ❞✐st✐♥❝t ❝❛s❡s ✿ ❡✐t❤❡r B(x 1 , r 1 ) ∩ X j 1 = B(x 1 , r 1 ) ∩ X N ♦r ✇❡ ❝❛♥ ❛ss✉♠❡ ❢♦r ✐♥st❛♥❝❡ t❤❛t t❤❡r❡ ❡①✐sts x 2 ∈ B(x 1 , r 1 ) ∩ X j 1 ✱ r 2 > 0 s✉❝❤ t❤❛t B(x 2 , r 2 ) ⊂ B(x 1 , r 1 ) ❛♥❞ B(x 2 , r 2 ) ∩ X N = ∅✳ ■♥ t❤❡ ✜rst ❝❛s❡✱ s✐♥❝❡ X N ❝♦✐♥❝✐❞❡s ✇✐t❤ X j 1 ♦♥ B(x 1 , r 1 )✱ ✇❡ ❤❛✈❡ st✐•• µ N i=1 X i = µ X j 1 ❢♦r ❢✉♥❝t✐♦♥s s✉♣♣♦rt❡❞ ✐♥ B(x 1 , r 1 ) × G d (E) ❛♥❞ ✇❡ t❛❦❡ x = x 1 ✱ ρ = r 1 ✱ j = j 1 ✳ ■♥ t❤❡ s❡❝♦♥❞ ❝❛s❡✱ ♦♥❡ ❝❛♥ t❛❦❡ x = x 2 ✱ ρ = r 2 ✱ j = j 1 ❛♥❞ t❤❡ r❡s✉•t ❤♦•❞s ❛s ✇❡••✳ ❚❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ✐s t❤❡♥ ❡①❛❝t•② s✐♠✐•❛r t♦ t❤❡ ❝❛s❡ ♦❢ ❛ ✉♥✐q✉❡ s✉❜♠❛♥✐❢♦•❞✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡r❡ ✐s ♥♦ ❣❡♥❡r❛• ✐♥•❡❝t✐✈✐t② ❢♦r t❤❡ ❞✉❛• ❛♣♣•✐❝❛t✐♦♥✱ ♣r♦♣♦s✐t✐♦♥ ✸✳✸✳✸ ❡♥s✉r❡s t❤❛t t❤❡ ♠❡tr✐❝s ✇❡ ✉s❡ ❛r❡ ❛t •❡❛st ❞✐st❛♥❝❡s ♦♥ t❤❡ s❡t ♦❢ ✜♥✐t❡ r❡✉♥✐♦♥ ♦❢ s✉❜♠❛♥✐❢♦•❞s✱ ✇❤✐❝❤ ✐s ✉s✉❛••② ❡♥♦✉❣❤ ❢♦r t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❡ ❛✐♠ ❛t✳ ❲❡ ❛•s♦ ❜❡•✐❡✈❡ t❤❛t t❤❡ r❡s✉•t ❝♦✉•❞ ❜❡ ❣❡♥❡r❛•✐③❡❞ t♦ ❛♥② r❡❝t✐✜❛❜•❡ s✉❜s❡ts ♦❢ E ✭❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❢♦r H d ✲❛•♠♦st ❛•• x ♦♥❡ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥✮ ❜✉t t❤❡ ♣r♦♦❢ ❞♦❡s ♥♦t ♥♦t s❡❡♠ t♦ ❢♦••♦✇ ❞✐r❡❝t•② ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ✿ t❤❡ ♣❛ss❛❣❡ ❢r♦♠ ✜♥✐t❡ r❡✉♥✐♦♥ t♦ ❝♦✉♥t❛❜•❡ r❡✉♥✐♦♥ ✐s ❢♦r ✐♥st❛♥❝❡ ♥♦t ♦❜✈✐♦✉s✳ ■♥ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ♦❢ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ G d (E) ❣✐✈❡♥ ✐♥ s❡❝t✐♦♥ ✸✳✸✳✶✱ ✇❡ ❝❛♥ ❛❝t✉❛••② r❡❝♦✈❡r t❤❡ ✐♥•❡❝t✐✈✐t② ♦❢ i * ♦♥ C 0 (E × G d (E)) ′ ✐ts❡•❢✱ ✇❤✐❝❤ ✐s st❛t❡❞ ✐♥ t❤❡ ♥❡①t ♣r♦♣♦s✐t✐♦♥ ✿ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳✹✳ ■❢ k = k e ⊗ k G ✇✐t❤ k e ❛ C 0 ✲✉♥✐✈❡rs❛• ❦❡r♥❡• ♦♥ E ❛♥❞ k G t❤❡ r❡str✐❝t✐♦♥ ♦❢ ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ L(E) ❣✐✈❡♥ ✐♥ ❡q✳✭✸✳✶✸✮✱ t❤❡♥ k ✐s ❛ C 0 ✲✉♥✐✈❡rs❛• ❦❡r♥❡• ♦♥ E × G d (E)✳ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ✐s ♠❛✐♥•② ❜❛s❡❞ ♦♥ t✇♦ r❡s✉•ts ❛❜♦✉t |❑❍❙✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ✈❡r② ✇❡••✲❦♥♦✇♥ ♣r♦♣❡rt② t❤❛t ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡ ✐s C 0 ✲✉♥✐✈❡rs❛•✳ ■♥ ♦✉r ❝♦♥t❡①t✱ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• K ❞❡✜♥❡❞ ♦♥ L(E) × L(E) ❜② ✶✵✻ K(l 1 , l 2 ) = e -|l 1 -l 2 | 2 σ 2 ✐s t❤✉s C 0 ✲✉♥✐✈❡rs❛• ✭❝❢ ❬✷✵(✮✳ ◆♦✇✱ t❤❡ ❦❡r♥❡• k G t❤❛t ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ ✐♥ ❡q✳✭✸✳✶✸✮ ✐s t❤❡ r❡str✐❝t✐♦♥ ♦❢ K t♦ t❤❡ s✉❜s❡t ♦❢ ❛•• ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥s ♦♥ d✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ✭✇❤✐❝❤ ✐s ✐❞❡♥t✐✜❡❞ t♦ G d (E)✮✳ ❚❤✐s s✉❜s❡t ✐s ❝•♦s❡❞ ✐♥ L(E) ❛♥❞ ✉s✐♥❣ ♣r♦♣❡rt② ✷✳✷✳✷ ❢r♦♠ ❝❤❛♣t❡r ✷ s❡❝t✐♦♥ ✷✳✷✳✷✱ ✇❡ ❞❡❞✉❝❡ t❤❛t k G ✐s t❤❡♥ ❛ C 0 ✲✉♥✐✈❡rs❛• ❦❡r♥❡•✳ ■❢ W e ⊂ C 0 (E, R) ❛♥❞ W t ⊂ C 0 (G d (E), R) ❛r❡ t❤❡ |❑❍❙ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❦❡r♥❡•s k e ❛♥❞ k G ✱ ✐t r❡s✉•ts t❤❛t ✇❡ ❤❛✈❡ W e ❞❡♥s❡ ✐♥ C 0 (E, R) ✭❜❡❝❛✉s❡ k e ✐s ❛ss✉♠❡❞ t♦ ❜❡ C 0 ✲✉♥✐✈❡rs❛•✮ ❛♥❞ W t ❞❡♥s❡ ✐♥ C 0 (G d (E), R)✳ ■t ✐s t❤❡♥ ❝•❡❛r t❤❛t W ✐s ❞❡♥s❡ ✐♥ C 0 (E × G d (E), R)✳ ■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡ ❡①♣•❛✐♥❡❞ ✐♥ t❤✐s s✉❜s❡❝t✐♦♥ ❤♦✇ t♦ ❞❡✜♥❡ ❍✐•❜❡rt ♠❡tr✐❝s ♦♥ ✈❛r✐❢♦•❞s t❤❛t ❛r❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❡♥s♦r ♣r♦❞✉❝ts ❜❡t✇❡❡♥ ❦❡r♥❡•s ♦♥ E ❛♥❞ ❦❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ❚❤✐s ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ♥❡❝❡ss❛r✐•② r❡❛• ❞✐st❛♥❝❡s ♦♥ ✈❛r✐❢♦•❞s ❜❡❝❛✉s❡ ♦❢ t❤❡ ♣♦ss✐❜•❡ ♥♦♥✲✐♥•❡❝t✐✈✐t② ♦❢ t❤❡ ❛♣♣•✐❝❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s ❛♥❞ t❤❡ ❞✉❛• ♦❢ t❤❡ |❑❍❙ W ✳ (❡t✱ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t ❢♦r ❛ ✈❡r② ✇✐❞❡ ❝•❛ss ♦❢ s✉❝❤ ❦❡r♥❡•s✱ t❤❡ r❡s✉•t✐♥❣ ❞✐st❛♥❝❡s ❛r❡ s❡♣❛r❛t✐♥❣ ✜♥✐t❡ ✉♥✐♦♥s ♦❢ s✉❜♠❛♥✐❢♦•❞s ❛♥❞ ✇❡ ❤❛✈❡ ❛r❣✉❡❞ ✐♥ ❢❛✈♦r ♦❢ •❛✉ss✐❛♥ ❦❡r♥❡•s ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❢♦r ✇❤✐❝❤ ✇❡ ♦❜t❛✐♥ t❤❡ r❡❛• ✐♥•❡❝t✐✈✐t② ❛♥❞ t❤✉s ❞✐st❛♥❝❡s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❛•• ✈❛r✐❢♦•❞s✳ ❲❡ ✇✐•• ❡•❛❜♦r❛t❡ ❛ •✐tt•❡ ♠♦r❡ ♦♥ t❤❡ ✐♥t❡r❡st ♦❢ s✉❝❤ •❛✉ss✐❛♥ ❦❡r♥❡•s ✐♥ s♣❡❝✐✜❝ s✐t✉❛t✐♦♥s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✭❝❢ ✜❣✉r❡ ✸✳✺✮✳ ✸✳✹ Pr♦♣❡rt✐❡s ♦❢ |❑❍❙ ♥♦r♠s ♦♥ ✈❛r✐❢♦•❞s ✸✳✹✳✶ ❉❡♣❡♥❞❡♥❝② ♦♥ t❤❡ s❝❛•❡s ♦❢ t❤❡ ❦❡r♥❡•s ❖♥❡ ♦❢ t❤❡ ♠❛•♦r ❛❞✈❛♥t❛❣❡ ♦❢ ❦❡r♥❡• ♠❡tr✐❝s ✐s t❤❛t t❤❡② ❛••♦✇ ❛♥❛•②s✐s ❛t ❞✐✛❡r❡♥t s❝❛•❡s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s✱ t❤✐s ❝❛♥ ❜❡ ❝♦♥tr♦••❡❞ ❜♦t❤ ✐♥ t❤❡ s♣❛❝❡ E ❛♥❞ t❤❡ •r❛ss♠❛♥♥✐❛♥ ❜② t❤❡ ✇✐❞t❤s σ e ❛♥❞ σ t ♦❢ t❤❡ •❛✉ss✐❛♥s ✐❢ ♦♥❡ t❛❦❡s •❛✉ss✐❛♥ ❦❡r♥❡•s✳ ❚❤❡ ♣❛r❛♠❡t❡r σ e ❝❛♥ ❜❡ t❤♦✉❣❤t ❛s r❡♣r❡s❡♥t✐♥❣ t❤❡ t②♣✐❝❛• ✐♥t❡r❛❝t✐♦♥ ❞✐st❛♥❝❡ ♦♥ t❤❡ ❉✐r❛❝s✬ ♣♦s✐t✐♦♥s ✐♥ E ✇❤❡r❡❛s σ t ❝♦♥tr♦•s t❤❡ ❞❡s✐r❡❞ ❝♦rr❡•❛t✐♦♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ s✉❜s♣❛❝❡s✳ ❆♥ ✐♥t❡r❡st✐♥❣ ❢❛❝t ✐s t❤❛t ❢♦r ✈❡r② •❛r❣❡ σ e ✭❜❛s✐❝❛••② ❛•• ♣♦✐♥ts s❡❡♥ ❛t t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ ❜② t❤❡ ❦❡r♥❡•✮✱ t❤❡ r❡s✉•t✐♥❣ ♠❡tr✐❝ ♦♥ ✈❛r✐❢♦•❞s ❞♦ ♥♦t ❜❡❝♦♠❡ ♥❡❝❡ss❛r✐•② tr✐✈✐❛• ❛s ✇✐t❤ ❝✉rr❡♥ts ✭❝❢ t❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ s❡❝t✐♦♥ ✸✳✶✳✷✮✳ ■♥❞❡❡❞ ❢♦r σ e → ∞✱ t❤❡ W ′ ✲❞✐st❛♥❝❡ ❜❡❝♦♠❡s ❛ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❦❡r♥❡• k t ✳ ▼♦r❡ s♣❡❝✐✜❝❛••② ✿ µ X 2 W ′ → X×X k t (T x X, T y X)dH d (x)dH d (y) = G d (E)×G d (E) k t (u, v)dν X (u)dν X (v) ✇❤❡r❡ ν X . = H d • T an -1 ✐s t❤❡ ✐♠❛❣❡ ♠❡❛s✉r❡ ♦❢ H d ❜② t❤❡ ❛♣♣•✐❝❛t✐♦♥ T an : X → G d (E), x → T x X✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛t ✐♥✜♥✐t❡ s❝❛•❡ ❢♦r k e ✱ µ X 2 W ′ r❡♣r❡s❡♥ts ❛ ♠❡tr✐❝ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♥♦♥✲♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡s t♦ t❤❡ s❤❛♣❡ X✱ ✇❤✐❝❤ ✇❡ ❝❛♥ s❡❡✱ ✐♥ ❛ ❝❡rt❛✐♥ ✇❛②✱ ❛s ❛ ❤✐st♦❣r❛♠ ♠❡tr✐❝ ♦♥ G d (E)✳ ❚❤❡r❡❢♦r❡✱ ❡✈❡♥ ❛t ✐♥✜♥✐t❡ s❝❛•❡ ❢♦r k e ✱ s♦♠❡t❤✐♥❣ ♦❢ t❤❡ s❤❛♣❡ st✐•• r❡♠❛✐♥s ✐♥ t❤❡ ✈❛r✐❢♦•❞ r❡♣r❡s❡♥t❛t✐♦♥ ❝♦♥tr❛r✐•② t♦ ❝✉rr❡♥ts✳ ■♥ t❤♦s❡ s✐t✉❛t✐♦♥s✱ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❦❡r♥❡• ♦♥ t❤❡ •r❛ss♠❛♥♥ ♣❛rt ♦❢ ✈❛r✐❢♦•❞s ✐s q✉✐t❡ ❞❡❝✐s✐✈❡✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ C 0 ✲✉♥✐✈❡rs❛•✐t② ✐ss✉❡ ❞✐s❝✉ss❡❞ ♣r❡✈✐♦✉s•② ❜❡❝♦♠❡s ✐♠♣♦rt❛♥t✳ ❲❡ s❤♦✇ ❛♥ ✐••✉str❛t✐♦♥ ♦❢ s✉❝❤ ♣❤❡♥♦♠❡♥♦♥ ✐♥ ✜❣✉r❡ ✸✳✺ ❢♦r t❤❡ •✐♥❡❛r ❦❡r♥❡• k L ✇❤✐❝❤ ✐s ♥♦t C 0 ✲✉♥✐✈❡rs❛• ❛♥❞ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• k G ✱ ✇❤✐❝❤ ✇❛s ♣r♦✈❡❞ t♦ ❜❡✳ ❚❤❡ ✜❣✉r❡ ❝♦♥✜r♠s ✇❤❛t ✇❛s ❛•r❡❛❞② ♥♦t✐❝❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ✶✵✼ ❋✐❣✉r❡ ✸✳✺✿ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ✈❛r✐❢♦•❞s✬ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ✷❉ ❣r✐❞✲•✐❦❡ s❤❛♣❡ ✭✐♥ ❜•❛❝❦✮ ❛♥❞ ✐ts r♦t❛t❡❞ ✈❡rs✐♦♥ ✭✐♥ ♠❛❣❡♥t❛✮ ❢♦r ✈❛r✐♦✉s r♦t❛t✐♦♥ ❛♥❣•❡s ❛♥❞ ❢♦r ❜♦t❤ t❤❡ •✐♥❡❛r ❦❡r♥❡• ❡q✳✭✸✳✶✷✮ ❛♥❞ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• ♦❢ ❡q✳✭✸✳✶✸✮ ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✱ ❛•• ✐♥ t❤❡ ❝❛s❡ ♦❢ •❛r❣❡✲s❝❛•❡ ❦❡r♥❡• ♦♥ t❤❡ s♣❛❝❡ E✳ ❚❤❡ ❣r❛♣❤ ❞✐s♣•❛②s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ✈❛r✐❢♦•❞ ❞✐st❛♥❝❡s ❛♥❞ t❤❡ ♥♦r♠ ♦❢ t❤❡ ♦r✐❣✐♥❛• s❤❛♣❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r♦t❛t✐♦♥ ❛♥❣•❡✳ ❲✐t❤ t❤❡ •✐♥❡❛r ❦❡r♥❡•✱ t❤❡ ✈❛•✉❡s r❡♠❛✐♥ ✈❡r② s♠❛•• ❢♦r ❛•• ❛♥❣•❡s✱ ❛♥❞ ✐t ✐s ♥❡❛r•② ✉♥❛❜•❡ t♦ ❞✐st✐♥❣✉✐s❤ ❜♦t❤ s❤❛♣❡s ✇❤❡r❡❛s t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• s❤♦✇s ❛♥ ❡①♣❡❝t❡❞ ❜❡❤❛✈✐♦r ✇✐t❤ ❛ ♠❛①✐♠✉♠ ❞✐st❛♥❝❡ ❢♦r ❛ ✹✺ • ❛♥❣•❡✳ s❡❝t✐♦♥ ✸✳✸✳✷ ❛❜♦✉t t❤❡ •✐♥❡❛r ❦❡r♥❡•✳ ❚❤❡ ❝♦♥❝•✉s✐♦♥ t❤❛t ✇❡ ❝❛♥ ❜❡ ❞r❛✇♥ ✐♥ ❣❡♥❡r❛• ✐s t❤❛t •❛✉ss✐❛♥ ❦❡r♥❡•s ♦♥ G d (E) ♣r♦✈✐❞❡ ♠❡tr✐❝s t❤❛t ❛r❡ ♦❜✈✐♦✉s•② ♠♦r❡ ❡✛❡❝t✐✈❡ ✐♥ s✐t✉❛t✐♦♥s ♦❢ ♠✉•t✐♣•❡ ❞✐r❡❝t✐♦♥s ❝r♦ss✐♥❣ ❛t ♥❡❛r❜② ♣♦✐♥ts✱ ❛s ♠✐❣❤t ❛♣♣❡❛r ✇❤❡♥ tr❡❛t✐♥❣ ✜❜❡rs ♦r tr❡❡✲•✐❦❡ str✉❝t✉r❡s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ ✇❡ •❡t σ t → ∞✱ ✇❡ s❡❡ t❤❛t k t (u, v) = 1 ❢♦r ❛•• u, v ∈ G d (E) ❛♥❞ t❤❡ |❑❍❙ ♥♦r♠ ✇❡ ✜♥❞ ♦♥•② t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❉✐r❛❝s ✿ t❤❡s❡ ❛r❡ ❡①❛❝t•② t❤❡ ♠❡❛s✉r❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ s❤❛♣❡s ✐♥tr♦❞✉❝❡❞ ❜② ••❛✉♥ès ✐♥ ❬✹✶( ❝❤❛♣ ✹✳ ✸✳✹✳✷ ❈♦♥tr♦• ♦❢ t❤❡ ✈♦•✉♠❡ ❜② W ′ ✲♥♦r♠ ❲❡ ♥♦✇ ❝♦♠❡ t♦ t❤❡ ♣r♦❜•❡♠ r❛✐s❡❞ ❜② ♦✉r ❞✐s❝✉ss✐♦♥ r❡•❛t❡❞ t♦ ✜❣✉r❡ ✸✳✸✳ ❲❡ s❤♦✇ ✐♥ ✇❤✐❝❤ ♣r❡❝✐s❡ s❡♥s❡ |❑❍❙ ♠❡tr✐❝s ♦♥ ✈❛r✐❢♦•❞s s✉❝❝❡ss❢✉••② ❛✈♦✐❞ t❤❡ ❝❛♥❝❡••❛t✐♦♥ ♣❤❡♥♦♠❡♥♦♥ ♦❢ ❝✉rr❡♥ts✳ ❚❤✐s ✐s ❢♦r♠✉•❛t❡❞ ✐♥ t❤❡ t❤❡♦r❡♠ ❜❡•♦✇ ❛♥❞ ✐ts ❝♦r♦••❛r② ✿ ❚❤❡♦r❡♠ ✸✳✹✳✶✳ ▲❡t µ ❜❡ ❛ ♣♦s✐t✐✈❡ d✲❞✐♠❡♥s✐♦♥❛• ✈❛r✐❢♦•❞ ♦❢ E s✉❝❤ t❤❛t Supp(µ) ⊂ B(0, 1) × G d (E)✳ ▲❡t k ❜❡ ❛ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ♦♥ E × G d (E) ❛♥❞ W ✐ts |❑❍❙✱ ❞❡✜♥❡❞ •✐❦❡ ✐♥ s❡❝t✐♦♥ ✸✳✸✳✷ ❜② k((x, u), (y, v)) = k e (x, y).k t (u, v) ✇❤❡r❡ k e ✐s ❛ r❡✲ ♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ♦♥ E ❛♥❞ k t ❛ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ♦♥ t❤❡ •r❛ss♠❛♥♥✐❛♥✳ ❲❡ ♠❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ❛❞❞✐t✐♦♥❛• ❛ss✉♠♣t✐♦♥s ✿ ✶✳ k e ✐s ❛ r❛❞✐❛• s❝❛•❛r ❦❡r♥❡• s♦ t❤❛t ✇❡ ❝❛♥ ✇r✐t❡ k e (x, y) = h e (|x -y|)✳ h e ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥✱ ✇✐t❤ h e (0) > 0✳ ✷✳ k t ✐s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ G d (E) × G d (E) s✉❝❤ t❤❛t ❢♦r ❛•• u ∈ G d (E)✱ k t (u, u) > 0✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t ❈t❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ µ ∈ W ′ s✉❝❤ t❤❛t ✿ µ W ′ ≥ ❈t❡ µ(E × G d (E)) . Pr♦♦❢✳ ❋♦r ❛♥② ✈❛r✐❢♦•❞ µ ∈ W ′ ✱ ✇❡ ❤❛✈❡ ✿ µ 2 W ′ = (E×G d (E)) 2 k e (x, y).k t (u, v)dµ(x, u)dµ(y, v) . ✭✸✳✷✺✮ ✶✵✽ ❋♦r t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✹✳✶✱ ✇❡ s❤❛•• ✜rst ❡①❛♠✐♥❡ t❤❡ ❝❛s❡ ♦❢ ❛ ❝♦♥st❛♥t ❦❡r♥❡• ❢♦r k e ✳ ❙t❡♣ ✶✿ ❲❡ ✜rst ❛ss✉♠❡ t❤❛t k e (x, y) = 1 ❢♦r ❛•• x, y✳ ▲❡t✬s ❞❡♥♦t❡ ❜② p : E × G d (E) → G d (E) t❤❡ ❛♣♣•✐❝❛t✐♦♥ (x, u) → u✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ✐♠❛❣❡ ♠❡❛s✉r❡ ν . = µ • p -1 ❞❡✜♥❡❞ ♦♥ G d (E)✳ ❚❤❡♥ ✿ µ 2 W ′ = G d (E)×G d (E) k t (u, v)dν(u)dν(v) . ◆♦t❡ t❤❛t ν(G d (E)) = µ(p -1 (G d (E))) = µ(E × G d E)✳ ◆♦✇✱ t❤❡ ❝♦♠♣❛❝t ❣r♦✉♣ SO(E) ♦❢ ❞✐r❡❝t ✐s♦♠❡tr✐❡s ♦❢ E ❛❝ts tr❛♥s✐t✐✈❡•② ♦♥ G d (E) ❜② t❤❡ r❡•❛t✐♦♥ g.❙♣❛♥(e 1 , .
., e d ) .

= ❙♣❛♥(g(e 1 ), .., g(e

d ))✱ ♠❛❦✐♥❣ G d (E) ❛ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡✳ ❙✐♥❝❡ SO(E) ✐s ❛ ❝♦♠♣❛❝t ❣r♦✉♣✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r ✐ts ✉♥✐q✉❡ ❜✐✲✐♥✈❛r✐❛♥t ❍❛❛r ♠❡❛s✉r❡ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ λ ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ λ(SO(E)) = 1✳ ■❢ ✇❡ ✜① ❛ ♣❛rt✐❝✉•❛r ❡•❡♠❡♥t u 0 ∈ G d (E)✱ λ ✐♥❞✉❝❡s ✐♥ t✉r♥ ❛ ♠❡❛s✉r❡ ♦♥ G d (E) ❞❡✜♥❡❞ ❢♦r ❛•• B ❜② λ Gr (B) = λ({g ∈ SO(E) | g.u 0 ∈ B})✳ ❖♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t t❤❡ r✐❣❤t✲✐♥✈❛r✐❛♥❝❡ ♦❢ λ ✐♠♣•✐❡s t❤❛t λ Gr ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ u 0 ∈ G d (E)✳ ■♥ t❤❡ s❛♠❡ ✇❛②✱ t❤❛♥❦s t♦ t❤❡ •❡❢t✲✐♥✈❛r✐❛♥❝❡ ♦❢ λ✱ λ Gr ✐s ✐♥✈❛r✐❛♥t ❜② t❤❡ ❛❝t✐♦♥ ♦❢ G ✐✳❡ λ Gr (g.B) = λ Gr (B) ❢♦r ❛•• g✳ ❆ s❡❝♦♥❞ ✐♠♣♦rt❛♥t ❡•❡♠❡♥t ✐s t❤❛t✱ ❛s ❛ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❛❝t ▲✐❡ ❣r♦✉♣ ✭♦♥ ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ❜✐✲✐♥✈❛r✐❛♥t ♠❡tr✐❝✮✱ G d (E) ❝❛♥ ❜❡ ❡q✉✐♣♣❡❞ ❜② ♣r♦•❡❝t✐♦♥ ✇✐t❤ ❛ •❡❢t✲✐♥✈❛r✐❛♥t ❞✐st❛♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛❝t✐♦♥ ♦❢ SO(E)✱ ✇❤✐❝❤ ✇❡ ✇✐•• ❞❡♥♦t❡ d✳ ▲❡t✬s ♥♦✇ ❝♦♥s✐❞❡r ❛ ❝♦♥t✐♥✉♦✉s✱ ♣♦s✐t✐✈❡ L 1 ✭❢♦r t❤❡ ♠❡❛s✉r❡ λ Gr ✮ ❢✉♥❝t✐♦♥ φ ♦♥ G d (E)✳ ❙✐♥❝❡ λ Gr (G d (E)) = λ(SO(E)) = 1 < +∞✱ φ ✐s ❛•s♦ ✐♥ L 2 ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❛t φ 2 = 1✳ ❲❡ ❝❛♥ ❛ss✉♠❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ s✉♣♣♦rt ❙✉♣♣(φ) ♦❢ φ ✐s ✐♥❝•✉❞❡❞ ✐♥ ❛ ❝❡rt❛✐♥ ❜❛•• ♦❢ r❛❞✐✉s δ ❝❡♥t❡r❡❞ ❛t u 0 ✳ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡ r❡❣✉•❛r✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ψ ♦♥ G d (E) × G d (E) ❞❡✜♥❡❞ ❜② ✿ ψ(u, v) = SO(E) φ(g.u)φ(g.v)dλ(g) . ■t✬s ♦❜✈✐♦✉s t❤❛t ψ ✐s ❛•s♦ ❛ ♣♦s✐t✐✈❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ ◆♦✇✱ t❤❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ✐♥✈❛r✐❛♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ λ Gr s❤♦✇ t❤❛t ❢♦r ❛•• u ∈ G d (E)✱ ✐❢ τ u : g → g.u t❤❡♥✱ λ • τ -1 u = λ Gr ❛♥❞ ✐t r❡s✉•ts t❤❛t ✿ SO(E) φ(g.u) 2 dλ(g) = G d (E) φ(v) 2 dλ Gr (v) = φ 2 2 = 1 , ❋r♦♠ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t②✱ ✇❡ ❤❛✈❡ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ∀(u, v) ∈ G d (E) 2 , ψ(u, v) ≤ SO(E) φ(g.u) 2 dλ(g) 1 2 SO(E) φ(g.v) 2 dλ(g) 1 2 ≤ φ 2 2 t❤❡r❡❢♦r❡ ψ(u, v) ≤ 1✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐❢ d(u, v) ≥ 2δ✱ ✇❡ ❤❛✈❡ d(g.u, g.v) ≥ 2δ ❢♦r ❛•• g s✐♥❝❡ d ✐s •❡❢t✲✐♥✈❛r✐❛♥t✳ ❚❤❡r❡❢♦r❡ d(g.u, u 0 ) + d(g.v, u 0 ) ≥ d(g.u, g.v) ≥ 2δ✳ ■t r❡s✉•ts t❤❛t ❡✐t❤❡r d(g.u, u 0 ) ≥ δ ♦r d(g.v, u 0 ) ≥ δ ❛♥❞ ❛♥②❤♦✇ φ(g.u)φ(g.v) = 0✳ ❈♦♥s❡q✉❡♥t•②✱ ψ ✈❛♥✐s❤❡s ❢♦r d(u, v) ≥ 2δ✳ ◆♦✇✱ s✐♥❝❡ k t ✐s ♣♦s✐t✐✈❡ ❛♥❞ 0 ≤ ψ ≤ 1 µ 2 W ′ ≥ G d (E)×G d (E) k t (u, v)ψ(u, v)dν(u)dν(v) . ✶✵✾ ▼♦r❡♦✈❡r✱ k t ✐s ❛•s♦ ❝♦♥t✐♥✉♦✉s ❛♥❞ k t (u, u) > 0 ❢♦r ❛•• u s♦✱ ❜② ❝♦♠♣❛❝t♥❡ss ♦❢ G d (E)✱ t❤❡r❡ ❡①✐sts α > 0 ❛♥❞ ❛♥ ♦♣❡♥ ❞♦♠❛✐♥ D ⊂ G d (E) × G d (E) ♦❢ t❤❡ ❢♦r♠ D = {(u, v) | d(u, v) < ǫ} s✉❝❤ t❤❛t ❢♦r ❛•• u, v ∈ D✱ k t (u, v) ≥ α✳ ❲❡ ❝❛♥ ❛•s♦ ❛ss✉♠❡✱ ❜② ❝❤♦♦s✐♥❣ ❛♣♣r♦♣r✐❛t❡•② t❤❡ ❢✉♥❝t✐♦♥ φ✱ t❤❛t δ ≤ ǫ 2 ❛♥❞ t❤❡r❡❢♦r❡ ψ(u, v) = 0 ♦✉ts✐❞❡ D✳ ■t r❡s✉•ts t❤❛t ✿ µ 2 W ′ ≥ α G d (E)×G d (E) ψ(u, v)dν(u)dν(v) . ✭✸✳✷✻✮ ❯s✐♥❣ ❋✉❜✐♥✐✬s t❤❡♦r❡♠ ✐♥ t❤❡ •❛st ✐♥t❡❣r❛•✱ ✇❡ ❝❛♥ ✇r✐t❡ G d (E)×G d (E) ψ(u, v)dν(u)dν(v) = SO(E) G d (E)×G d (E) φ(g.u)φ(g.v)dν(u)dν(v) dλ(g) = SO(E) G d (E) φ(g.u)dν(u) 2 dλ(g) ≥ SO(E) G d (E) φ(g.u)dν(u)dλ(g) 2 ✭✸✳✷✼✮ t❤❡ •❛st ❡st✐♠❛t❡ r❡s✉•t✐♥❣ ❢r♦♠ ❈❛✉❝❤②✲❙❝❤✇❛rt③ ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡ ❢❛❝t t❤❛t λ(SO(E)) = 1✳ ❯s✐♥❣ ❛❣❛✐♥ ❋✉❜✐♥✐✬s t❤❡♦r❡♠✱ SO(E) G d (E) φ(g.u)dν(u)dλ(g) = G d (E) SO(E)
φ(g.u)dλ(g) dν(u) .

▼❛❦✐♥❣ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ τ u : g → g.u ✐♥ t❤❡ ✐♥s✐❞❡ ✐♥t❡❣r❛• ❛♥❞ ✉s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ♣r❡✈✐♦✉s•②✱ ✇❡ ♦❜t❛✐♥ t❤❛t SO(E) φ(g.u)dλ(g) = G d (E) φ(w)dλ Gr (w) = φ 1 > 0 ■♥s❡rt✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✐♥ ❡q✉❛t✐♦♥ ✭✸✳✷✼✮✱ G d (E)×G d (E) ψ(u, v)dν(u)dν(v) ≥ φ 2 1 ν(G d (E)) 2 = φ 2 1 µ(E × G d E) 2 . ❙♦ t❤❛t ✇❡ ❡✈❡♥t✉❛••② ♦❜t❛✐♥ ✿ µ 2 W ′ ≥ α φ 2 1 .µ(E × G d E) 2 ❛♥❞ s✐♥❝❡ β . = α φ 2 1 ✐s ❛ ❝♦♥st❛♥t t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ µ✱ t❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ✐♥ t❤❛t ❝❛s❡✳ ❙t❡♣ ✷✿ ❲❡ ♥♦✇ ♠♦✈❡ t♦ t❤❡ ♣r♦♦❢ ❢♦r ❛ ❣❡♥❡r❛• ❦❡r♥❡• k e ✳ ❋r♦♠ t❤❡ ❤②♣♦t❤❡s❡s✱ t❤❡ s✉♣♣♦rt ♦❢ µ ✐s ✐♥❝•✉❞❡❞ ✐♥ B(0, 1) × G d (E) ❛♥❞ k e ✐s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♣♦s✐t✐✈❡ r❛❞✐❛• s❝❛•❛r ❦❡r♥❡• ♦♥ E✳ ❚❤✉s✱ t❤❡r❡ ❡①✐sts ❛ r❡❛• ♥✉♠❜❡r δ < 1 s✉❝❤ t❤❛t ❢♦r ❛♥② x, y ∈ E ✇✐t❤ |x -y| ≤ δ✱ ✇❡ ❤❛✈❡ k e (x, y) ≥ h e (0)/2 . = κ✳ ▲❡t✬s ♥♦✇ ❝♦♥s✐❞❡r ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ✇✐t❤ ❛ s❡t ♦❢ ❝✉❜❡s {C i } i=1,..,M ♦❢ ❞✐❛♠❡t❡r s♠❛••❡r t❤❛♥ δ s✉❝❤ ✶✶✵ t❤❛t X ⊂ M i=1 C i ✳ ❚❤❡♥ t❤❡ s✉❜s❡ts S i . = (C i × G d (E)) ∩ ❙✉♣♣(µ) ❢♦r♠ ❛ ♣❛rt✐t✐♦♥ ♦❢ ❙✉♣♣(µ)✳ ▼♦r❡♦✈❡r✱ µ 2 W ′ = (E×G d (E)) 2 k e (x, y) k t (u, v)dµ(x, u)dµ(y, v) ≥ M i=1 S i ×S i k e (x, y) k t (u, v)dµ(x, u)dµ(y, v) ≥ κ M i=1 S i ×S i k t (u, v)dµ(x, u)dµ(y, v) . ❲❡ ❝❛♥ ❛♣♣•② t❤❡ r❡s✉•t ♦❢ st❡♣ ✶ t♦ ❡❛❝❤ ♦❢ t❤❡s❡ ✐♥t❡❣r❛•s✱ t❤✉s ♦❜t❛✐♥✐♥❣ µ 2 W ′ ≥ βκ M i=1 µ(S i ) 2 ≥ βκ. 1 M M i=1 µ(S i ) 2 ≥ βκ M µ(E × G d E) 2 , t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t② ❜❡✐♥❣ ❛ ❞✐s❝r❡t❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③ ❛♥❞ t❤❡ •❛st ♦♥❡ r❡s✉•t✐♥❣ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t {S i } ✐s ❛ ♣❛rt✐t✐♦♥ ♦❢ ❙✉♣♣(µ)✳ ❚❤✐s ❡♥❞s t❤❡ ♣r♦♦❢ ❜❡❝❛✉s❡ t❤❡ ❝♦♥st❛♥ts β ❛♥❞ κ ❛r❡ ❜♦t❤ ✐♥❞❡♣❡♥❞❡♥t ♦❢ µ✱ ❛♥❞ s♦ ✐s M ✇❤✐❝❤ ♦♥•② ❞❡♣❡♥❞s ♦♥ t❤❡ ❦❡r♥❡• k e ✳ ❚❤❡ ❤②♣♦t❤❡s❡s ♦❢ t❤❡♦r❡♠ ✸✳✹✳✶ ♦♥ t❤❡ ❦❡r♥❡•s ❛r❡ ♠♦st•② t❡❝❤♥✐❝❛• ❜✉t ❛r❡ ♥♦t ♣❛rt✐❝✉•❛r•② r❡str✐❝t✐✈❡ ✐♥ ♣r❛❝t✐❝❡ s✐♥❝❡ ♦♥❡ ❝❛♥ ❝❤❡❝❦ ❡❛s✐•② t❤❛t ❛•• t❤❡ ❡①❛♠♣•❡s ♦❢ ❦❡r♥❡•s ❣✐✈❡♥ ✐♥ s❡❝t✐♦♥ ✸✳✸✳✶ ❝♦♠♣•② t♦ t❤❡ r❡q✉✐r❡♠❡♥ts ♦♥ k t ✳ ❆s ❛ ❞✐r❡❝t ❝♦r♦••❛r②✱ ✇❡ ❛•s♦ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✿ ❈♦r♦••❛r② ✸✳✹✳✶✳ ▲❡t X ❜❡ ❛ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ E ✐♥❝•✉❞❡❞ ✐♥ t❤❡ ✉♥✐t ❜❛••✳ ❲❡ ♠❛❦❡ t❤❡ s❛♠❡ ❤②♣♦t❤❡s❡s ♦♥ t❤❡ ❦❡r♥❡• ❛s ✐♥ t❤❡♦r❡♠ ✸✳✹✳✶✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t ❈t❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ X s✉❝❤ t❤❛t µ X W ′ ≥ ❈t❡ H d (X) . Pr♦♦❢✳ ❚❤✐s ✐s ❡ss❡♥t✐❛••② ❛ s♣❡❝✐❛• ❝❛s❡ ♦❢ t❤❡♦r❡♠ ✸✳✹✳✶ ❢♦r r❡❝t✐✜❛❜•❡ ✈❛r✐❢♦•❞s✳ ■♥❞❡❡❞✱ t❤❡ s✉♣♣♦rt ♦❢ µ X ✐s ✐♥❝•✉❞❡❞ ✐♥ B(0, 1) × G d (E) ❛♥❞ t❤❛♥❦s t♦ t❤❡ t❤❡♦r❡♠✱ µ X W ′ ≥ ❈t❡ µ X (E × G d (E))✳ ■♥ ❛❞❞✐t✐♦♥✱ ❢r♦♠ t❤❡ ✈❡r② ❞❡✜♥✐t✐♦♥ ♦❢ µ X ✱ µ X (E × G d (E)) = H d (X)✳ ❚❤✐s r❡s✉•t ✐s t❤❡♦r❡t✐❝❛••② ❡ss❡♥t✐❛• ❜❡❝❛✉s❡ ✐t s❤♦✇s t❤❛t ♣❛t❤♦•♦❣✐❝❛• ❝❛s❡s ❛s t❤❡ ♦♥❡ ♦❢ ✜❣✉r❡ ✸✳✸ ❝❛♥♥♦t ❤❛♣♣❡♥ ✇❤❡♥ s❤❛♣❡s ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ✈❛r✐❢♦•❞s✳ ■♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✱ ✐t ❡♥s✉r❡s t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❦❡r♥❡• ♥♦r♠ ✇❡ ✉s❡ ✇✐t❤ t❤❡ ❛❝t✉❛• ✈♦•✉♠❡ ♦❢ t❤❡ s❤❛♣❡s s♦ t❤❛t ❛rt✐✜❝✐❛• ❡•✐♠✐♥❛t✐♦♥ ♦❢ ♠❛ss ❞✉r✐♥❣ r❡❣✐str❛t✐♦♥ ♣r♦❝❡ss ♦r t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❛r❡ ♥♦t •✐❦❡•② t♦ ♦❝❝✉r ✐♥ t❤✐s s❡tt✐♥❣✳ ❲❡ ✇✐•• s❤♦✇ ♠♦r❡ s♣❡❝✐✜❝ ❡①❛♠♣•❡s ♦❢ t❤✐s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ◆♦t❡ t❤❛t ✇❤❡♥ X ✐s ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ❛♥❞ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❜♦✉♥❞❛r② ♦❢ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥✱ t❤❡ ♥♦t✐♦♥ ♦❢ ✈♦•✉♠❡ ♦❢ ❛ s❤❛♣❡ X ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥❞ ♥♦t t❤❡ ✈♦•✉♠❡ ♦❢ t❤❡ ❞♦♠❛✐♥✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❝♦r♦••❛r② s❛②s t❤❛t ✇❤❡♥ ❛ ♣•❛✐♥ s❤❛♣❡ ✐s r❡♣r❡s❡♥t❡❞ ❜② ✐ts ❜♦✉♥❞❛r②✱ ✐ts ♥♦r♠ ❢♦r ❣❡♥❡r✐❝ ✈❛r✐❢♦•❞ ❦❡r♥❡• ✐s •❛r❣❡r t❤❛♥ t❤❡ ❛r❡❛ ♦❢ ✐ts ❜♦✉♥❞❛r②✳ ✶✶✶ ✸✳✹✳✸ ▼❡tr✐❝ ✈❛r✐❛t✐♦♥ ❢♦r♠✉•❛ ❆ •❛st ❢✉♥❞❛♠❡♥t❛• ✐ss✉❡ t♦ ♠❡♥t✐♦♥ ✐s t❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ❦❡r♥❡• ♥♦r♠s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ♦❢ s❤❛♣❡s✱ s✐♥❝❡ r❡❣✐str❛t✐♦♥ ❛♥❞ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s r❡•② ♦♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❣r❛❞✐❡♥ts ♦❢ s✉❝❤ ♥♦r♠s✳ ❲❡ s❤❛•• ❞❡t❛✐• t❤❡ t❡❝❤♥✐❝❛• ❞✐s❝r❡t❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ s✉❝❤ ❣r❛❞✐❡♥ts ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ❜✉t ✐t ✐s ❛•s♦ ✈❡r② ✈❛•✉❛❜•❡ t♦ ❤❛✈❡ ❛ t❤❡♦r❡t✐❝❛• ❛♥❞ ❣❡♥❡r❛• ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❤♦✇ t❤❡ ♠❡tr✐❝ ♦♥ s❤❛♣❡ ✈❛r✐❡s ✇❤❡♥ t❤❡ s✉♣♣♦rt ✐s ❞❡❢♦r♠❡❞✳ ◆❛♠❡•②✱ ✐❢ X ✐s ❛ s❤❛♣❡ ✭❤❡r❡ ✇❡ ✇✐•• ❛ss✉♠❡ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞✮ ❛♥❞ ❛ |❑❍❙ W ♦♥ ✈❛r✐❢♦•❞s ✐s s❡t✱ ✇❡ ✇❛♥t t♦ ❡①♣r❡ss t❤❡ ✈❛r✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ X ♦❢ t❡r♠s •✐❦❡ µ X , µ ′ W ′ ✇✐t❤ µ ′ ❛ ❝❡rt❛✐♥ ❡•❡♠❡♥t ♦❢ W ′ ✳ ❙✐♥❝❡ µ X , µ ′ W ′ = µ X (K W µ ′ ) ✭❝❢ s❡❝t✐♦♥ ✸✳✸✳✷✮✱ ✇❡ ❝❛♥ •♦♦❦ ❡q✉✐✈❛•❡♥t•② ❢♦r ✈❛r✐❛t✐♦♥s ♦❢ µ X (ω) ❢♦r ❛ ✜①❡❞ ω ∈ W ✳ ❲❤❛t ✇❡ ✇✐s❤ t♦ ❞❡r✐✈❡ ✐s t❤✉s ❛ r❡s✉•t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❝✉r✲ r❡♥ts✳ ❚❤❡ ♣r♦♣❡r ❢♦r♠✉•❛t✐♦♥ ✐s t♦ ❡①♣r❡ss t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ µ X (ω) ❢♦r s♠❛•• ✈❛r✐❛✲ t✐♦♥s ♦❢ X ♦❜t❛✐♥❡❞ ❜② ✢♦✇✐♥❣ ❛ ✈❡❝t♦r ✜❡•❞ ❢r♦♠ X✳ ❲✐t❤ s♦♠❡ ❤②♣♦t❤❡s❡s ♦♥ X ❛♥❞ W ✱ ✐t ✐s ♣♦ss✐❜•❡ t♦ ❞❡r✐✈❡ ❛ ❢♦r♠✉•❛ t❤❛t ❣❡♥❡r❛•✐③❡s ✐♥ ♦✉r ❝♦♥t❡①t t❤❡ ♥♦t✐♦♥ ✜rst ✈❛r✐❛t✐♦♥ ♦❢ ❛ ✈❛r✐❢♦•❞ st✉❞✐❡❞ ✐♥ ❬✹(✳ ❚❤❡ r❡s✉•t ✐s s✉♠♠❡❞ ✉♣ ❜❡•♦✇ ✿ ❚❤❡♦r❡♠ ✸✳✹✳✷✳ ▲❡t X ❜❡ ❛♥ ♦r✐❡♥t❛❜•❡ ❝♦♠♣❛❝t s✉❜♠❛♥✐❢♦•❞ ❛♥❞ µ X ✐ts ❛ss♦❝✐❛t❡❞ ✈❛r✐❢♦•❞✳ ▲❡t v ❜❡ ❛♥② C 1 ✈❡❝t♦r ✜❡•❞ ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt ❞❡✜♥❡❞ ♦♥ E ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• s✉❜❣r♦✉♣ t → φ t ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✢♦✇ ♦❢ v✳ ❚❤❡♥✱ ✐❢ X t . = φ t (X) ✐s t❤❡ tr❛♥s♣♦rt❡❞ ♠❛♥✐❢♦•❞✱ ✇❡ ❤❛✈❡ ❢♦r ❛♥② C 1 ❢✉♥❝t✐♦♥ (x, V ) → ω(x, V ) ♦♥ E × G d (E) ✿ d dt t=0 µ Xt (ω) = X ∂ω ∂x -div X ∂ω ∂V -ωH X |v ⊥ + ∂X ν, ωv ⊤ + ∂ω ∂V |v ⊥ ✇❤❡r❡ v ⊤ ❛♥❞ v ⊥ ❞❡♥♦t❡ t❤❡ t❛♥❣❡♥t✐❛• ❛♥❞ ♥♦r♠❛• ♣❛rt ♦❢ v ❛•♦♥❣ X✱ ν ✐s t❤❡ ✉♥✐t ♦✉t✇❛r❞ ♥♦r♠❛• ❛•♦♥❣ ∂X✱ ❛♥❞ H X t❤❡ ♠❡❛♥ ❝✉r✈❛t✉r❡ ✈❡❝t♦r t♦ X✳ Pr♦♦❢✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ♦❜•❡❝t✐✈❡ ✐s s✐♠✐•❛r t♦ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❝✉rr❡♥ts✱ t❤❡ ♣r♦♦❢ ✐s q✉✐t❡ ❞✐✛❡r❡♥t ✐♥ t❤❡ ❞❡t❛✐•s✳ ❲❡ ❛ss✉♠❡ t❤❛t X ✐s ❛ s♠♦♦t❤ ❝♦♠♣❛❝t ♦r✐❡♥t❛❜•❡ s✉❜♠❛♥✐❢♦•❞ ♦❢ E s♦ t❤❛t ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❝❛♥♦♥✐❝❛• ✈♦•✉♠❡ ❢♦r♠ σ X ♦♥ X✳ •✐✈❡♥ ❛ C 1 ✈❡❝t♦r ✜❡•❞ v ♦♥ E ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ 1✲ ♣❛r❛♠❡t❡r ❣r♦✉♣ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s φ t ✇✐t❤ φ 0 = Id ❛♥❞ ∂ t | t=0 φ t = v✳ ❚❤❡ ✇❤♦•❡ ♣♦✐♥t ♦❢ t❤❡ t❤❡♦r❡♠ ✐s t♦ ❝♦♠♣✉t❡ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ µ X (ω) ✇❤❡♥ X ✐s ❞❡❢♦r♠❡❞ ❜② ✈❡❝t♦r ✜❡•❞ v✱ t❤❛t ✐s t♦ s❛② ✿ d dt t=0 µ φt(X) (ω) = d dt t=0 Xt ω(x, T x X t )dH d (x) ✭✸✳✷✽✮ ✇❤❡r❡ X t = φ t (X)✳ ◆♦t❡ t❤❛t t❤✐s ❣❡♥❡r❛•✐③❡s t❤❡ ✜rst ✈❛r✐❛t✐♦♥ ♦❢ ❛ ✈❛r✐❢♦•❞ ❝♦♠✲ ♣✉t❡❞ ✐♥ ❬✹(✱ ❢♦r ✇❤✐❝❤ ω = 1✳ ❍❡r❡✱ ω ✐s ❛♥② C 1 ❢✉♥❝t✐♦♥ ♦♥ E × G d (E)✳ ❆s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✷✳✷✱ ✇❡ ❤❛✈❡ µ φt(X) (ω) = µ X (φ * t ω) ❛♥❞ ✿ d dt t=0 µ φt(X) (ω) = X d dt t=0 (φ * t ω)(x, T x X)dH d (x) = X £ v ω(x, T x X)dH d (x) ✭✸✳✷✾✮ ✇❤❡r❡ £ v ω st❛♥❞s ❢♦r t❤❡ ▲✐❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ω(x, V ) ❛•♦♥❣ v✳ ❍❡r❡❛❢t❡r✱ t♦ s✐♠♣•✐❢② t❤❡ ♥♦t❛t✐♦♥✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ (x, V ) → f (x, V )✱ ✇❡ ✇✐•• s✐♠♣•② ✇r✐t❡ X f ✐♥st❡❛❞ ♦❢ X f (x, T x X)dH d (x)✳ ◆♦✇✱ (φ * t ω)(x, T x X) = |d x φ t .T x X|ω(φ t (x), d x φ t (T x X))✳ ▲❡t ✉s r❡❝❛•• t❤❛t J t = |d x φ t .T x X| ✐s t❤❡ ✈♦•✉♠❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T x X✳ ❚❛❦✐♥❣ ✶✶✷ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛• ❛t t = 0 •❡❛❞s t♦ t❤r❡❡ t❡r♠s ✿ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❢✉♥❝t✐♦♥ ω ✇✐t❤ r❡s♣❡❝t t♦ ♣♦s✐t✐♦♥✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ✈♦•✉♠❡ ❝❤❛♥❣❡✳ ❚❤❡ ✜rst t❡r♠ ✐s t❤❡ s✐♠♣•❡st ♦♥❡ ❛♥❞ ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♠♣✉t❛t✐♦♥ s❤♦✇s t❤❛t ✐t ❡q✉❛•s ∂ω ∂x |v ✳ ❚❤❡ t✇♦ ♦t❤❡rs ❛r❡ ♠♦r❡ ✐♥✈♦•✈❡❞✳ ❉❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✈♦•✉♠❡ ❝❤❛♥❣❡ ✿ ❋♦r ❛♥② ✈❡❝t♦r ✜❡•❞ u ❞❡✜♥❡❞ ♦♥ X✱ ✇❡ s❤❛•• ❞❡♥♦t❡ ❜② u ⊤ ❛♥❞ u ⊥ t❤❡ t❛♥❣❡♥t✐❛• ❛♥❞ ♥♦r♠❛• ❝♦♠♣♦♥❡♥ts ♦❢ u ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ♦❢ X ❛t ❡❛❝❤ ♣♦✐♥t✳ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥♥❡❝t✐♦♥ ∇ • • ♦♥ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡ ❛♥❞ ❛♥ ♦rt❤♦♥♦r♠❛• ❢r❛♠❡ ♦❢ t❛♥❣❡♥t ✈❡❝t♦r ✜❡•❞s (e i ) i=1,..,d ♦♥ X✳ ◆♦✇ J t = det( d x φ t (e i ), d x φ t (e j )) i,j s♦ ❛ s✐♠♣•❡ ❝❛•❝✉•❛t✐♦♥ s❤♦✇s t❤❛t ✿ d dt t=0 J t = d i=1 e i , ∇ e i v ❲r✐t✐♥❣ v = v ⊤ + v ⊥ ♣r♦✈✐❞❡s ❛ ✜rst t❡r♠ d i=1 e i , ∇ e i v ⊤ ✇❤✐❝❤ ✐s t❤❡ t❛♥❣❡♥t✐❛• ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✈❡❝t♦r ✜❡•❞ v ⊤ ❞❡♥♦t❡❞ ✉s✉❛••② div X (v ⊤ )✳ ❚❤❡ s❡❝♦♥❞ t❡r♠ ❜❡❝♦♠❡s d i=1 e i , ∇ e i v ⊥ ✳ ❋♦r ❛•• i = 1, .., d✱ ✇❡ ❤❛✈❡ e i , v ⊥ = 0 s♦ t❤❛t ❛❢t❡r ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❡ ✜♥❞ t❤❛t e i , ∇ e i v ⊥ = -∇ e i e i , v ⊥ ✳ ❚❤❡r❡❢♦r❡ ✿ d i=1 e i , ∇ e i v ⊥ = - d i=1 ∇ e i e i , v ⊥ = - d i=1 ∇ e i e i ⊥ , v ⊥ ■♥ t❤✐s •❛st ❡①♣r❡ss✐♦♥✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ ♠❡❛♥ ❝✉r✈❛t✉r❡ ✈❡❝t♦r t♦ t❤❡ s✉❜♠❛♥✐❢♦•❞ X✱ ✇❤✐❝❤ ✐s t❤❡ tr❛❝❡ ♦❢ t❤❡ ❲❡✐♥❣❛rt❡♥ ♠❛♣ ❛♥❞ ✐s ❞❡♥♦t❡❞ H X ✳ ❆s ❛ r❡s✉•t✱ ✇❡ ✜♥❞ t❤❛t ✿ X ω d dt t=0 J t = X ω div X (v ⊤ ) - X ω H X , v ⊥ ◆♦✇✱ ✇❡ ✇✐•• s❤♦✇ t❤❛t t❤❡ ✜rst t❡r♠ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❛ ❜♦✉♥❞❛r② ✐♥t❡❣r❛•✳ ■♥❞❡❡❞✱ ✐❢ ✇❡ ❞❡♥♦t❡ ❜② ω t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ X ❜② ω(x) = ω(x, T x X)✱ ✇❡ ❤❛✈❡ div X (ωv ⊤ ) = ω div X (v ⊤ ) + ∇ω|v ⊤ ✳ ❆♣♣•②✐♥❣ t❤❡ ❞✐✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ♦♥ t❤❡ ♦r✐❡♥t❛❜•❡ ♠❛♥✐❢♦•❞ X ❣✐✈❡s ✿ X ω div X (v ⊤ ) = - X ∇ω|v ⊤ + ∂X ω ν, v ⊤ ✇❤❡r❡ ν ✐s t❤❡ ✉♥✐t ♦✉t✇❛r❞ ♥♦r♠❛• t♦ t❤❡ ❜♦✉♥❞❛r②✳ ❉❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ t❛♥❣❡♥t s♣❛❝❡s ✿ ❲❡ ♥♦✇ ❝♦♠❡ t♦ t❤❡ •❛st t❡r♠ ✐♥ ❡q✉❛t✐♦♥ ✭✸✳✷✾✮✱ ✇❤✐❝❤ ✐s t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ω ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ✐♥ G d (E)✳ ❆s ❡①♣•❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✷✳✶✱ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② t❤❡ t❛♥❣❡♥t s♣❛❝❡ t♦ G d (E) ❛t V ✇✐t❤ t❤❡ s♣❛❝❡ ♦❢ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥s L(V, V ⊥ )✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐❢ ✇❡ s❡t V t = d x φ t (T x X)✱ ✇❡ ❤❛✈❡✱ t❤❛♥❦s t♦ ❡q✳✭✸✳✸✮ ✿ d dt t=0 V t = p TxX ⊥ • ∇v| TxX ∈ L(T x X, (T x X) ⊥ ) ✇❤✐❝❤ ✇❡ ✇✐•• ♥♦t❡ ♠♦r❡ ❝♦♥❝✐s❡•② d dt t=0 V t = ∇ ⊥ v✳ ❚❤❡ s♣❛❝❡ L(T x X, (T x X) ⊥ ) ❜❡✐♥❣ tr✐✈✐❛••② ✐s♦♠♦r♣❤✐❝ t♦ T x X ⊥ ⊗ (T x X) * ✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r ∇ ⊥ v ❜❡✐♥❣ ❛♥ ❡•❡♠❡♥t ✶✶✸ ♦❢ t❤❛t s♣❛❝❡ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ ∂ω ∂V ✇❤✐❝❤ ✇❡ t❤❡r❡❢♦r❡ ✐❞❡♥t✐❢② t♦ ❛♥ ❡•❡♠❡♥t ♦❢ (T x X ⊥ ) * ⊗ T x X ✿ ∂ω ∂V = n j=d+1 η * j ⊗ α j ❢♦r (η d+1 , .., η n ) ❛♥ ♦rt❤♦♥♦r♠❛• ❢r❛♠❡ ♦❢ T x X ⊥ ❛♥❞ (α j ) ✈❡❝t♦rs ♦❢ T x X ✭❛s ✉s✉❛• η * ❞❡♥♦t❡s t❤❡ •✐♥❡❛r ❢♦r♠ η, . ✮✳ ❚❤❡♥ t❤❡ ✈❛r✐❛t✐♦♥ ✇❡ ✇✐s❤ t♦ ❝♦♠♣✉t❡ ✐s ✿ ∂ω ∂V |∇ ⊥ v = ∂ω ∂V |∇v = n j=d+1 η j , ∇ α j v ■❢ ✇❡ ✐♥tr♦❞✉❝❡ ∂ω ∂V |v = n j=d+1 η * j (v)α j = n j=d+1 η j , v α j ✇❤✐❝❤ ✐s ❛ t❛♥❣❡♥t ✈❡❝t♦r ✜❡•❞ ♦♥ X✱ ✇❡ ❤❛✈❡ ✿ div X ∂ω ∂V |v = d i=1 n j=d+1 ( e i , ∇ e i α j η j , v + e i , ∇ e i η j , v α j + e i , η j , ∇ e i v α j ) ❚❤❡ •❛st t❡r♠ ✐♥ t❤❡ s✉♠ ✐s ❛•s♦ n j=d+1 η j , ∇ α j v ✱ ✇❤✐❝❤ ✐s ♥♦t❤✐♥❣ ❡•s❡ t❤❛♥ ∂ω ∂V |∇v ✳ ❆s ❢♦r t❤❡ t✇♦ ♦t❤❡r t❡r♠s ✐♥ t❤❡ s✉♠✱ ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t ✐t ❡q✉❛•s ✿   d i=1 e i , ∇ e i n j=d+1 η * j ⊗ α j |v   = (div X ∂ω ∂V |v) ❙♦ ❣❡t ❡✈❡♥t✉❛••② t❤❛t ✿ ∂ω ∂V |∇v = div X ∂ω ∂V |v -(div X ∂ω ∂V |v) ✭✸✳✸✵✮ ■♥t❡❣r❛t✐♥❣ ❡q✉❛t✐♦♥ ✭✸✳✸✵✮ ♦✈❡r t❤❡ s✉❜♠❛♥✐❢♦•❞ X ❛♥❞ ✉s✐♥❣ ❛s ♣r❡✈✐♦✉s•② t❤❡ ❞✐✈❡r❣❡♥❝❡ t❤❡♦r❡♠✱ ✇❡ ✜♥❞ t❤❛t ✿ X ∂ω ∂V |∇v = ∂X ν, ∂ω ∂V |v - X (div X ∂ω ∂V |v) ✭✸✳✸✶✮ ❙②♥t❤❡s✐s ✿ •r♦✉♣✐♥❣ ❛•• t❤❡ ❞✐✛❡r❡♥t t❡r♠s ♦❜t❛✐♥❡❞ s♦ ❢❛r✱ ✇❡ ❣❡t t❤❡ ❢♦••♦✇✐♥❣ ✿ X £ v ω = X ∂ω ∂x -div X ∂ω ∂V |v - X ∇ω|v ⊤ + ω H X |v ⊥ + ∂X ν, ∂ω ∂V |v + ωv ⊤ ❲❡ r❡♠✐♥❞ t❤❛t ω(x) = ω(x, T x X) s♦ (∇ω|v ⊤ ) = ∂ω ∂x |v ⊤ + ∂ω ∂V |∇v ⊤ ❛♥❞ ❛♣♣•②✐♥❣ t❤❡ r❡s✉•t ♦❢ ❡q✉❛t✐♦♥ ✭✸✳✸✵✮ t♦ v ⊤ ✿ ∂ω ∂V |∇v ⊤ = div X ∂ω ∂V |v ⊤ -(div X ∂ω ∂V |v ⊤ ) ◆♦t✐❝✐♥❣ t❤❛t ∂ω ∂V |v ⊤ = 0 ❜② t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ ∂ω ∂V ❛♥❞ ✉s✐♥❣ t❤❡ ❡q✉❛•✐t② v = v ⊤ + v ⊥ ✇❡ ✜♥❞ ❡✈❡♥t✉❛••② t❤❛t ✿ X £ v ω = X ∂ω ∂x -div X ∂ω ∂V -ωH X |v ⊥ + ∂X ν, ∂ω ∂V |v + ωv ⊤ ✇❤✐❝❤ ♣r♦✈❡s t❤❡ r❡s✉•t ♦❢ t❤❡♦r❡♠ ✸✳✹✳✷✳ ✶✶✹ ❲❡ ✇✐•• •✉st ♠❛❦❡ ❛ ❢❡✇ q✉❛•✐t❛t✐✈❡ ❝♦♠♠❡♥ts s✐♥❝❡ t❤❡r❡ ❛r❡ ❛ ❢❡✇ ♥♦t❡✇♦rt❤② ❝♦♥s❡✲ q✉❡♥❝❡s ♦❢ t❤❡♦r❡♠ ✸✳✹✳✷ t♦ ♠❡♥t✐♦♥✳ ❚❤❡ ✜rst ✐♠♣♦rt❛♥t r❡♠❛r❦ ✐s t❤❛t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ✈❛r✐❢♦•❞ ♠❡tr✐❝ ✐s ❝♦♥tr♦••❡❞ ♦♥•② ❜② t❤❡ ✈❡❝t♦r ✜❡•❞ v ❛♥❞ ♥♦t ✐ts ❞❡r✐✈❛t✐✈❡s✳ ❚❤✐s ✐s ♥♦t str❛✐❣❤t❢♦r✇❛r❞ s✐♥❝❡ ❛ ✈❛r✐❢♦•❞ ❦❡r♥❡• k((x, V ), (x ′ , V ′ )) ✐s ❡♥❝♦❞✐♥❣ ✐♥ ❛♥ ❛r❜✐✲ tr❛r② ♥♦♥✲•✐♥❡❛r ✇❛② ✜rst ♦r❞❡r ✐♥❢♦r♠❛t✐♦♥ t❤r♦✉❣❤ t❤❡ ✐♥❝•✉s✐♦♥ ♦❢ •r❛ss♠❛♥♥✐❛♥ ✈❛r✐❛❜•❡s V ❛♥❞ V ′ ✳ ■♥t❡r❡st✐♥❣•②✱ t❤❡ r❡s✉•t ✐s ✈❛•✐❞ ❡✈❡♥ ✐❢ t❤❡ ❞✐♠❡♥s✐♦♥ ♦r t❤❡ ❝♦❞✐♠❡♥s✐♦♥ ♦❢ X ✐s ❞✐✛❡r❡♥t ❢r♦♠ 1✳ ■♥ r❡t✉r♥✱ ✇❡ s❡❡ t❤❛t t❤❡ ❢♦r♠✉•❛ ✐♥✈♦•✈❡s s♦♠❡ t❡r♠s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ X✱ ♦♥❡ ♦❢ t❤❡♠ ❡①♣r❡ss✐♥❣ t❤❡ t❛♥❣❡♥t✐❛• ❡①t❡♥s✐♦♥ ♦❢ X ❛•♦♥❣ ✐ts ❜♦✉♥❞❛r② ❛♥❞ ❛ s❡❝♦♥❞ ♦♥❡ r❡•❛t❡❞ t♦ t❤❡ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥ ♦♥ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ♥♦r♠❛• ❞✐r❡❝t✐♦♥ v ⊥ ✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡s❡ ❜♦✉♥❞❛r② t❡r♠s ✇✐•• ❛♣♣❡❛r ♦❢ ♣❛rt✐❝✉•❛r ✐♠♣♦rt❛♥❝❡ ✐♥ ❝❤❛♣t❡r ✹✳ ■♥ t❤❡ ✐♥t❡r✐♦r ♦❢ X✱ ✇❡ s❡❡ t❤❛t t❤❡ ✈❛r✐❛t✐♦♥ ❞❡♣❡♥❞s ♦♥•② ♦♥ t❤❡ ♦rt❤♦❣♦♥❛• ❝♦♠♣♦♥❡♥t v ⊥ ♦❢ v✱ ✇❤✐❝❤✱ ✐♥ ♦t❤❡r t❡r♠s✱ s❤♦✇s t❤❛t t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ✐s ♦rt❤♦❣✲ ♦♥❛• t♦ t❤❡ s❤❛♣❡✳ ◆♦t❡ ❛•s♦ t❤❛t ✇❡ ❤❛✈❡ ❛ss✉♠❡❞ X ✐s ♦r✐❡♥t❡❞ ❡ss❡♥t✐❛••② ❜❡❝❛✉s❡ t❤❡ ♣r♦♦❢ ♠❛❦❡s ✉s❡ ♦❢ t❤❡ ❞✐✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ✭t❤❛t r❡•✐❡s ♦♥ ❙t♦❦❡s t❤❡♦r❡♠✮✳ (❡t✱ s✐♥❝❡ ✇❡ ✇♦r❦ ✇✐t❤ ✈❛r✐❢♦•❞s✱ t❤❡ ✐♥t❡❣r❛• ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ X ❛♥❞ s♦ ♦♥❡ ❝♦✉•❞ ♣r❡s✉♠❛❜•② r❡♠♦✈❡ t❤✐s ❤②♣♦t❤❡s✐s ❜② •✉st ❝♦♥s✐❞❡r✐♥❣ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ❛♥❞ •♦❝❛• ♦r✐❡♥t❛t✐♦♥s✳ ✸✳✺ ▲❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠ ❢♦r ✉♥♦r✐✲ ❡♥t❡❞ s❤❛♣❡s ❆t t❤✐s ♣♦✐♥t✱ ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ ❛ t❤❡♦r❡t✐❝❛• ❜❛❝❦❣r♦✉♥❞ t♦ r❡♣r❡s❡♥t ❛♥❞ ❝♦♠♣❛r❡ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s t❤r♦✉❣❤ ✈❛r✐❢♦•❞s ❛♥❞ ❦❡r♥❡•s ♦♥ ✈❛r✐❢♦•❞s✳ ❲❤❛t ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ✐s ❛ ❞✐st❛♥❝❡ ✭❛❝t✉❛••② ❛ ✇❤♦•❡ ❝•❛ss ♦❢ ❞✐st❛♥❝❡s ♣r♦✈✐❞❡❞ ❜② ❞✐✛❡r❡♥t ❦❡r♥❡•s✮ ❜❡✲ t✇❡❡♥ t❤❡ ♦❜•❡❝ts t❤❛t ❝❛♥ s❡r✈❡ ❛s ❛♥ ❛tt❛❝❤♠❡♥t t❡r♠ ✐♥ ♣r❛❝t✐❝❛••② ❛♥② ♠❛t❝❤✐♥❣ ♣r♦❝❡ss✳ ❆s ✇❡ ❤❛✈❡ ❞♦♥❡ s♦ ❢❛r ✐♥ t❤✐s t❤❡s✐s✱ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ▲❉❉▼▼ ♠♦❞❡• ❢♦r •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥s✱ ✇❤✐❝❤ ✇❛s ❛•r❡❛❞② ❡①♣♦s❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ✐s t♦ ❞❡r✐✈❡ t❤❡ ❡q✉❛t✐♦♥s ♥❡❡❞❡❞ ❢♦r t❤❡ ♥✉♠❡r✐❝❛• ✐♠♣•❡✲ ♠❡♥t❛t✐♦♥ ♦❢ ▲❉❉▼▼ ♦♥ ✈❛r✐❢♦•❞s ❛♥❞ ♣r❡s❡♥t ❛ ❢❡✇ ❡①❛♠♣•❡s ♦❢ ♠❛t❝❤✐♥❣s ❜❛s❡❞ ♦♥ t❤✐s ❢r❛♠❡✇♦r❦✳ ■♥ ❛•• t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ♥♦✇ r❡str✐❝t ♦✉rs❡•✈❡s t♦ t❤❡ ✉s✉❛• ❝❛s❡s ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s •✐✈✐♥❣ ✐♥ t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛• ❡✉❝•✐❞❡❛♥ s♣❛❝❡ R 3 ✳ ✸✳✺✳✶ ❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ❲❡ ❤❛✈❡ ❛•r❡❛❞② ♣r❡s❡♥t❡❞ ▲❉❉▼▼ r❡❣✐str❛t✐♦♥ ❜♦t❤ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝✉rr❡♥ts ✐♥ ❝❤❛♣t❡r ✶ ❛♥❞ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥ ❝❤❛♣t❡r ✷✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s✱ t❤❡ ♣r✐♥❝✐♣•❡s ❛r❡ ❜❛s✐❝❛••② t❤❡ s❛♠❡✳ ❲❡ ❛❣❛✐♥ ❝♦♥s✐❞❡r ❛ |❑❍❙ s♣❛❝❡ V ♦❢ ❛❞♠✐ss✐❜•❡ ✈❡❝t♦r ✜❡•❞s ♦♥ E ❛♥❞ ✢♦✇s ♦❢ L 2 t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞s v ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ r✐❣❤t✲✐♥✈❛r✐❛♥t ♠❡tr✐❝ 1 0 |v t | 2 V dt✳ ■♥ ♦✉r s✐t✉❛t✐♦♥s✱ ✇❡ s❤❛•• ❢♦❝✉s ♦♥ |❑❍❙ V t❤❛t ❛r❡ ❣❡♥❡r❛t❡❞ ❜② •❛✉ss✐❛♥ ❦❡r♥❡•s ♦❢ ❝❡rt❛✐♥ s❝❛•❡ σ V ♦r ❜② s✉♠s ♦❢ s✉❝❤ ❦❡r♥❡•s✱ ❛s ❡①♣•❛✐♥❡❞ ✐♥ ♠♦r❡ ❞❡t❛✐• ✐♥ ❝❤❛♣t❡r ✶✳ ❚❤❡ ❣❡♦❞❡s✐❝s ❢♦r t❤❛t ♠❡tr✐❝ ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡✐r ✐♥✐t✐❛• ♠♦♠❡♥t✉♠ p 0 ∈ V * ✳ ❚❤❡♥✱ ❝♦♥s✐❞❡r✐♥❣ t✇♦ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s X ✭t❤❡ s♦✉r❝❡✮ ❛♥❞ Y ✭t❤❡ t❛r❣❡t✮ ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ µ X ❛♥❞ µ Y ✐♥ ❛ |❑❍❙ ♦❢ ✈❛r✐❢♦•❞s W ′ ✱ ✇❡ ❝❛♥ ❢♦r♠✉•❛t❡ s✐♠✐•❛r•② t❤❡ r❡❣✐str❛t✐♦♥ ♣r♦❜•❡♠ ❜❡t✇❡❡♥ X ❛♥❞ Y ❛s t❤❡ ✶✶✺ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ✿      minp 0 J(p 0 ) . = E(p 0 ) + γ µ φ 1 (X) -µ Y 2 W ′ ∂φ t ∂t = v t • φ t ✭✸✳✸✷✮ ❯♣ t♦ t❤❡ ❝❤❛♥❣❡ ♦♥ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ ✇❤✐❝❤ ✐s ♥♦✇ ❛ ✈❛r✐❢♦•❞ ❦❡r♥❡• ❞✐st❛♥❝❡✱ t❤✐s ♣r♦❜•❡♠ ❤❛s ❛♥ ✐❞❡♥t✐❝❛• ❢♦r♠✉•❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ r❡❣✐str❛t✐♦♥ ♦♥ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✭❝❢ ✐♥tr♦❞✉❝t✐♦♥ s❡❝t✐♦♥ ✶✳✸✳✷✮✳ ❲✐t❤ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ J ✇✐t❤ r❡s♣❡❝t t♦ p 0 ✐s t❤❡r❡❢♦r❡ ♦❜t❛✐♥❡❞ ❛❣❛✐♥ ❜② ✜rst ✢♦✇✐♥❣ t❤❡ ❢♦r✇❛r❞ ❍❛♠✐•t♦♥✐❛♥ ❡q✉❛t✐♦♥ ✇❤✐❝❤ ❣✐✈❡s t❤❡ ♣♦✐♥t ♣♦s✐t✐♦♥s q t ❛♥❞ t❤❡ ♠♦♠❡♥t✉♠ p t ❛t ❛•• t✐♠❡s t ∈ [0, 1] ❛♥❞ t❤❡♥ ❜② ✿ ∇ p 0 J = K V (q, q)p 0 -P 0 ✇✐t❤ P 0 ❛♥ ❛❞•♦✐♥t ✈❛r✐❛❜•❡ ♦❜t❛✐♥❡❞ ❜② ✢♦✇✐♥❣ ❜❛❝❦✇❛r❞ ✐♥ t✐♠❡ t❤❡ ❝♦✉♣•❡❞ ❡q✉❛✲ t✐♦♥s ✿    Qt = -(∂ 2 q,p H r (q t , p t )) * Q t + (∂ 2 q,q H r (q t , p t )) * P t Ṗt = -(∂ 2 p,p H r (q t , p t )) * Q t + (∂ 2 q,p H r (q t , p t )) * P t Q 1 = -∇ q 1 A(q 1 ), P 1 = 0 ✭✸✳✸✸✮ ✇❤❡r❡ ∇ q 1 A(q 1 ) ✐s ❜❛s✐❝❛••② t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞❡❢♦r♠❡❞ s♦✉r❝❡ s❤❛♣❡ φ 1 (X) ❛♥❞ t❤❡ t❛r❣❡t Y ✳ ❚❤❡ ♦♥•② ❝❤❛♥❣❡ ✇✐t❤ r❡s♣❡❝t t♦ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠ ♦♥ ❝✉rr❡♥ts ✐s t❤❡ ❛tt❛❝❤♠❡♥t q✉❛♥t✐t② A(q 1 ) ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ t♦t❛• ❡♥❡r❣② ❛♥❞ ✐ts ❣r❛❞✐❡♥t ∇A(q 1 ) ❢♦r t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ P ✐♥ t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥✳ ❲❡ ❞❡t❛✐• t❤♦s❡ ❝♦♠♣✉t❛t✐♦♥s ❢♦r t❤❡ ❝❛s❡ ♦❢ ❡♠❜❡❞❞❡❞ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✐♥ R 3 ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✳ ❚❤❡ r❡st ♦❢ t❤❡ ❛•❣♦r✐t❤♠✱ s✐♠✐•❛r•② t♦ ▲❉❉▼▼ ❢♦r •❛♥❞♠❛r❦s ♦r ❝✉rr❡♥ts✱ ✐s ❛♥ ❛❞❛♣t✐✈❡ st❡♣ ❣r❛❞✐❡♥t ❞❡s❝❡♥t s❝❤❡♠❡ ♦♥ p 0 ✳ ✸✳✺✳✷ ❉✐s❝r❡t❡ ❡①♣r❡ss✐♦♥s ❢♦r ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ▲❡t✬s ♥♦✇ ❢♦❝✉s ♦♥ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ t❡r♠ A(q 1 )✳ ■♥ ♦✉r ❝♦♥t❡①t ♦❢ ✉♥♦r✐✲ ❡♥t❡❞ s❤❛♣❡s ♠♦❞❡•❡❞ ❛s ✈❛r✐❢♦•❞s✱ ✇❡ ✉s❡ ❞✐st❛♥❝❡s ♣r♦✈✐❞❡❞ ❜② ❦❡r♥❡•s ❛♥❞ t❤❡✐r ❛ss♦❝✐❛t❡❞ |❑❍❙ ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✸✳ ❚❤♦s❡ ❛r❡ t❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ❛ ❦❡r♥❡• k e ♦♥ t❤❡ s♣❛❝❡ R 3 ❛♥❞ ❛ ❦❡r♥❡• k t ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ■❢ W ✐s t❤❡ |❑❍❙ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ k = k e ⊗ k t ✱ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ✇❡ ❤❛✈❡ ✐s t❤❡♥ ✿ A = µ φ 1 (X) -µ Y 2 W ′ ✭✸✳✸✹✮ ❙✐♥❝❡ φ 1 (X) ✐s t❤❡ tr❛♥s♣♦rt❡❞ s♦✉r❝❡ ♦❜•❡❝t ❜② t❤❡ ✈❡❝t♦r ✜❡•❞ ❛t t✐♠❡ 1 ✭r❡♣r❡✲ s❡♥t❡❞ ❜② t❤❡ s❡t ♦❢ ♣♦✐♥ts q 1 ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ♥♦t❛t✐♦♥s✮✱ ❢♦r♠❛••② t❤❡ ♣r♦❜•❡♠ r❡❞✉❝❡s t♦ ❡①♣r❡ss ❡①♣•✐❝✐t•② q✉❛♥t✐t✐❡s •✐❦❡ µ X -µ Y 2 W * ✇❤❡r❡ X ❛♥❞ Y ❛r❡ t✇♦ s❤❛♣❡s ♦❢ t❤❡ s❛♠❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ s✉❝❤ t❡r♠s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣♦✐♥ts ♦❢ X✳ ❲❡ ❛ss✉♠❡ ♥♦✇ t❤❛t X ❛♥❞ Y ❛r❡ t✇♦ ✉♥♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• s❤❛♣❡s ✭♣♦ss✐❜•② ❞✐s❝♦♥♥❡❝t❡❞✮ ❣✐✈❡♥ r❡s♣❡❝t✐✈❡•② ❛s s❡ts ♦❢ ✈❡rt❡① (q k ) k=1,..,N ✱ (y l ) l=1,..,M t♦❣❡t❤❡r ✇✐t❤ s❡ts ♦❢ ✉♥♦r✐❡♥t❡❞ s✐♠♣•❡①❡s (f 1 i , .., f d i ) i=1,..,n ❛♥❞ (g 1 j , .., g d j ) j=1,..,m ✳ ❚❤❡ ❝♦♠✲ ♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛r✐❢♦•❞ r❡♣r❡s❡♥t❛t✐♦♥s µ X ❛♥❞ µ Y ♦❢ ❡❛❝❤ s❤❛♣❡ ✇❛s ❡①♣•❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✷✳✷ ❛♥❞ ♣r♦✈✐❞❡❞ ❜② ❡q✉❛t✐♦♥s ✭✸✳✻✮ ❢♦r ❝✉r✈❡s✱ ✭✸✳✼✮ ❢♦r s✉r❢❛❝❡s✳ ❲❡ ✶✶✻ ✇r✐t❡ µ X = n i=1 l i .δ (p i ,U i ) ❛♥❞ µ Y = m j=1 λ j .δ (q j ,V j ) ✳ ❚❤❡♥ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❜❡❝♦♠❡s ✿ A = µ X 2 W * -2 µ X , µ Y W * + µ Y 2 W * = i,j=1..n l i l j k e (p i , p j ).k t (U i , U j ) -2 i=1..n j=1..m l i λ j k e (p i , q j ).k t (U i , V j ) + i,j=1..n λ i λ j k e (q i , q j ).k t (V i , V j ) ✭✸✳✸✺✮ ❚❤✉s✱ ❝♦♠♣✉t✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ X ❛♥❞ Y ❝♦♥s✐sts ❜❛s✐❝❛••② ✐♥ ❝♦♠♣✉t✐♥❣ t❤❡ ✈❛r✐❢♦•❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s❤❛♣❡s ❢r♦♠ t❤❡✐r s❡ts ♦❢ ♣♦✐♥ts ❛♥❞ ♠❡s❤❡s ❛♥❞ ❝♦♠♣✉t❡ ❦❡r♥❡• ❡✈❛•✉❛t✐♦♥s ❜❡t✇❡❡♥ ♣♦✐♥t ♣♦s✐t✐♦♥s ❛♥❞ t❛♥❣❡♥ts✳ ■t✬s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t✱ ✐♥ t❤❡ ❝❛s❡s ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❡r♠s •✐❦❡ k t (U, V ) ✐s ❛❝t✉❛••② ♠✉❝❤ s✐♠♣•✐✜❡❞ ❛♥❞ ❞♦❡s ♥♦t r❡q✉✐r❡ t♦ ❣♦ t❤r♦✉❣❤ ♣r♦•❡❝t✐♦♥ ♠❛tr✐❝❡s✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❞✐♠❡♥s✐♦♥ ♦r ❝♦❞✐♠❡♥s✐♦♥ ✐s ❛•✇❛②s ♦♥❡ ❛♥❞ s♦ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❦❡r♥❡•s ✇✐t❤ t❤❡ ♣r✐♥❝✐♣❛• ❛♥❣•❡s ✐♥ ✸✳✸✳✶ s❤♦✇ t❤❛t ♦♥❡ s✐♠♣•② ❤❛s ✿ k P (U, V ) = ξ U , ξ V 2k k G (U, V ) = e -2 σ 2 (1-ξ U ,ξ V 2 ) ✇❤❡r❡ ξ U ✐s t❤❡ ✉♥✐t t❛♥❣❡♥t ✈❡❝t♦r t♦ t❤❡ s❡❣♠❡♥t ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝✉r✈❡s ♦r t❤❡ ✉♥✐t ♥♦r♠❛• t♦ t❤❡ tr✐❛♥❣•❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ s✉r❢❛❝❡s✱ ♥♦ ♠❛tt❡r ✇❤❛t ♦r✐❡♥t❛t✐♦♥ t♦ ξ U ✐s ❣✐✈❡♥✳ ❚❤❡ ♠♦r❡ t❡❝❤♥✐❝❛• ♣❛rt ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞✐s✲ t❛♥❝❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ (x k )✬s✱ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ ✜rst s❤❛♣❡✳ ■♥ ❡q✉❛t✐♦♥ ✭✸✳✸✺✮✱ t❤❡ ❛tt❛❝❤♠❡♥t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ p i ✱ l i ❛♥❞ U i ❜✉t ❛•• t❤❡s❡ t❡r♠s ❛r❡ t❤❡♠s❡•✈❡s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ (x k )✬s✱ r❡s♣❡❝t✐✈❡•② ❛s t❤❡ ❝❡♥t❡rs✱ ✈♦•✉♠❡s ❛♥❞ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥s t♦ ❡✈❡r② ❝❡•• ♦❢ X✳ ❚❤❡ r❡s✉•t✐♥❣ ❢✉♥❝t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ ❞❡♥♦t❡❞ A((x k )) ❝❛♥ ❜❡ t❤❡r❡❢♦r❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥ ✿ ∂ x k A = n i=1 (∂ p i A • ∂ x k p i + ∂ l i A • ∂ x k l i + ∂ U i A • ∂ x k U i ) ✭✸✳✸✻✮ ❚❤❡ ❞✐✛❡r❡♥t✐❛•s ♦❢ A ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ p i ✱ l i ❛♥❞ U i ❛r❡ ❡❛s✐•② ❡①♣r❡ss❡❞ ❢r♦♠ ❡q✳✭✸✳✸✺✮✱ ✐♥✈♦•✈✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛•s ♦❢ t❤❡ ❦❡r♥❡•s k e ❛♥❞ k t ✳                          ∂ p i A = n j=1 l i l j [∂ 1 k e (p i , p j ) + ∂ 2 k e (p j , p i )].k t (U i , U j ) -2 m j=1 l i λ j ∂ 1 k e (p i , q j ).k t (U i , V j ) ∂ U i A = n j=1 l i l j k e (p i , p j ).[∂ 1 k t (U i , U j ) + ∂ 2 k e (U j , U i )] -2 m j=1 l i λ j k e (p i , q j ).∂ 1 k t (U i , V j ) ∂ l i A = 2 n j=1 l j k e (p i , p j ).k t (U i , U j ) -2 m j=1 λ j k e (p i , q j ).k t (U i , V j ) ✭✸✳✸✼✮ ✇❤❡r❡ ∂ 1 ❛♥❞ ∂ 2 ❞❡♥♦t❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❛r❣✉♠❡♥t✳ ✶✶✼ ◆♦✇✱ t❤❡ •❛st st❡♣ ❝♦♥s✐sts ✐♥ ❝♦♠♣✉t✐♥❣ ❞❡r✐✈❛t✐✈❡s ♦❢ p i ✱ l i ❛♥❞ U i ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣♦✐♥ts ♦❢ X✳ ❚❤❡s❡ ❞❡♣❡♥❞ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❝♦❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s❤❛♣❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ❡♠❜❡❞❞❡❞ ✐♥ R 3 ✱ ✐t ❝❛♥ ❜❡ s✐♠♣•② ❞♦♥❡ ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ ❡q✉❛t✐♦♥s ✭✸✳✻✮ ❢♦r ❝✉r✈❡s ♦r ❡q✉❛t✐♦♥s ✭✸✳✼✮ ❢♦r s✉r❢❛❝❡s✳ ❲❡ ❣✐✈❡ t❤❡ ❞❡t❛✐•s ❢♦r t❤❡s❡ t✇♦ ❝❛s❡s ❜❡•♦✇✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❝✉r✈❡s✱ ✇❡ ❤❛✈❡ p i = (x f 1 i +x f 2 i ) 2 ✱ l i = |x f 2 i -x f 1 i |✳ ❚❤❡ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ s✐♥❣•❡ ♥♦r♠❛•✐③❡❞ ✈❡❝t♦r u i = x f 2 i -x f 1 i |x f 2 i -x f 1 i | ✱ ✇❤✐❝❤ ❣✐✈❡s ❢♦r ❛•• i ∈ {1, .., n} ❛♥❞ k ∈ {1, .., N }                                    ∂ x k,s p i = 1 2 (δ {k=f 1 i } + δ {k=f 2 i } ) ∂ x k,s u i = 1 |x f 2 i -x f 1 i | e s - x f 2 i ,s -x f 1 i ,s |x f 2 i -x f 1 i | u i δ {k=f 2 i } - 1 |x f 2 i -x f 1 i | e s - x f 2 i ,s -x f 1 i ,s |x f 2 i -x f 1 i | u i δ {k=f 1 i } ∂ x k,s l i = x f 2 i ,s -x f 1 i ,s |x f 2 i -x f 1 i | δ {k=f 2 i } - x f 2 i ,s -x f 1 i ,s |x f 2 i -x f 1 i | δ {k=f 1 i } ✭✸✳✸✽✮ ■♥ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥s✱ x k,s ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r s ❝♦♦r❞✐♥❛t❡ ♦❢ x k ✐♥ t❤❡ ❡♠✲ ❜❡❞❞✐♥❣ s♣❛❝❡✬s ❝❛♥♦♥✐❝❛• ❜❛s✐s (e s )✳ ❚❤❡s❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ s✐♠♣•② ✐♥t❡r♣r❡t❡❞ ❛♥❞ ✐♠♣•❡♠❡♥t❡❞ ❛s t❤❡ ✇❛② t♦ ❞✐str✐❜✉t❡ t❤❡ ❣r❛❞✐❡♥t ❝♦♠♣✉t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❡❣♠❡♥ts ♦✈❡r t❤❡ ♣♦✐♥ts ♦❢ t❤❡ s❤❛♣❡ X✳ ■♥ t❤❡ ❝❛s❡ ♦❢ tr✐❛♥❣✉•❛t❡❞ s✉r❢❛❝❡s ✐♥ R 3 ✱ ✇❡ ❤❛✈❡ p i = (x f 1 i +x f 2 i +x f 3 i ) 3 ✱ l i = |(x f 2 i -x f 1 i ) ∧ (x f 3 i -x f 1 i )| ❛♥❞ ❛❣❛✐♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ✐s s✐♠♣•② ❡♥❝♦❞❡❞ ❜② t❤❡ ♥♦r♠❛•✐③❡❞ ♥♦r♠❛• ✈❡❝t♦r u i = (x f 1 i -x f 2 i )∧(x f 2 i -x f 3 i ) |(x f 1 i -x f 2 i )∧(x f 2 i -x f 3 i )| s♦ t❤❛t ❢♦r ❛•• i ∈ {1, .., n} ❛♥❞ k ∈ {1, .., N }                        ∂ x k,s p i = 1 3 (δ {k=f 1 i } + δ {k=f 2 i } + δ {k=f 3 i } ) ∂ x k,s u i = 1 l i e s ∧ (x f 2 i -x f 3 i ) -e s ∧ (x f 2 i -x f 3 i ) , u i u i δ {k=f 1 i } +... ∂ x k,s l i = e s ∧ (x f 2 i -x f 3 i ) , u i δ {k=f 1 i } + ... ✭✸✳✸✾✮ ❲❡ ✇r♦t❡ ✏ +...✏ t♦ ♠❡❛♥ t❤❛t s✐♠✐•❛r t❡r♠s ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r k = f 1 i ❛♥❞ k = f 2 i ✳ ❈♦♠❜✐♥✐♥❣ t❤❡ •❛st r❡•❛t✐♦♥s ✇✐t❤ ❡q✉❛t✐♦♥s ✭✸✳✸✻✮ ❛♥❞ ✭✸✳✸✼✮ ♣r♦✈✐❞❡s ❛ ❢✉•• ❞❡s❝r✐♣✲ t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡✳ ✸✳✺✳✸ ◆✉♠❡r✐❝❛• ❝♦♠♣•❡①✐t② ❋r♦♠ ❛ ♣✉r❡•② ❝♦♠♣✉t❛t✐♦♥❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✇❤❛t ❛♣♣❡❛rs ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❞✐s❝r❡t❡ ❡①♣r❡ss✐♦♥s ✐s t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❝✉rr❡♥ts ❛♥❞ ✈❛r✐❢♦•❞s✱ ✐♥ t❤❡ ❝❛s❡s ♦❢ ✸❉ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ •✐❡ ❡♥t✐r❡•② ✐♥ t❤❡ ❦❡r♥❡•s t❤❛t ♦♣❡r❛t❡ ♦♥ t❤❡ t❛♥❣❡♥t ♦r ♥♦r♠❛• ✈❡❝t♦rs✳ ❇❛s✐❝❛••②✱ t❤✐s ❦❡r♥❡• ✐♥ ❝✉rr❡♥ts ✐s ❛•✇❛②s t❤❡ •✐♥❡❛r ♦♥❡ ♦♥ t❤❡ ✶✶✽ t❛♥❣❡♥ts✱ ✐✳❡ ξ, ξ ′ ✇❤❡r❡❛s✱ ✇✐t❤ ✈❛r✐❢♦•❞s ♦✛❡r ♠❛♥② ♦t❤❡r ♣♦ss✐❜✐•✐t✐❡s •✐❦❡ ❢♦r ✐♥st❛♥❝❡ k P = ξ, ξ ′ 2k ♦r k G = e -2 σ 2 (1-ξ,ξ ′ 2 ) ✳ ■♥ ♣r❛❝t✐❝❡✱ t❤❡s❡ t✇♦ ♠❡t❤♦❞s ❛r❡ t❤✉s ✈❡r② ❝•♦s❡ t♦❣❡t❤❡r ❛♥❞ ✐t ✐s ❡❛s② t♦ ❞❡s✐❣♥ ♦♥❡ ❝♦♠♠♦♥ ❝♦❞❡ str✉❝t✉r❡ t♦ ✐♥❝•✉❞❡ ❜♦t❤✳ ❆♥♦t❤❡r ❝♦♥s❡q✉❡♥❝❡ ✐s t❤❡ ❢❛❝t t❤❛t t❤❡ ❝♦♠♣•❡①✐t② ❢♦r ❞♦✐♥❣ ✈❛r✐❢♦•❞ r❡❣✐str❛t✐♦♥ ✐s ♦❢ t❤❡ ❡①❛❝t s❛♠❡ ♦r❞❡r t❤❛♥ t❤❡ ♦♥❡ ♦❢ ❝✉rr❡♥ts✳ ❈♦♠♣✉t✐♥❣ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥❛•s ❛♥❞ ✐ts ❣r❛❞✐❡♥ts ❜② ❜r✉t❡ ❢♦r❝❡ ♠❡t❤♦❞ ✐s ❛•s♦ O(N 2 ) ❢♦r s❤❛♣❡s ✇✐t❤ N ♣♦✐♥ts✳ ❏✉st ❛s ❢♦r ❝✉rr❡♥ts✬ ♠❡tr✐❝s✱ t❤❡ ❡ss❡♥t✐❛• ♣❛rt ♦❢ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❦❡r♥❡• ❝♦♥✈♦•✉t✐♦♥s ✭❤❡r❡ ✐♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ R 3 ×G 1 (R 3 )✮✳ ❚❤✉s t❤❡ ❛•t❡r♥❛t✐✈❡ ♥✉♠❡r✐❝❛• s❝❤❡♠❡s ❛♥❞ tr✐❝❦s ❢♦r ❢❛st❡r ❦❡r♥❡• ❡✈❛•✉❛t✐♦♥s✱ ♣r❡s❡♥t❡❞ ✐♥ ❝❤❛♣ ✷✳ s❡❝t✐♦♥ ✷✳✸✳✸✱ ❝♦✉•❞ ❛•s♦ ❜❡ tr❛♥s♣♦s❡❞ t♦ ✈❛r✐❢♦•❞s ❛s ❢♦r ✐♥st❛♥❝❡ ❋•❚ ❛♥❞ •P❯ ❝♦♠♣✉t❛t✐♦♥s✳ ❲❡ ❤❛❞ s❤♦✇♥ ❡♠♣✐r✐❝❛••② t❤❛t •P❯ ❝❛♥ ♣❡r❢♦r♠ ❡①tr❡♠❡•② ✇❡•• ✉♣ t♦ t❤❡ •✐♠✐t ♦❢ ✶✵✵✵✵✲✷✵✵✵✵ ♣♦✐♥ts✱ ❛♥❞ t❤✐s r❡♠❛✐♥s ♣❡r❢❡❝t•② tr✉❡ ❢♦r ❛•• ✈❛r✐❢♦•❞ ❦❡r♥❡•s ❛s ✇❡••✳ ❋♦r ✈❡r② ❤✐❣❤ N ✱ ✇❡ ❤❛❞ ♠❡♥t✐♦♥❡❞ t❤❛t t❤❡ ❋❋❚ ❝♦♥✈♦•✉t✐♦♥ ♠❡t❤♦❞ ♦♥ ❣r✐❞s ✭❛s ✐♥tr♦❞✉❝❡❞ ❛♥❞ ✐♠♣•❡♠❡♥t❡❞ ✐♥ ❬✷✽(✮ ✐s ♣r♦❜❛❜•② t❤❡ ♠♦st ❛❞❛♣t❡❞✳ ❋♦r ❝✉rr❡♥ts✱ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ ❡♥❡r❣② ❛♥❞ ❣r❛❞✐❡♥ts ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✉s✐♥❣ ♦♥•② ❞✐s❝r❡t❡ ❝♦♥✲ ✈♦•✉t✐♦♥s ✐♥ R 3 ♦❢ t❤❡ ❢♦r♠ j k e (x i , x)ξ i ✳ ❚❤✐s ❤♦•❞s ❡ss❡♥t✐❛••② ❜❡❝❛✉s❡ t❤❡ ❦❡r♥❡• ξ, ξ ′ ❝❛♥ ❜❡ ❡①♣❛♥❞❡❞ ✐♥ ❝♦♦r❞✐♥❛t❡s ❛s ❛ ♣r♦❞✉❝t ♦❢ s❡♣❛r❛t❡ t❡r♠s ✐♥ ξ ❛♥❞ ξ ′ ✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s✱ t❤✐s r❡♠❛✐♥s tr✉❡ ❢♦r t❤❡ ♣♦•②♥♦♠✐❛• ❦❡r♥❡•s k P ❛♥❞ ❣r✐❞ str❛t❡✲ ❣✐❡s ❛r❡ ✇♦rt❤ ✐♠♣•❡♠❡♥t✐♥❣ ✐♥ t❤❛t ❝❛s❡✳ ❍♦✇❡✈❡r✱ ✐t ❜r❡❛❦s ❞♦✇♥ ❢♦r t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• k G ✭✉♥•❡ss ❜② tr✉♥❝❛t✐♥❣ t❤❡ ❚❛②•♦r s❡r✐❡s✮ ❢♦r ✇❤✐❝❤ ♦♥❡ ✇♦✉•❞ ♥❡❡❞ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥❛• ❣r✐❞s✱ ✉s✉❛••② t♦♦ ♠❡♠♦r② ❛♥❞ t✐♠❡✲❝♦♥s✉♠✐♥❣ t♦ ❜❡ r❡❛••② ✐♥t❡r❡st✐♥❣✱ ❛s ♠❡♥t✐♦♥❡❞ ✐♥ ❝❤❛♣ ✷✳ ✷✳✸✳✸✳ ❋✐♥❛••②✱ t♦ ❡①t❡♥❞ ❛ •✐tt•❡ ❜✐t t❤❡ ❞✐s❝✉ss✐♦♥ t♦ •❛r❣❡r ❞✐♠❡♥s✐♦♥s ❛♥❞ ❝♦❞✐♠❡♥s✐♦♥s ✭❛•t❤♦✉❣❤ t❤❡s❡ ❤❛✈❡ ♥♦t ❜❡❡♥ s✉❜•❡❝t t♦ ✐♠♣•❡♠❡♥t❛t✐♦♥ ②❡t✮✱ ✐t✬s ✇♦rt❤ ♣♦✐♥t✐♥❣ ♦✉t t❤❛t t❤❡ s♣❛❝❡ Λ d E✱ ✇❤✐❝❤ ✐s ✉s❡❞ t♦ r❡♣r❡s❡♥t •♦❝❛• ❣❡♦♠❡tr② ✐♥ t❤❡ ❝✉rr❡♥ts✬ s❡tt✐♥❣✱ ❜❡❝♦♠❡s ✈❡r② ❤✐❣❤ ❞✐♠❡♥s✐♦♥❛• ✇❤❡♥ ✇♦r❦✐♥❣ ✇✐t❤ s✉❜♠❛♥✐❢♦•❞s ❡♠❜❡❞❞❡❞ ✐♥ ❛ •❛r❣❡ ✈❡❝t♦r s♣❛❝❡ E✳ ❚❤✐s s♣❛❝❡ ✐s ♠✉❝❤ ❜✐❣❣❡r t❤❛♥ t❤❡ s♣❛❝❡ ♦❢ ❛•• ♣♦ss✐❜•❡ t❛♥❣❡♥t s♣❛❝❡s✱ ✇❤✐❝❤ ✐s ♥♦t t❤❡ ❝❛s❡ ♦❢ t❤❡ •r❛ss♠❛♥♥✐❛♥✳ ❲❡ ❛r❣✉❡ t❤❛t t❤✐s ✇♦✉•❞ r❡s✉•t ✐♥ ❛ ♠✉❝❤ ♠♦r❡ ♥✉♠❡r✐❝❛••② ✐♥t❡♥s✐✈❡ ✉s❡ ♦❢ ❝✉rr❡♥ts t❤❛♥ ✈❛r✐❢♦•❞s ✐♥ t❤♦s❡ ❝❛s❡s✳ ✸✳✺✳✹ ❙✐♠✉•❛t✐♦♥s ❛♥❞ r❡s✉•ts ❲❡ ♥♦✇ ❝♦♠❡ t♦ ❛ ❢❡✇ r❡s✉•ts ♦❢ ✈❛r✐❢♦•❞ ▲❉❉▼▼ ❛•❣♦r✐t❤♠✱ ✐♥ ✇❤✐❝❤ ✇❡ ✇❛♥t t♦ ❡♠♣❤❛s✐③❡ t❤❡ ❜❡♥❡✜t ✐♥ s✐t✉❛t✐♦♥s t❤❛t tr❛❞✐t✐♦♥❛••② ✐♥✈♦•✈❡ ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s ✇❤❡♥ ♦❜•❡❝ts ❛r❡ ♠❛t❝❤❡❞ ✇✐t❤ t❤❡ ❝✉rr❡♥ts✬ ❢r❛♠❡✇♦r❦✳ ■♥ ❛•• t❤❡ ❢♦••♦✇✐♥❣ ❡①♣❡r✐♠❡♥ts✱ ✇❡ ❝❤♦s❡ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• k e (x, y) = e - |x-y| 2 σ 2 e ♦♥ t❤❡ s♣❛❝❡ E✳ ❚❤❡r❡ ❛r❡ ❛•s♦ ♠❛♥② ♣♦ss✐❜✐•✐t✐❡s ❢♦r t❤❡ ❦❡r♥❡• k t ♦♥ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ✿ ✇❡ ❤❛✈❡ ❢♦❝✉s❡❞ ❡ss❡♥t✐❛••② ♦♥ t❤❡ •✐♥❡❛r ❦❡r♥❡• k L ❛♥❞ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• k G ✳ ❊✈❡♥ t❤♦✉❣❤ ❜♦t❤ ❦❡r♥❡•s ✇❡r❡ ♣r♦✈❡❞ t♦ ✐♥❞✉❝❡ ❛ ❞✐st❛♥❝❡ ♦♥ t❤❡ s❡t ♦❢ r❡✉♥✐♦♥ ♦❢ s✉❜♠❛♥✐❢♦•❞s✱ ✇❡ ❤❛✈❡ ❛•s♦ ❡①♣•❛✐♥❡❞ ✇❤② t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• ❤❛s ❜❡tt❡r s❡♣❛r❛t✐♦♥ ♣r♦♣❡rt✐❡s✱ ❛s ❛r❣✉❡❞ ❜❡❢♦r❡✱ ❡s♣❡❝✐❛••② ✇❤❡♥ t❤❡ s❝❛•❡ σ e ♦❢ t❤❡ ❦❡r♥❡• ♦♥ E ✐s ❝❤♦s❡♥ t♦ ❜❡ •❛r❣❡✳ ❙t✐••✱ t❤❡ ♣♦•②♥♦♠✐❛• ❦❡r♥❡• ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♥♦t ✐♥tr♦❞✉❝✐♥❣ ❛♥ ❛❞❞✐t✐♦♥❛• s❝❛•❡ ♣❛r❛♠❡t❡r ❛♥❞ ❛•s♦ •❡❛❞s t♦ ❢❛st ♥✉♠❡r✐❝❛• ❝♦♠♣✉t❛t✐♦♥s ❢♦r s❤❛♣❡s ✇✐t❤ ❤✐❣❤ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✳ ❚❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✸✳✻ s❤♦✇s ❛ s✐t✉❛t✐♦♥ ✇✐t❤ ♣✐❦❡s✬ str✉❝t✉r❡s t❤❛t ❛r❡ t②♣✐❝❛••② ♥♦t ✇❡•• ♠❛t❝❤❡❞ ✉s✐♥❣ ▲❉❉▼▼ ✇✐t❤ ❝✉rr❡♥ts✳ ❚❤✐s ❡①❛♠♣•❡ ❝❛♥ ❜❡ ❛•s♦ ✉s❡❞ t♦ s❤♦✇ t❤❡ r♦❜✉st♥❡ss ♦❢ ✈❛r✐❢♦•❞s ✇❤❡♥ ✇❡ ✐♥❝r❡❛s❡ t❤❡ s❝❛•❡ ♣❛r❛♠❡t❡r ♦❢ t❤❡ s♣❛t✐❛• ✶✶✾ ❋✐❣✉r❡ ✸✳✻✿ ❊①❛♠♣•❡ ♦❢ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ ✷❉ ❝✉r✈❡s✳ ❚❤❡ s♦✉r❝❡ s❤❛♣❡ ✐s t❤❡ ❜•✉❡ ❝✐r❝•❡✱ t❤❡ t❛r❣❡t ✐s t❤❡ r❡❞ st❛r ✇✐t❤ ❢♦✉r ♥❛rr♦✇ ❜r❛♥❝❤❡s✳ ❖♥ t❤❡ •❡❢t✱ t❤❡ ♠❛t❝❤✐♥❣ ✐s ♣❡r❢♦r♠❡❞ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❝✉rr❡♥ts✳ ❖♥ t❤❡ r✐❣❤t✱ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ✈❛r✐❢♦•❞s ❡①♣♦s❡❞ ✐♥ s❡❝t✐♦♥ ✸✳✺✳✶
♥ ❞❡❛• ❡✣❝✐❡♥t•② ✇✐t❤ ❛♥② ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t ❝♦♠♣♦♥❡♥ts✳ ❦❡r♥❡• k e ✱ ❛s ✐••✉str❛t❡❞ ✐♥ ✜❣✉r❡ ✸✳✼✳ ■t ✐s ♣❛rt✐❝✉•❛r•② r❡♠❛r❦❛❜•❡ t❤❛t ❡✈❡♥ ❛t s❝❛•❡s ♠✉❝❤ •❛r❣❡r t❤❛♥ t❤❡ s✐③❡ ♦❢ t❤❡ ♦❜•❡❝t✱ t❤❡ ❛•❣♦r✐t❤♠ ✐s st✐•• ❛❜•❡ t♦ ❝❛♣t✉r❡ t❤❡ ♦✈❡r❛•• ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ t❛r❣❡t s❤❛♣❡✱ s✐♠♣•② ❜❛s❡❞ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❛♥❣❡♥t s♣❛❝❡s ✭❝❢ ❞✐s❝✉ss✐♦♥ ♦❢ s❡❝t✐♦♥ ✸✳✹✳✶✮✳ ■♥ ✜❣✉r❡ ✸✳✽✱ ✇❡ s❤♦✇ ❛♥ ❡①❛♠♣•❡ ✇❤❡r❡ s❤❛♣❡s ❛r❡ ❝♦♥st✐t✉t❡❞ ♦❢ ♠❛♥② ❞✐s❝♦♥✲ ♥❡❝t❡❞ ♣✐❡❝❡s ♦❢ ❝✉r✈❡s ✇❤✐❝❤ ❛r❡ ♥♦t ❝♦♥s✐st❡♥t•② ♦r✐❡♥t❡❞✱ ❛♥❞ s❤♦✇ t❤❡ r❡s✉•ts ♦❢ ❜♦t❤ ♠❡t❤♦❞s ✭❝✉rr❡♥ts ❛♥❞ ✈❛r✐❢♦•❞s✮✳ ❖t❤❡r ✐♥t❡r❡st✐♥❣ s✐t✉❛t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② tr❡❡✲•✐❦❡ str✉❝t✉r❡s ❢♦r ✇❤✐❝❤ ♦r✐❡♥t❛t✐♥❣ t❤❡ ❞✐✛❡r❡♥t ❜r❛♥❝❤❡s ❝♦♥s✐st❡♥t•② ❢♦r r❡❣✐s✲ tr❛t✐♦♥ ❝❛♥ ❜❡❝♦♠❡ ❛ ♥❡❛r•② ✐♥tr❛❝t❛❜•❡ ♣r♦❜•❡♠ ✇❤❡♥ ❢❛❝❡❞ ✇✐t❤ ♠❛♥② ✐♥t❡rs❡❝t✐♥❣ ❜r❛♥❝❤❡s✳ ❲❡ ❣✐✈❡ ❛ s✐♠♣•❡ ✐♥s✐❣❤t ♦❢ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐♥ ✜❣✉r❡ ✸✳✾✳ ❲❡ ❛•s♦ ♣r❡s❡♥t s♦♠❡ r❡s✉•ts ♦❢ s✉r❢❛❝❡ ♠❛t❝❤✐♥❣ ❞♦♥❡ ✇✐t❤ ♦✉r ✈❛r✐❢♦•❞ ❛•❣♦r✐t❤♠✳ ■♥ ✜❣✉r❡ ✸✳✶✵✱ ❛ s✉r❢❛❝❡ r❡❣✐str❛t✐♦♥ ✐s ❛♣♣•✐❡❞ ❜❡t✇❡❡♥ ❛ st❛♥❞❛r❞ s♣❤❡r❡ ❛♥❞ t❤❡ ❙t❛♥❢♦r❞ ❜✉♥♥② ♠♦❞❡• ✇✐t❤ s❛♠❡ ♣❛r❛♠❡t❡rs ❢♦r ❜♦t❤ ♠❡t❤♦❞s✳ ❲❡ ♦❜s❡r✈❡ s♦♠❡ ♠✐s♠❛t❝❤ ❛rt✐❢❛❝ts ✐♥ t❤❡ ❝✉rr❡♥ts✬s r❡s✉•t ✿ t❤❡ ❡❛rs ❛r❡ ♥♦t ❢✉••② r❡❝♦✈❡r❡❞ ❛s ✇❡•• ❛s s♦♠❡ s♠❛•• ❞❡t❛✐•s ✐♥ t❤❡ ❤❡❛❞ ❛♥❞ ✇❡ s❡❡ t❤❡ ❛♣♣❛r✐t✐♦♥ ♦❢ ✉♥❞❡s✐r❛❜•❡ ♠❡♠❜r❛♥❡s ❛t t❤❡ ❜❛s✐s ♦❢ t❤❡ ❡❛rs✳ ❚❤✐s ✐s ❞✉❡ t♦ ❛ ✬✢❛tt❡♥✐♥❣ ❡✛❡❝t✬ ♦❢ ❝✉rr❡♥ts t❤❛t ♦❢t❡♥ sq✉❡❡③❡ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ s✉r❢❛❝❡ t♦❣❡t❤❡r ✐♥ ♦r❞❡r t♦ ❛rt✐✜❝✐❛••② ❡•✐♠✐♥❛t❡ ✉♥♠❛t❝❤❡❞ ♣❛rts ♦❢ t❤❡ s❤❛♣❡s✳ ❚❤✐s ❤❛s ❜❡❡♥ ❛ ❝♦♥s✐st❡♥t •❛❝❦ ♦❢ ❝✉rr❡♥ts ❢♦r ♠❛♥② ❛♣♣•✐❝❛t✐♦♥s✳ ■♥ ❝♦♥tr❛st✱ ✇❡ s❡❡ t❤❛t t❤❡ ✈❛r✐❢♦•❞ ❛•❣♦r✐t❤♠ ♠❛t❝❤❡s ❛•♠♦st ♣❡r❢❡❝t•② ❛•• ♣❛rts ♦❢ t❤❡ s❤❛♣❡s✳ ✸✳✻ ❉✐s❝✉ss✐♦♥✱ ❡①t❡♥s✐♦♥s ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ❇② ✇❛② ♦❢ ❝♦♥❝•✉s✐♦♥ t♦ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐s❤ t♦ ♣r♦✈✐❞❡ ❛ ❢❡✇ ❛❞❞✐t✐♦♥❛• ❝♦♠♠❡♥ts ❛♥❞ ♣❡rs♣❡❝t✐✈❡s t❤❛t ♥❛t✉r❛••② ❛r✐s❡ ❢r♦♠ ❛•• t❤❡ ♣r❡✈✐♦✉s ♠❛t❡r✐❛•✳ ❲❡ ❞✐✈✐❞❡ ✐t ✐♥t♦ t❤r❡❡ ♠❛✐♥ t♦♣✐❝s✳ ❲❡ ✜rst ❡•❛❜♦r❛t❡ ❛ •✐tt•❡ ♠♦r❡ ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❝✉rr❡♥ts ❛♥❞ ✈❛r✐❢♦•❞s ✇✐t❤ r❡s♣❡❝t t♦ ❦❡r♥❡• s❝❛•❡s ❛♥❞ t❤❡ ✐ss✉❡ ♦❢ ♥♦✐s❡✳ ❲❡ t❤❡♥ ♣r♦♣♦s❡✱ ❛s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✐❞❡❛s ♦❢ ❝❤❛♣t❡r ✷ ❛♥❞ ✸✱ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✈❛r✐❢♦•❞s t♦ ♠♦❞❡• ✉♥♦r✐❡♥t❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❆♥❞ ✜♥❛••②✱ ✇❡ ❜r✐❡✢② ♠❡♥t✐♦♥ ❛♥ ❛•t❡r♥❛t✐✈❡ ♥♦t✐♦♥ ✶✷✶ ❋✐❣✉r❡ ✸✳✾✿ ❊①❛♠♣•❡ ♦❢ r❡❣✐str❛t✐♦♥ ♦♥ tr❡❡s ✇✐t❤ ❝✉rr❡♥ts ✭•❡❢t✮ ❛♥❞ ✈❛r✐❢♦•❞ ✭r✐❣❤t✮✳ ❖❜s❡r✈❡ ❛❣❛✐♥ t❤❛t✱ ✇✐t❤ ❝✉rr❡♥ts✱ ❝♦♥s✐st❡♥t ♦r✐❡♥t❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s♦✉r❝❡ ❛♥❞ t❤❡ t❛r❣❡t ✐s ♥❡❝❡ss❛r② t♦ ❛✈♦✐❞ t❤❡ ✉♥♥❛t✉r❛• ❞❡❢♦r♠❛t✐♦♥ s❤♦✇♥ ♦♥ t❤❡ ✜❣✉r❡✳ ♦❢ ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞ t❤❛t ❝♦✉•❞ ❜❡ ✐♥t❡r❡st✐♥❣ ✐♥ ❢✉t✉r❡ st✉❞✐❡s ❛s ✇❡••✳ ✸✳✻✳✶ ❙❝❛•❡ ❛♥❞ ♥♦✐s❡ ❆♥ ✐♠♣♦rt❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❝✉rr❡♥ts ❛♥❞ ✈❛r✐❢♦•❞s t❤❛t ✇❡ ❤❛✈❡ ❡♠♣❤❛s✐③❡❞ s❡✈❡r❛• t✐♠❡s t❤r♦✉❣❤♦✉t t❤✐s ❝❤❛♣t❡r ✐s t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❦❡r♥❡• ♠❡tr✐❝s ✇❤❡♥ ✇❡ ✐♥❝r❡❛s❡ t❤❡ ✇✐❞t❤ σ e ♦❢ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ t❤❡ s♣❛❝❡ E✳ ❚❤❡ ❝❛♥❝❡••❛t✐♦♥ ❡✛❡❝ts ♦❢ ❝✉rr❡♥ts ♠❛❦❡ s✉❝❤ ♥♦r♠s ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✐♥s❡♥s✐t✐✈❡ t♦ ❛♥②t❤✐♥❣ ❡•s❡ t❤❛♥ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ♦❜•❡❝ts ❛s ♥♦t✐❝❡❞ ✐♥ ✸✳✹✳✶✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇✐t❤ ✈❛r✐❢♦•❞s✱ ✇❡ ♦❜t❛✐♥ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❛♥❣❡♥t s♣❛❝❡s ♦❢ t❤❡ t✇♦ s❤❛♣❡s✳ ❲❡ s❤♦✇❡❞ t❤❛t✱ ❛t ✐♥✜♥✐t❡ s❝❛•❡ σ e ✱ ✇❤❛t r❡♠❛✐♥s ❢r♦♠ ❛ s❤❛♣❡ X ✐s t❤❡ ✐♠❛❣❡ ♠❡❛s✉r❡ H d • T an -1 ✳ ❋r♦♠ t❤❡ ❞✐s❝r❡t❡ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡s❡ ❝❛♥ ❜❡ t❤♦✉❣❤t ❛s ❤✐st♦❣r❛♠s ♦♥ G d (E)✳ ■❢ ✇❡ ❢♦❝✉s ♦♥ t❤❡ s♣❡❝✐❛• ❝❛s❡ ♦❢ ♣•❛♥❛r ❝✉r✈❡s✱ G 1 (R 2 ) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ✉♥✐t s❡♠✐✲❝✐r❝•❡ ✭✇✐t❤ ✐❞❡♥t✐✜❡❞ ❡①tr❡♠✐t✐❡s✮ ❛♥❞ s✉❝❤ ❤✐st♦❣r❛♠s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ♥✉♠❡r✐❝❛••② ❛♥❞ ✈✐s✉❛•✐③❡❞✱ ❛s ✇❡ s❤♦✇ ♦♥ ❛ ❢❡✇ ❡①❛♠♣•❡s ✐♥ ✜❣✉r❡ ✸✳✶✶✳ ❚❤❡r❡ ✐s ♦❢ ❝♦✉rs❡ ♥♦ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❤✐st♦❣r❛♠s ❛♥❞ t❤❡ s❡t ♦❢ ❛•• s❤❛♣❡s s✐♥❝❡ t❤❡s❡ ♦❜•❡❝ts ❛r❡ t♦t❛••② ❜•✐♥❞ t♦ t❤❡ •♦❝❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡s ✐♥ E✳ ❍♦✇❡✈❡r✱ ✜①✐♥❣ ❛ ❣✐✈❡♥ ❤✐st♦❣r❛♠ st✐•• ✐♠♣♦s❡s ❝♦♥str❛✐♥ts ♦♥ s❤❛♣❡✳ ❲❤❡♥ ❝♦♥str❛✐♥ts ❛r❡ ❛•s♦ ❛❞❞❡❞ ♦♥ t❤❡ s❡t ♦❢ ♣♦ss✐❜•❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❛s ✇❡ ❞♦ ✐♥ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s ❢♦r ✐♥st❛♥❝❡✱ ✐t ❜❡❝♦♠❡s ❛ ♣❛rt✐❝✉•❛r•② ✐♥t❡r❡st✐♥❣ q✉❡st✐♦♥ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ♠✉❝❤ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ❜❛s❡❞ ♦♥•② ♦♥ ❤✐st♦❣r❛♠s ❝♦✉•❞ ❞r✐✈❡ r❡❣✐str❛t✐♦♥s ❀ ❛ t♦♣✐❝ t❤❛t ✇❡ •✉st s❦✐♠♠❡❞ t❤r♦✉❣❤ ✇✐t❤ t❤❡ ❡①❛♠♣•❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✼✳ ❚❤❡r❡ ✐s ❤♦✇❡✈❡r ❛ ♥❡❣❛t✐✈❡ ❝♦✉♥t❡r♣❛rt t♦ t❤❡ ❢❛❝t t❤❛t ✈❛r✐❢♦•❞s r❡♠❛✐♥ s❡♥s✐t✐✈❡ t♦ t❛♥❣❡♥t s♣❛❝❡s ❞✐str✐❜✉t✐♦♥s ❡✈❡♥ ✐❢ t❤❡ s❝❛•❡ σ e ❜❡❝♦♠❡s •❛r❣❡✳ ❚❤✐s ✐s t❤❡ ♣r♦❜•❡♠ ♦❢ ♥♦✐s② s❤❛♣❡s t❤❛t ❝❛♥ ✐♥✈♦•✈❡ ✐♠♣♦rt❛♥t ♣❡rt✉r❜❛t✐♦♥s ✐♥ t❤❡ •r❛ss♠❛♥♥✐❛♥ ❞♦♠❛✐♥✳ ❖♥ t❤❡ ♦♣♣♦s✐t❡✱ ❝✉rr❡♥ts ❤❛✈❡ ❛ r❡❣✉•❛r✐③✐♥❣ ❡✛❡❝t t❤❛t ♠❛❦❡s t❤❡ ❛♣♣r♦❛❝❤ r♦❜✉st t♦ ♥♦✐s❡ ❢♦r ✇❡••✲❝❤♦s❡♥ σ e ✳ ❲❡ s❤♦✇ t❤❡ r❡s✉•t ♦❢ ❛ ♠❛t❝❤✐♥❣ ♣❡r❢♦r♠❡❞ ✉♥❞❡r ♥♦✐s❡ ✐♥ ✜❣✉r❡ ✸✳✶✷✳ ❆s ✇❡ s❡❡✱ ❝✉rr❡♥ts ❛r❡ ❛❜•❡ t♦ ✐❣♥♦r❡ ♥♦✐s❡ q✉✐t❡ ✇❡•• s✐♥❝❡ t❤❡ ❤✐❣❤✲ ❢r❡q✉❡♥❝② ✈❛r✐❛t✐♦♥s ❛r❡ ❝❛♥❝❡••✐♥❣ ❢♦r t❤❡ ❝❤♦s❡♥ s❝❛•❡ σ e ✳ ❋♦r ✈❛r✐❢♦•❞s✱ t❤✐♥❣s ❛r❡ ✶✷✷ ❋✐❣✉r❡ ✸✳✶✵✿ ❙✉r❢❛❝❡ ♠❛t❝❤✐♥❣ r❡s✉•ts ❜❡t✇❡❡♥ ❛ s♣❤❡r❡ ❛♥❞ t❤❡ ❙t❛♥❢♦r❞ ❜✉♥♥②✳ ❆t t♦♣ r♦✇✱ t❤❡ s♦✉r❝❡ s♣❤❡r❡ ❛♥❞ t❤❡ t❛r❣❡t ❜✉♥♥② s❤❛♣❡✳ ■♥ t❤❡ ♠✐❞❞•❡ r♦✇✱ t❤❡ r❡s✉•t ❛❢t❡r ♠❛t❝❤✐♥❣ ✇✐t❤ ❝✉rr❡♥ts✳ ❆t ❜♦tt♦♠ r♦✇✱ ✇✐t❤ t❤❡ ✈❛r✐❢♦•❞ ❛♣♣r♦❛❝❤✳ ✶✷✸ ❙❤❛♣❡s ❉✐r❡❝t✐♦♥s✬ ❤✐st♦❣r❛♠s ❋✐❣✉r❡ ✸✳✶✶✿ ❙❡✈❡r❛• ♣•❛♥❡ ❝✉r✈❡s ❛♥❞ t❤❡ ❤✐st♦❣r❛♠s ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❞✐r❡❝t✐♦♥s✳ ❋♦r ✈✐s✉❛•✐③❛t✐♦♥ ♣✉r♣♦s❡s✱ ✇❡ ❤❛✈❡ ✐❞❡♥t✐✜❡❞ t❤❡ •r❛ss♠❛♥♥✐❛♥ G 1 (R 2 ) ✇✐t❤ t❤❡ ✉♥✐t s❡♠✐✲❝✐r❝•❡✳ ❚❤❡ ❝♦•♦r ♦♥ ❡❛❝❤ ♣♦✐♥t ♦❢ t❤❡ s❡♠✐✲❝✐r❝•❡ r❡♣r❡s❡♥ts t❤❡ ❞❡♥s✐t② ✭♣♦ss✐❜•② s✐♥❣✉•❛r✮ ❢♦r t❤❡ ❍❛✉s❞♦r✛ ♠❡❛s✉r❡ H 1 ♦❢ ♣♦✐♥ts ❤❛✈✐♥❣ ❛ t❛♥❣❡♥t ♣♦✐♥t✐♥❣ ✐♥ t❤❛t ♣❛rt✐❝✉•❛r ❞✐r❡❝t✐♦♥✳ ❚❤❡ s✉♠ ♦❢ ❛•• t❤❡s❡ ❞❡♥s✐t✐❡s ✐s ♣r❡❝✐s❡•② t❤❡ t♦t❛• •❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡✳ ✶✷✹ ❈✉rr❡♥ts ❱❛r✐❢♦•❞s ✇✐t❤ σ t = 2 ❱❛r✐❢♦•❞s ✇✐t❤ σ t = 5 ❱❛r✐❢♦•❞s ✇✐t❤ σ t = 10 ❋✐❣✉r❡ ✸✳✶✷✿ |❡❣✐str❛t✐♦♥ ♦❢ t❤❡ ❜•✉❡ s❡❣♠❡♥t ♦♥ t❤❡ ♥♦✐s② s❤❛♣❡ ✐♥ r❡❞✳ ❲❡ s❤♦✇ t❤❡ ❞❡❢♦r♠❡❞ s♦✉r❝❡ s❤❛♣❡ ✐♥ ❞❛s❤❡❞ •✐♥❡✳ ❈✉rr❡♥t ❛♥❞ ✈❛r✐❢♦•❞ ❛•❣♦r✐t❤♠s ✇❡r❡ ❛♣♣•✐❡❞ ✇✐t❤ ❜♦t❤ ❛ •❛✉ss✐❛♥ ❦❡r♥❡• ♦♥ E ♦❢ ❝♦♠♠♦♥ σ e = 0.4✳ ❋♦r ✈❛r✐❢♦•❞s✱ ✇❡ ✉s❡❞ t❤❡ •❛✉ss✐❛♥ ❦❡r♥❡• k G ♦❢ s❝❛•❡ σ t ♦♥ t❤❡ •r❛ss♠❛♥♥ ♣❛rt ❛♥❞ ✇❡ s❤♦✇ t❤❡ ❜❡❤❛✈✐♦r ❛s ✇❡ ✐♥❝r❡❛s❡ σ t ✳ ✶✷✺ ♠♦r❡ ❝♦♠♣•❡① ❛♥❞ ❞❡♣❡♥❞ ♦♥ t❤❡ s❝❛•❡ ❝❤♦s❡♥ ❢♦r t❤❡ •❛✉ss✐❛♥ ♦♥ G d (E)✳ ■❢ σ t ✐s t♦♦ s♠❛••✱ t❤❡ ❛•❣♦r✐t❤♠ ❜❡❝♦♠❡s s❡♥s✐t✐✈❡ t♦ t❤❡ ❞✐st♦rt❡❞ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ t❛♥❣❡♥ts ❛♥❞ ❝r❡❛t❡s ❛ ♣r♦♠✐♥❡♥❝❡ t♦ ❛❝❝♦✉♥t ❢♦r t❤❡s❡ ❞✐r❡❝t✐♦♥s✳ ❖♥❡ ♠✉st t❤❡r❡❢♦r❡ r❡❞✉❝❡ t❤✐s s❡♥s✐t✐✈✐t② ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ ✇✐❞t❤ σ t ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ r♦❜✉st♥❡ss t♦ ♥♦✐s❡✳ ■♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✱ t❤✐s ❝♦✉•❞ ❜❡ ❛ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ✈❛r✐❢♦•❞ ❛♣♣r♦❛❝❤✱ t②♣✐❝❛••② ✐❢ ♦♥•② s♦♠❡ ♣❛rts ♦❢ s❤❛♣❡s ❛r❡ ❛✛❡❝t❡❞ ❜② ♥♦✐s❡ ✇❤✐•❡ ♦t❤❡rs ❛r❡ s♠♦♦t❤ ❜❡❝❛✉s❡ ✐t ✇♦✉•❞ ❡♥❢♦r❝❡ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ •❛r❣❡ σ t ✱ ♣r♦✈✐❞✐♥❣ t❤✉s ♦♥•② ❛ ❣r♦ss ❛♥❛•②s✐s ♦❢ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥s ❡✈❡r②✇❤❡r❡✳ ❲❡ ❝♦✉•❞ st✐•• ❛r❣✉❡ t❤❛t t❤❡ ✐ss✉❡ ♦❢ ♥♦✐s❡ ❝❛♥ ❜❡ ❛•s♦ tr❡❛t❡❞ ✐♥ ♣r❛❝t✐❝❡ ❢r♦♠ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❢♦r ✐♥st❛♥❝❡ ❜② ✐♠♣♦s✐♥❣ ♠♦r❡ r❡❣✉•❛r✐t② t♦ t❤❡ ✈❡❝t♦r ✜❡•❞ v ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥✳ ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ s✐♠♣•② ❜② r❡❞✉❝✐♥❣ t❤❡ tr❛❞❡♦✛ γ ♦r ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ t②♣✐❝❛• s❝❛•❡ σ V ♦❢ ✈❡❝t♦r ✜❡•❞s✳ (❡t t❤❡ q✉❡st✐♦♥ ♦❢ ✜♥❞✐♥❣✱ ✐♥ t❤✐s ❝♦♥t❡①t✱ r❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❛tt❛❝❤♠❡♥t ♠❡tr✐❝s t❤❛t ❝♦✉•❞ s✐♠✉•t❛♥❡♦✉s•② ❛✈♦✐❞ ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s ✇❤✐•❡ ♣r♦✈✐❞✐♥❣ ❛ ♥❛t✉r❛• r♦❜✉st♥❡ss t♦ ♥♦✐s❡ ✐s✱ t♦ ♦✉r ❦♥♦✇♥•❡❞❣❡✱ st✐•• ❛♥ ♦♣❡♥ q✉❡st✐♦♥✳ ✸✳✻✳✷ ❋✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ❚❤❡ ✈❛r✐❢♦•❞ ✐❞❡❛ t♦ r❡♣r❡s❡♥t ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❝❛♥ ❜❡ ❛•s♦ ❛❞❛♣t❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ s❤❛♣❡s ❝❛rr②✐♥❣ s✐❣♥❛•s✱ ❢♦••♦✇✐♥❣ ❛ s✐♠✐•❛r ♣❛t❤ ❛s ❝❤❛♣t❡r ✷ ✇✐t❤ ❝✉rr❡♥ts✳ ■♥ ❢❛❝t✱ •✐♥❦✐♥❣ t❤❡ ✐❞❡❛s t❤✐s ❝❤❛♣t❡r ✇✐t❤ t❤❡ ♦♥❡s ♦❢ ❝❤❛♣t❡r ✷✱ ✐t ✐s ❡✈❡♥ ❛ q✉✐t❡ ❡❛s② ❡①t❡♥s✐♦♥ ♦❢ ✈❛r✐❢♦•❞s✳ ❲❡ ✇✐•• •✉st ❜r✐❡✢② s✉♠ ✉♣ t❤❡ ♠❛✐♥ •✐♥❡s✳ ❲✐t❤ t❤❡ ♥♦t❛t✐♦♥s ❛♥❞ ❞❡✜♥✐t✐♦♥s ♦❢ ❝❤❛♣t❡r ✷✱ ✇❡ ❛r❡ ♥❛t✉r❛••② •❡❞ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ✿ ❉❡✜♥✐t✐♦♥ ✸✳✻✳✶✳ ❲❡ s❛② t❤❛t µ ✐s ❛ d✲ ❞✐♠❡♥s✐♦♥❛• ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞ ♦♥ (E, M )✱ E ❜❡✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ M t❤❡ s✐❣♥❛• ♠❛♥✐❢♦•❞✱ ✐❢ µ ✐s ❛ ❇♦r❡• ✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ t❤❡ s♣❛❝❡ E × G d (E) × M ♦r ❡q✉✐✈❛•❡♥t•② ✐❢ µ ∈ C 0 (E × G d (E) × M ) ′ ✳ ❚❤✐s ✐s s✐♠♣•② ❛✉❣♠❡♥t✐♥❣ t❤❡ ✈❛r✐❢♦•❞ r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ ❛ s✐❣♥❛• ❝♦♠♣♦♥❡♥t✳ ◆♦✇✱ ❛♥② ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f )✱ ✇✐t❤ X ❛ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡t✱ ❤❛s ❛♥ ❡q✉✐✈✲ ❛•❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ❣✐✈❡♥ ❜② ✿ µ (X,f ) (ω) = X ω(x, T x X, f (x))dH d (x) ✭✸✳✹✵✮ ❚❤❡ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤✐s r❡•❛t✐♦♥ ❢♦r ❛ ♣♦•②❤❡❞r❛• s❤❛♣❡ ✇r✐t❡s n i=1 m i .δ (x i ,V i ,f i ) ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ✐♥ ♣r❡✈✐♦✉s s❡❝t✐♦♥s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❦❡r♥❡• ♠❡tr✐❝s ♦♥ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ✐s ❛•s♦ ✈❡r② ♥❛t✉r❛•✳ ❖♥❡ ❝❛♥ •✉st ❝♦♥s✐❞❡r ❦❡r♥❡•s ♦❢ t❤❡ ❢♦r♠ k e ⊗ k t ⊗ k f ✱ ✇❤❡r❡ k e ✱ k t ❛♥❞ k f ❛r❡ r❡❛•✲✈❛•✉❡❞ ♣♦s✐t✐✈❡ ❦❡r♥❡•s r❡s♣❡❝t✐✈❡•② ♦♥ E✱ G d (E) ❛♥❞ M s❛t✐s❢②✐♥❣ ❡♥♦✉❣❤ r❡❣✉•❛r✐t② ♣r♦♣❡rt✐❡s✳ ❲❡ t❤❡♥ ♦❜t❛✐♥ ❛ ❍✐•❜❡rt ♠❡tr✐❝ ❜❡t✇❡❡♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t❤❛t ✇r✐t❡s ✿ µ (X,f ) , µ (Y,g) 2 W ′ = X×Y k e (x, y) k t (T x X, T y Y ) k f (f (x), g(y))dH d (x)dH d (y) ✭✸✳✹✶✮ ❚❤❡ r✐❣♦r♦✉s st✉❞② ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ ♥♦r♠s✱ ❛s ✇❡ ❞✐❞ ✐♥ ❝❤❛♣t❡r ✷ s❡❝t✐♦♥ ✷✳✷✳✸ ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ✐s ♣♦st♣♦♥❡❞ t♦ ❛ ❢✉t✉r❡ ✇♦r❦✱ ❜✉t ❞♦❡s ♥♦t s❡❡♠✱ ❛ ♣r✐♦r✐✱ t♦ ❜r✐♥❣ ❛❞❞✐t✐♦♥❛• ❞✐✣❝✉•t✐❡s✳ ❚❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s✬ ♠❡tr✐❝ ❢♦r ❝♦♠♣✉t✐♥❣ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✬ ❞✐st❛♥❝❡s ❛♥❞ r❡❣✐str❛t✐♦♥ ✐s ✈❡r② ❝♦♥✈❡♥✐❡♥t st❛rt✐♥❣ ❢r♦♠ t❤❡ s✐♠✐•❛r ❝♦❞❡s ♦♥ ✈❛r✲ ✐❢♦•❞s✳ ■♥❞❡❡❞✱ ❛•• ✐t r❡q✉✐r❡s ✐s t♦ ❛❞❞ t❤❡ ♣♦♥❞❡r❛t✐♦♥s ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ❦❡r♥❡• k f ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ❛♥❞ ❝♦♠♣✉t❡ t❤❡ ♥❡✇ ❣r❛❞✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ ♣♦✐♥t ♣♦s✐t✐♦♥s ❛♥❞ s✐❣♥❛•s✳ ❚❤✉s t❤✐s ❢r❛♠❡✇♦r❦ ✭✐♥ t❤❡ ❝❛s❡ M = R✮ ✇❛s ❛•s♦ ✐♥❝•✉❞❡❞ ✐♥ ♦✉r ❝♦❞❡s ❛s ❛♥ ❛•t❡r♥❛t✐✈❡ t♦ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❋✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s✱ ❛s ❢❛r ❛s ♥✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s ❝❛♥ t❡•• ✉♣ t♦ ♥♦✇✱ s✉❝❝❡ss❢✉••② ❝♦♠❜✐♥❡ t❤❡ ✐♥t❡r❡st ♦❢ ♥♦♥✲♦r✐❡♥t❛t✐♦♥ ✭❝♦♠♣❛r❡❞ t♦ ❝✉rr❡♥ts✮ ✇✐t❤ t❤❡ ♣♦ss✐❜✐•✐t② t♦ ✉s❡ s✐❣♥❛•s ✐♥ r❡❣✐str❛t✐♦♥ ❛♥❞ ❡st✐♠❛t❡ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s✳ ❚♦ •✐♥❦ t❤❛t ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ❞✐s❝✉ss✐♦♥ ♦♥ t❤❡ ❞❡♣❡♥❞❡♥❝② ♦❢ t❤❡ ♠❡tr✐❝s ✇✐t❤ r❡s♣❡❝t t♦ ❦❡r♥❡•s✬ s❝❛•❡s✱ ✐t ❛•s♦ ❛♣♣❡❛rs t❤❛t ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s✱ ❜② ♣•❛②✐♥❣ ✇✐t❤ t❤❡ t❤r❡❡ ♣♦ss✐❜•❡ ♣❛r❛♠❡t❡rs σ e ✱ σ t ✱ σ f ✭✐♥ t❤❡ ❝❛s❡ ♦❢ •❛✉ss✐❛♥ ❦❡r♥❡•s✮✱ ❡♥❝♦♠♣❛ss ✐♥ ❛ ✉♥✐✜❡❞ ❢r❛♠❡✇♦r❦ ❛ ✈❡r② ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♠❡tr✐❝s ❜❡t✇❡❡♥ ♦❜•❡❝ts ✿ ♠❡❛s✉r❡s ✭σ t , σ f → ∞✮✱ ✈❛r✐❢♦•❞s ✭σ f → ∞✮✱ ❞✐r❡❝t✐♦♥s✬ ❤✐st♦❣r❛♠s ✭σ e , σ f → ∞✮✱ s✐❣♥❛•s✬ ❤✐st♦❣r❛♠s ✭σ e , σ t → ∞✮✳ ✸✳✻✳✸ ❖r✐❡♥t❡❞ ✈❛r✐❢♦•❞s ▲❡t✬s ✜♥❛••② ♠❡♥t✐♦♥ ❛ •❛st ♣♦ss✐❜•❡ tr❛❝❦ t❤❛t✱ ✇❡ ❜❡•✐❡✈❡✱ ❝♦✉•❞ •❡❛❞ t♦ ✐♥t❡r❡st✐♥❣ ❞❡✈❡•♦♣♠❡♥ts ✐♥ s❤❛♣❡ ♠♦❞❡••✐♥❣ ❛♥❞ ❛♥❛•②s✐s✱ ✇❤✐❝❤ ✐s ❝❛••❡❞ ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞s✳ ❚❤✐s ❝♦♥❝❡♣t✱ ❛•t❤♦✉❣❤ •❡ss ❝♦♠♠♦♥ ✐♥ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r②✱ ❤❛s ❛ ❞❡✜♥✐t✐♦♥ ✈❡r② ❝•♦s❡ t♦ ✈❛r✐❢♦•❞s✳ ■t ♠❛❦❡s ✉s❡ ♦❢ t❤❡ ♦r✐❡♥t❡❞ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛•• ♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ♦❢ E✱ ❞❡♥♦t❡❞ ❣❡♥❡r❛••② G d (E)✳ ■♥ ❛ s✐♠✐•❛r ✇❛② ❛s t❤❡ P•ü❝❦❡r ❡♠❜❡❞❞✐♥❣✱ G d (E) ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ t❤❡ ✉♥✐t s♣❤❡r❡ ♦❢ Λ d E✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ d = 1 ♦r d = n -1✱ ♦r✐❡♥t❡❞ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞ ❝❛♥ ❜❡ ❛❝t✉❛••② ✐❞❡♥t✐✜❡❞ t♦ t❤❡ ✉♥✐t s♣❤❡r❡ ♦❢ E ✱ ✇❤✐❝❤ ♠❛❦❡s ✐t ❛ ♣❛rt✐❝✉•❛r•② s✐♠♣•❡ s♣❛❝❡ t♦ ✉s❡ ✐♥ ♣r❛❝t✐❝❡✳ ❖r✐❡♥t❡❞ ✈❛r✐❢♦•❞s ❛r❡ t❤❡♥ ❞❡✜♥❡❞ s✐♠✐•❛r•② t♦ ✈❛r✐❢♦•❞s ❛s ✿ ❉❡✜♥✐t✐♦♥ ✸✳✻✳✷✳ ❆♥ ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞ ✐s ❛ ❇♦r❡• ✜♥✐t❡ ♠❡❛s✉r❡ ♦♥ E × G d (E)✱ ✐✳❡ ❛♥ ❡•❡♠❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• s♣❛❝❡ C 0 (E × G d (E)) ′ ✳ ◆♦✇✱ ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞s ♣r♦✈✐❞❡ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥② ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts X ♦❢ E ❜② ✐♥t❡❣r❛t✐♥❣ ❛s ✉s✉❛• ❛♥② ❢✉♥❝t✐♦♥ ♦❢ C 0 (E × G d (E)) ♦♥ X✱ ✇❤✐❝❤ ❛ss♦❝✐❛t❡ t♦ X t❤❡ ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞ ❞❡♥♦t❡❞ µ X ✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♥♦t ❡q✉✐✈❛•❡♥t ♥❡✐t❤❡r t♦ ❝✉rr❡♥ts ♥♦r t♦ ✈❛r✐❢♦•❞s ✐♥ ❣❡♥❡r❛•✳ ■❢ X ✐s ❢♦r ✐♥st❛♥❝❡ ❛ ❝♦♥♥❡❝t❡❞ ♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞ ❛♥❞ X t❤❡ ♦♥❡ ✇✐t❤ ♦♣♣♦s✐t❡ ♦r✐❡♥t❛t✐♦♥✱ ✐♥ t❤❛t s❡tt✐♥❣✱ µ X = -µ X ❛♥❞ µ X = µ X ✳ ❆•t❤♦✉❣❤ ✐t ❤❛s ♥♦t ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t ②❡t✱ ♦♥❡ ❝♦✉•❞ ♦❜✈✐♦✉s•② ❞❡✜♥❡ ❦❡r♥❡•s ❛s ✇❡ ❞✐❞ ✇✐t❤ ✈❛r✐❢♦•❞s ✉s✐♥❣ ✐♥ t❤❛t ❝❛s❡ t❤❡ ❡♠❜❡❞❞✐♥❣ ♦❢ G d (E) ✐♥t♦ t❤❡ s♣❤❡r❡ ♦❢ Λ d E ❛♥❞ ♦❜t❛✐♥ |❑❍❙ ❞✐st❛♥❝❡s ✐♥ t❤❛t ❝♦♥t❡①t✳ ❙✉❝❤ ❛ ❢r❛♠❡✇♦r❦✱ ❛s ✇❡ ❝❛♥ ❡①♣❡❝t✱ ✇♦✉•❞ ♥♦t s✉✛❡r ❢r♦♠ t❤❡ ❝❛♥❝❡••❛t✐♦♥ ✐ss✉❡s ❛♣♣❡❛r✐♥❣ ✇✐t❤ ❝✉rr❡♥ts ✭❛t t❤❡ ♣r✐❝❡ ♦❢ ❛ •♦✇❡r r♦❜✉st♥❡ss t♦ ♥♦✐s❡✮✳ ❈♦♠♣❛r❡❞ t♦ ✈❛r✐❢♦•❞s✱ ✐t ✇♦✉•❞ •✐❦❡•② ✐♠♣r♦✈❡ r❡s✉•ts ✇❤❡♥ ✇♦r❦✐♥❣ ♦♥ s❤❛♣❡s ✇✐t❤ ❛ ❝•❡❛r•② ❞❡✜♥❡❞ ❛♥❞ ✉s❡❢✉• ♦r✐❡♥t❛t✐♦♥ ✐♥❢♦r♠❛t✐♦♥ ❜✉t ✐t r❡♠❛✐♥s ❝•❡❛r•② ✐♥❡✛❡❝t✐✈❡ ✐♥ tr❡❛t✐♥❣ ❝❛s❡s ✇❤❡r❡ ♦r✐❡♥t❛t✐♦♥ ✐s ♥♦t ✇❡••✲❞❡✜♥❡❞ ♦r r❡•❡✈❛♥t✳ ✸✳✻✳✹ ❈♦♥❝•✉s✐♦♥ ❍❛✈✐♥❣ ❡①♣•♦r❡❞ ✐♥ t❤❡ ✜rst ♣•❛❝❡ t❤❡ ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s t❤❛t ❝♦♠❡ ❛•♦♥❣ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❝✉rr❡♥ts ✐♥ s❤❛♣❡ ♠♦❞❡••✐♥❣✱ ✇❡ ❤❛✈❡ ❛❞❛♣t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r t❤❡ ❛•t❡r♥❛t✐✈❡ ❝♦♥❝❡♣t ♦❢ ✈❛r✐❢♦•❞s✱ ❜② r❡♣r❡s❡♥t✐♥❣ s❤❛♣❡s ❛s s♣❛t✐❛••② s♣r❡❛❞ ❞✐str✐❜✉t✐♦♥s ♦❢ ✉♥✲ ♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡s t❤❡♠s❡•✈❡s ♠♦❞❡•❡❞ ❛s ❡•❡♠❡♥ts ♦❢ t❤❡ •r❛ss♠❛♥♥ ♠❛♥✐❢♦•❞✳ ❆s ❢♦r ❝✉rr❡♥ts✱ ✇❡ ❡♠❜❡❞ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♦❜•❡❝ts •✐❦❡ s✉❜♠❛♥✐❢♦•❞s ❛♥❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ✐♥ ♦♥❡ ❝♦♠♠♦♥ ❢✉♥❝t✐♦♥❛• s♣❛❝❡✳ ❚❤❡ ✜rst ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ♦✉r ✇♦r❦ ✇❛s t♦ ❞❡✜♥❡ ♠❡tr✐❝s t❤❛t s❤❛r❡ t❤❡♦r❡t✐❝❛• r❡q✉✐r❡♠❡♥ts ❛♥❞ ❡♥❛❜•❡ ❝♦♥✈❡♥✐❡♥t ♥✉♠❡r✐❝❛• ❝♦♠♣✉t❛t✐♦♥s✳ ❲❡ ❤❛✈❡ s❤♦✇♥ ❤♦✇ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❞❡✜♥✐♥❣ ❍✐•❜❡rt str✉❝t✉r❡s ✶✷✼ ♦♥ ✈❛r✐❢♦•❞s t❤r♦✉❣❤ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡•s ❛♥❞ ♣r♦♣♦s❡❞ ❛ ❣❡♥❡r❛• ❝♦♥str✉❝t✐♦♥ ♣r♦❝❡ss ❢♦r s✉❝❤ ❦❡r♥❡•s✳ ❋r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• s✐❞❡✱ ✇❡ ♣r♦✈❡❞ t❤❛t t❤❡s❡ ♠❡tr✐❝s✱ ❛s ♦♣♣♦s❡❞ t♦ t❤❡ ♣r❡✈✐♦✉s r❡♣r❡s❡♥t❛t✐♦♥ ❜② ❝✉rr❡♥ts✱ ❞♦ ♥♦t ❛rt✐✜❝✐❛••② ✬❡•✐♠✐♥❛t❡✬ s❤❛♣❡ ✈♦•✲ ✉♠❡ ❞✉❡ t♦ ♦r✐❡♥t❛t✐♦♥✳ ❲❡ ❛•s♦ ❝♦♠♣✉t❡❞ ❛ ✈❛r✐❛t✐♦♥ ❢♦r♠✉•❛ ♦❢ t❤❡ ♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ s❤❛♣❡s✳ ■♥ t❡r♠s ♦❢ ♥✉♠❡r✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✱ ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛♥ ❛❞❛♣✲ t❛t✐♦♥ ♦❢ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ s❤❛♣❡ r❡❣✐str❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤✐s ❢r❛♠❡✇♦r❦✳ ❚❤❡ r❡s✉•ts ✇❡ ♣r❡s❡♥t❡❞ ❛r❡ ♠❛✐♥•② ❢♦❝✉s❡❞ ♦♥ s②♥t❤❡t✐❝ ❡①❛♠♣•❡s ❢♦r t❤❡ t✐♠❡ ❜❡✐♥❣✱ ❜✉t ✇❡ ✇✐•• s❤♦✇ ❛ ❢❡✇ ❛♣♣•✐❝❛t✐♦♥s t♦ r❡❛• ❞❛t❛s❡ts ✐♥ ❝❤❛♣t❡r ✹✱ ✐♥ ✇❤✐❝❤ ✇❡ ❡①t❡♥❞ t❤❡ ✈❛r✐❢♦•❞ ❢r❛♠❡✇♦r❦ t♦ ❛t•❛s ❡st✐♠❛t✐♦♥✳ ❆s ✇❡ st❛rt❡❞ t♦ ♠❡♥t✐♦♥ ✐♥ t❤✐s •❛st s❡❝t✐♦♥✱ t❤✐s ✇♦r❦ ✐s ❛•s♦ ♣❛✈✐♥❣ t❤❡ ✇❛② t♦ s❡✈❡r❛• ❡①t❡♥s✐♦♥s ❛♥❞ ♥❡✇ ✐ss✉❡s✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ✐❞❡❛s ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ♦r ♦r✐❡♥t❡❞ ✈❛r✐❢♦•❞s ❝❛♥ ❜❡ ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞•② ✐♠♣•❡♠❡♥t❡❞ ❛♥❞ ♥♦✇ ♥❡❡❞ t♦ ❜❡ ❡①♣❡r✐♠❡♥t❡❞ ♦♥ •❛r❣❡r s❡ts ♦❢ ❛♣♣•✐❝❛t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❞✐s❝✉ss✐♦♥ ♦♥ t❤❡ r❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ✈❛r✐❢♦•❞s ❛♥❞ t❛♥❣❡♥t s♣❛❝❡s✬ ❤✐st♦❣r❛♠s✱ t❤❛t ✇❛s ♦♥•② ✐♥✐t✐❛t❡❞ ✐♥ ✸✳✻✳✶✱ ✇♦✉•❞ ♣r♦❜❛❜•② ❞❡s❡r✈❡ ♠♦r❡ ✐♥✲❞❡♣t❤ st✉❞②✱ ♣❡r❤❛♣s ❜② •✐♥❦✐♥❣ t❤❛t t♦ s❤❛♣❡ ❝♦♠♣❛r✐s♦♥ ❜❛s❡❞ ♦♥ ❞❡s❝r✐♣t♦rs✱ ❢♦r ✐♥st❛♥❝❡ s❤❛♣❡ ❝♦♥t❡①ts ✐♥ ✷❉ ❛♥❞ ✸❉ s❤❛♣❡s ❬✶✺( ♦r ❙■❋❚ ❞❡s❝r✐♣t♦rs ✐♥ ✐♠❛❣❡s✳ ✶✷✽ ❈❤❛♣t❡r ✹ ❙t❛t✐st✐❝❛• ❆t•❛s ❡st✐♠❛t✐♦♥ ❛♥❞ ❛♥❛•②s✐s ❈♦♥t❡♥ts ✹✳✶ ❋♦r✇❛r❞ ❛t•❛s ♠♦❞❡•s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✵ ✹✳✷ ❆t•❛s ❡st✐♠❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷ ✹✳✷✳✶ ❆ ✜rst ♠♦❞❡• ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷ ✹✳✷✳✷ ❊st✐♠❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷ ✹✳✷✳✸ |❡s✉•ts ♦♥ t❤❡ r❡t✐♥❛ ❞❛t❛s❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼ ✹✳✷✳✹ ❆ ✇❡••✲♣♦s❡❞ ❝♦♥t✐♥✉♦✉s ❢r❛♠❡✇♦r❦ ❢♦r ❛t•❛s ❡st✐♠❛t✐♦♥ ✳ ✳ ✶✺✶ ✹✳✸ ❙t❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ ♠♦♠❡♥t❛ ❛♥❞ r❡s✐❞✉❛•s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✸ ✹✳✸✳✶ ❉✐♠❡♥s✐♦♥❛•✐t② r❡❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✸ ✹✳✸✳✷ •❡♥❡r❛•✐③❡❞ ▲❉❆ ♦♥ ❍✐•❜❡rt s♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✼ ✹✳✸✳✸ ▼♦❞❡s ❛♥❞ ❝•❛ss✐✜❝❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥❛• ❛t•❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾ ✹✳✸✳✹ ❈•❛ss✐✜❝❛t✐♦♥ r❡s✉•ts ♦♥ t❤❡ ❖❈❚ ❞❛t❛s❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✷ ✹✳✹ ❆t•❛s ♦♥ ✈❛r✐❢♦•❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✹✳✹✳✶ ❆•❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✹✳✹✳✷ ❲♦r❦s ♦♥ ❡♠❜r②♦♥✐❝ ♠♦✉s❡ ❤❡❛rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✹✳✹✳✸ ❚❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♦♥ ❈♦❝❤•❡❛❡ ❞❛t❛s❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✷ ✹✳✺ ❈♦♥❝•✉s✐♦♥ ❛♥❞ ❡①t❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✺ ✶✷✾ ✹✳✶ ❋♦r✇❛r❞ ❛t•❛s ❝♦♥str✉❝t✐♦♥ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ■♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ✜rst r❛♥❣❡ ♦❢ ❛♣♣•✐❝❛t✐♦♥s r❡•❛t❡❞ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛♥❛t♦♠✐❝❛• st✉❞✐❡s✱ t❤❡ ❡st✐♠❛t❡❞ ❞❡❢♦r♠❛t✐♦♥s ❛r❡ •✐❦❡•② t♦ ♠❛❦❡ ♣♦ss✐❜•❡ t❤❡ ❞❡t❡❝t✐♦♥ ♦❢ ❛❜♥♦r♠❛•✐t✐❡s ✐♥ t❤❡ s❤❛♣❡ ♦❢ s✉❜•❡❝ts✱ •❡❛❞✐♥❣ t♦ ♣♦t❡♥t✐❛• ♠❡❞✐❝❛• ❞✐❛❣♥♦s✐s ❢♦r ❝❡rt❛✐♥ ❞✐s❡❛s❡s✳ ❚♦ ❤❛✈❡ ❛♥② ♠❡❛♥✐♥❣ ❢r♦♠ t❤❡ st❛t✐st✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ r❡❣✐str❛t✐♦♥ ♥❡❡❞ t♦ ❜❡ tr❛♥s♣♦s❡❞ t♦ •❛r❣❡r ❞❛t❛❜❛s❡ ♦❢ s✉❜•❡❝ts✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❞✐s❝✉ss t❤❡ ♣r♦❜•❡♠ ♦❢ ❣r♦✉♣✲st✉❞②✱ ♥❛♠❡•② ❤♦✇ t♦ ❛♥❛•②③❡ ✐♥ ❛ st❛t✐st✐❝❛• ✇❛② ♣♦t❡♥t✐❛••② •❛r❣❡ ❣r♦✉♣s ♦❢ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❛t❛s❡ts✳ ❚❤✐s ♣r♦❜•❡♠ ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡•② st✉❞✐❡❞ ✐♥ t❤❡ ♣❛st✱ ♠♦st•② ✐♥ t❤❡ ❝❛s❡ ♦❢ ✷❉ ❛♥❞ ✸❉ ✐♠❛❣❡s ❬✶✶✱ ✻✱ ✹✾( ❛♥❞ ♠♦r❡ r❡❝❡♥t•② ❣❡♥❡r❛•✐③❡❞ t♦ s❤❛♣❡s •✐❦❡ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ❬✷✽✱ ✸✷(✳ ❚❤❡ ♠❛✐♥ ❝❤❛••❡♥❣❡✱ st❛rt✐♥❣ ❢r♦♠ r❡❣✐str❛t✐♦♥ ♦❢ t✇♦ s✉❜•❡❝ts✱ ✐s ♣r❡❝✐s❡•② t♦ ❛✈♦✐❞ t❤❡ ❜✐❛s t❤❛t ✇♦✉•❞ ✐♥❡✈✐t❛❜•② ♦❝❝✉r ✐❢ ✇❡ ❝❤♦s❡ ♦♥❡ s♣❡❝✐✜❝ s✉❜•❡❝t ✐♥ t❤❡ ❞❛t❛❜❛s❡ ❛♥❞ r❡❣✐st❡r ❛•• ♦t❤❡r ✐♥❞✐✈✐❞✉❛•s ♦♥ t❤✐s ♦♥❡✳ ■♥st❡❛❞ ♦❢ t❤✐s ♥❛✐✈❡ ❛♣♣r♦❛❝❤✱ r❡❝❡♥t ♠❡t❤♦❞s ❛r❡ ❣❡♥❡r❛••② ❜❛s❡❞ ♦♥ ❛•❣♦r✐t❤♠s t❤❛t tr② t♦ ❡st✐♠❛t❡ ❛♥ ❛t•❛s ❢r♦♠ t❤❡ ❞❛t❛✳ ■♥ s❤❛♣❡ ❛♥❛•②s✐s✱ ❛♥ ❛t•❛s ✐s ♠❛❞❡ ♦❢ ❛ t❡♠♣•❛t❡ s❤❛♣❡✱ ♦r ❛ ♠❡❛♥ s❤❛♣❡✱ t❤❛t ❝❛♣t✉r❡s t❤❡ ❝♦♠♠♦♥ ❢❡❛t✉r❡s ♦❢ ❛•• s✉❜•❡❝ts✱ t♦❣❡t❤❡r ✇✐t❤ ❛ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t❡♠♣•❛t❡ ❛♥❞ ✐♥❞✐✈✐❞✉❛•s✱ ✇❤✐❝❤✱ ♣r♦♣❡r•② ❛♥❛•②③❡❞✱ s❤♦✉•❞ tr❛♥s❝r✐❜❡ t❤❡ s❤❛♣❡ ✈❛r✐❛❜✐•✐t② ❛♠♦♥❣ t❤❡ ♣♦♣✉•❛t✐♦♥✳ ❚❡❝❤♥✐❝❛••② s♣❡❛❦✐♥❣✱ ❣✐✈❡♥ ❛ ♣♦♣✉•❛t✐♦♥ ♦❢ N s✉❜•❡❝ts {S i } i=1,..,N ✱ t❤❡ ♦✉t♣✉t ✐s ❛ t❡♠♣•❛t❡ S ❛♥❞ N ❞❡❢♦r♠❛t✐♦♥s {φ i } i=1,..,N ✳ ❚❤❡s❡ ❞❡❢♦r♠❛t✐♦♥s ❝❛♥ ❡✐t❤❡r ♣✉•• t❤❡ s✉❜•❡❝ts ❜❛❝❦ ♦♥ t❤❡ t❡♠♣•❛t❡ ❡q✳✭✹✳✶✮✱ ✇❤✐❝❤ ✐s t❤✉s ❝❛••❡❞ t❤❡ ❜❛❝❦✇❛r❞ ♠♦❞❡•✱ ♦r tr❛♥s♣♦rt t❤❡ t❡♠♣•❛t❡ t♦✇❛r❞ t❤❡ ✐♥❞✐✈✐❞✉❛•s ❡q✳✭✹✳✷✮✱ ✇❤✐❝❤ ✐s ❦♥♦✇♥ ❛s t❤❡ ❢♦r✇❛r❞ ♠♦❞❡•✳ φ i (S i ) = S + ǫ i ⇐⇒ S i = φ -1 i (S) + φ -1 i .ǫ i . ✭✹✳✶✮ S i = φ i (S) + ǫ i . ✭✹✳✷✮ ■♥ ❜♦t❤ t❤❡s❡ ♠♦❞❡•s✱ t❤❡ ǫ i ❛r❡ s❤❛♣❡ r❡s✐❞✉❛•s ❢♦r ❡❛❝❤ s✉❜•❡❝t✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ s✉❜•❡❝t✳ ❆s ❛•r❡❛❞② ♠❡♥t✐♦♥❡❞ ❢♦r r❡❣✐str❛t✐♦♥✱ t❤❡s❡ ❛r❡ ♥❡❝❡ss❛r② t♦ ❛❝❝♦✉♥t ❢♦r ❛•• t❤❛t ❝❛♥♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♠♦❞❡• ✐♥❝•✉❞✐♥❣ ♥♦✐s❡✱ t♦♣♦•♦❣✐❝❛• ❝❤❛♥❣❡s ❛♥❞ ❤✐❣❤✲❢r❡q✉❡♥❝② ✈❛r✐❛✲ t✐♦♥s✳ ◆♦t❡ ❤♦✇❡✈❡r t❤❛t ✐♥ t❤❡ ❜❛❝❦✇❛r❞ ♠♦❞❡•✱ t❤❡ r❡s✐❞✉❛•s ❛r❡ •✐✈✐♥❣ ♦♥ t❤❡ t❡♠♣•❛t❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛♥❞ ❛r❡ tr❛♥s♣♦rt❡❞ ❜② ❡❛❝❤ ❞❡❢♦r♠❛t✐♦♥ ✇❤❡r❡❛s ✐♥ t❤❡ ❢♦r✇❛r❞ ♠♦❞❡•✱ t❤❡② ❛r❡ ✐♥tr✐♥s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s✉❜•❡❝t✳ ❚❤✐s ✐s s✉♠♠❡❞ ✉♣ ❜② ❋✐❣✉r❡ ✹✳✶✳ ■t ✐s ❛r❣✉❡❞ ✐♥ ❬✷✽( t❤❛t t❤❡ ❢♦r✇❛r❞ ♠♦❞❡• ✐s ❜❡tt❡r ✜tt❡❞ t♦ ♠♦❞❡••✐♥❣ ❝♦♥s✐❞❡r❛t✐♦♥s ❛♥❞ ♠♦r❡ ❛❞❛♣t❡❞ t♦ st❛t✐st✐❝❛• ♣✉r♣♦s❡s ✿ ✐t ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❢r♦♠ ❛ ❝♦♥s✐st❡♥t ♠❛①✐♠✉♠ ❛ ♣♦st❡r✐♦r✐ ✭▼❆P✮ ❡st✐♠❛t♦r ❛♥❞ ❝♦♠♣✉t✐♥❣ t❤❡ •✐❦❡•✐❤♦♦❞ ♦❢ ❛ ♥❡✇ s✉❜•❡❝t ✐s ♠✉❝❤ ❡❛s✐❡r t❤❛♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❜❛❝❦✇❛r❞ ♠♦❞❡•✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ✐♠♣•❡♠❡♥t❡❞ ❢♦r ✷❉ ❛♥❞ ✸❉ ❝✉rr❡♥ts ❛♥❞ ❢♦✉♥❞ s✉❝❝❡ss❢✉• ❛♣♣•✐❝❛t✐♦♥s t♦ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦❢ ✇❤✐t❡ ♠❛tt❡r ✜❜❡r ❜✉♥❞•❡s ❬✸✵( ♦r ❞❡❡♣ ❜r❛✐♥ str✉❝t✉r❡s✬ s✉r❢❛❝❡s ❬✸✻(✳ ❇❡❢♦r❡ ❝♦♠✐♥❣ t♦ t❤❛t✱ ✇❡ ❜r✐❡✢② s✉♠ ✉♣ t❤❡ ❣❡♥❡r❛• ❞❡r✐✈❛t✐♦♥ ♦❢ ▼❆P ❡st✐♠❛t♦r ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✉s✉❛• ❝✉rr❡♥ts✱ ❢♦••♦✇✐♥❣ ❬✻( ❛♥❞ ❬✷✽(✳ ❲❡ ✐♥s✐st ♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ ❛r❡ ♦♥•② ❢♦r♠❛• ❞❡r✐✈❛t✐♦♥s s✐♥❝❡ t❤❡② ❛r❡ ❞♦♥❡ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡s✳ ❍♦✇❡✈❡r✱ ❛s ❞❡t❛✐•❡❞ ✐♥ ❬✷✽(✱ ✐t ❝❛♥ ❜❡ tr❛♥s•❛t❡❞ ✐♥ t❡r♠s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ♠♦❞✲ ❡•s✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✉s✉❛• ❛♣♣•✐❝❛t✐♦♥s✳ ❋r♦♠ ❛ ❇❛②❡s✐❛♥ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❣✐✈❡♥ t❤❡ s❤❛♣❡ ♦❜s❡r✈❛t✐♦♥s S i r❡♣r❡s❡♥t❡❞ ❛s ❝✉rr❡♥ts ✐♥ ❛ |❑❍❙ W ′ ✱ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ✶✸✵ ❋✐❣✉r❡ ✹✳✶✿ ❋♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❛t•❛s ❡st✐♠❛t✐♦♥✬s ♠♦❞❡•s ❬✷✽(✳ r❡❞✉❝❡s t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ ✜♥❞✐♥❣ ❛ t❡♠♣•❛t❡ S t❤❛t ✐♥t✉✐t✐✈❡•② ♠❛①✐♠✐③❡s t❤❡ •✐❦❡✲ •✐❤♦♦❞ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s ❦♥♦✇✐♥❣ t❤❡ t❡♠♣•❛t❡✱ ✐✳❡ ❢♦r ✐♥❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s✱ N i=1 p(S i |S)✱ ✇❤❡r❡ t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ i ✐♥ ❡q✳✭✹✳✷✮ ❛r❡ ❤✐❞❞❡♥ ✈❛r✐❛❜•❡s✳ ❚♦ ❣✐✈❡ ❛ ♣r❡❝✐s❡ ♠❡❛♥✐♥❣ t♦ t❤❛t✱ ♦♥❡ ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ ♣r♦❜❛❜✐•✐st✐❝ ♠♦❞❡•s ❛♥❞ ♣r✐♦rs ❢♦r t❤❡ r❡s✐❞✉❛•s ❛♥❞ ❞❡❢♦r♠❛t✐♦♥s✳ |❡s✐❞✉❛•s ǫ i ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ✇✐t❤ •❛✉ss✐❛♥ ♣r♦❜❛❜✐•✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✿ p r (ǫ) = C r . exp(-ǫ 2 W ′ /(2σ 2 W )) . ❉❡❢♦r♠❛t✐♦♥s ❛r❡ t♦t❛••② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡✐r ✐♥✐t✐❛• ✈❡•♦❝✐t② ✜❡•❞ ✐♥ v 0 ∈ V t❤❛♥❦s t♦ t❤❡ ❣❡♦❞❡s✐❝ ❡q✉❛t✐♦♥s✱ s♦ t❤❛t ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ ❛ ♣r✐♦r ♦♥ ❞❡❢♦r♠❛t✐♦♥ ✇✐t❤ ❛ ♣r♦❜❛❜✐•✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✿ p φ (v 0 ) = C φ . exp(-v 0 2 V /(2σ 2 V )) ✭✹✳✸✮ ◆♦✇✱ ♦♥❡ ❝❛♥ ❡①♣r❡ss ❢♦r♠❛••② t❤❡ ♣r♦❜❛❜✐•✐t② ♦❢ ❛♥ ♦❜s❡r✈❛t✐♦♥ S i ❦♥♦✇✐♥❣ t❤❡ t❡♠♣•❛t❡ ❜② ✉s✐♥❣ t❤❡ ❇❛②❡s r✉•❡ ✿ p(S i |S) = p(S i |S, v i 0 )p φ (v i 0 )dv i 0 = p r (S i -(φ v i 0 ) * S)p φ (v i 0 )dv 0 ✭✹✳✹✮ ❚❤❡ t❡r♠ p r (S i -(φ v i 0 ) * S) ❞❡♣❡♥❞s ♦♥ t❤❡ ✈❡❝t♦r ✜❡•❞ v 0 ❜② ❛ ❣❡♦❞❡s✐❝ s❤♦♦t✲ ✐♥❣ ♣r♦❝❡❞✉r❡ ✇❤✐❝❤ ♣r♦✈✐❞❡s ♥♦ ❝•♦s❡❞ ❢♦r♠ ❢♦r ♠❛①✐♠✐③✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✐♥t❡❣r❛•✳ ❍❡♥❝❡✱ ✇❡ ❛♣♣•② ❛ st❛♥❞❛r❞ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❝❛••❡❞ ❋❛st ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ▼♦❞❡s ✭❋❆▼✱ ❝❢ ❬✻(✮✱ r❡♣•❛❝✐♥❣ t❤❡ ✐♥t❡❣r❛• ♦❢ ❡q✳✭✹✳✹✮ ❜② t❤❡ ♠❛①✐♠✉♠ ♦❢ ✐ts ✐♥t❡❣r❛♥❞ ✿ p(S i |S) ≈ max v i 0 p r (S i -(φ v i 0 ) * S)p φ (v i 0 )✳ ❚❤❡♥✱ t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♦❢ t❤❡ •✐❦❡•✐❤♦♦❞ ♦r ❡q✉✐✈❛•❡♥t•② t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ •♦❣✲•✐❦❡•✐❤♦♦❞ •❡❛❞s t♦ t❤❡ ❢♦••♦✇✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ✿ (S, v i 0 ) = ❛r❣♠✐♥ (S,v i 0 ) N i=1 1 2σ 2 V v i 0 2 V + 1 2σ 2 W S i -(φ v i 0 ) * S 2 W ′ ✭✹✳✺✮ ❆❣❛✐♥✱ t❤✐s ♠✐♥✐♠✐③❛t✐♦♥ ✐s •✉st ❢♦r♠❛• ❛t t❤✐s ♣♦✐♥t ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t s♣❡❝✐❢② ✐♥ ✇❤❛t s♣❛❝❡ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ S ✐s ♦♣t✐♠✐③❡❞ ♦♥✳ ❖♥•② ✐♥ t❤❡ ❝❛s❡ ♦❢ ❞✐s❝r❡t❡ s❤❛♣❡s ❝❛♥ ❡q✳✭✹✳✺✮ ❜❡ ❛❝t✉❛••② ❣✐✈❡♥ ❛ r❡❛• ♠❡❛♥✐♥❣✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ t❡♠♣•❛t❡ ✐s ♠❛❞❡ ♦❢ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ❛♥❞ t❤❡ ✈❛r✐❛❜•❡ S •✐✈❡s ✐♥ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• s♣❛❝❡✳ ❚❤❡ r❡s♦•✉t✐♦♥ ♦❢ t❤✐s ♣r♦❜•❡♠ t❤❡♥ ❣✐✈❡s ❜♦t❤ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ S ❛♥❞ t❤❡ ✶✸✶ ❞❡❢♦r♠❛t✐♦♥s φ v i 0 ❢r♦♠ t❤❡ t❡♠♣•❛t❡ t♦ ❡❛❝❤ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥✱ ✇❤✐❝❤ ✭st✐•• ✐♥ t❤❡ ❞✐s❝r❡t❡ s✐t✉❛t✐♦♥✮ ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ❢r♦♠ t❤❡ ✐♥✐t✐❛• ✈❡❝t♦r ✜❡•❞s v i 0 ✳ ◆♦t❡ t❤❛t✱ ❢♦r ❛ ✜①❡❞ S✱ t❤❡ ♣r❡✈✐♦✉s ♣r♦❜•❡♠ ❝♦♥s✐sts ❡①❛❝t•② ✐♥ s♦•✈✐♥❣ N ♠❛t❝❤✐♥❣ ♣r♦❜•❡♠s ❜❡t✇❡❡♥ S ❛♥❞ t❤❡ S i ✳ ❈♦♥✈❡rs❡•②✱ ❢♦r ❣✐✈❡♥ φ v i 0 ✱ ♦♣t✐♠✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ S ✉s❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ❣✐✈❡♥ ❜② ❝✉rr❡♥ts✳ ❚❤✉s✱ ❡st✐♠❛t✐♥❣ ❛♥ ❛t•❛s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝✉rr❡♥ts ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❛ s✐♠✉•t❛♥❡♦✉s ❣r❛❞✐❡♥t ❞❡s❝❡♥t ♦♥ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥s✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ✐♠♣•❡♠❡♥t❡❞ ♦♥ ✈❛r✐♦✉s s❤❛♣❡ ❞❛t❛s❡ts✱ ❛s ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✸✸✱ ✸✹(✳ ✹✳✷ ❆t•❛s ❡st✐♠❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❍❛✈✐♥❣ ♦✉t•✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ t❤❡ ✐❞❡❛ ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦❢ s❤❛♣❡s r❡♣✲ r❡s❡♥t❡❞ ❜② ❝✉rr❡♥ts t❤r♦✉❣❤ ❢♦r✇❛r❞ s❝❤❡♠❡✱ ✇❡ ♥♦✇ ✐♥t❡♥❞ t♦ ♣r♦✈✐❞❡ ❛ s✐♠✐•❛r ❢r❛♠❡✇♦r❦ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❖✉r ❛♣♣r♦❛❝❤ ♠❛❦❡s ♣r♦✜t ♦♥ t❤❡ ♠❡tr✐❝s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✬ s❡tt✐♥❣ t❤❛t ✇❛s ❞❡✈❡•♦♣❡❞ ✐♥ ❝❤❛♣t❡r ✷ ❛♥❞ t❤❛t ❞❡♠♦♥str❛t❡❞ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ s✐♠♣•❡r ♣r♦❜•❡♠ ♦❢ r❡❣✐st❡r✐♥❣ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ ♣r❡s❡♥t ❛ ♠♦❞❡• t❤❛t ❞✐r❡❝t•② tr❛♥s❝r✐❜❡s ✐♥ t❡r♠s ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❜•❡♠ ♦❢ ❡q✳✭✹✳✺✮ ❛♥❞ ✇❡ ✇✐•• s❤♦✇ ✐♥ ✇❤❛t r❡s♣❡❝t ❝♦♠❜✐♥✐♥❣ ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛• ❝❛♥ ♣r♦✈✐❞❡ ♠♦r❡ r❡•❡✈❛♥t t❡♠♣•❛t❡s✳ ❲❡ s❤❛•• t❤❡♥ ♣r♦♣♦s❡ ❛♥ ❡♥r✐❝❤❡❞ ♠♦❞❡• ✇❤❡r❡ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ♦♥ ❡❛❝❤ s✉❜•❡❝t ❛r❡ ❡st✐♠❛t❡❞ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥s✳ ❋✐♥❛••②✱ ✐♥ •✐❣❤t ♦❢ ♠♦r❡ r❡❝❡♥t ❜✉t st✐•• ♣r♦s♣❡❝t✐✈❡ r❡✢❡①✐♦♥s ❛t t❤❡ t✐♠❡✱ ✇❡ ❞✐s❝✉ss t❤❡ ♣♦ss✐❜✐•✐t② t♦ ❢♦r♠❛•✐③❡✱ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s s❡tt✐♥❣✱ ❛ ✇❡••✲♣♦s❡❞ ♣r♦❜•❡♠ t❤❛t ❝♦✉•❞ ❣✉❛r❛♥t❡❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐♥✐♠✐③❡rs ❢♦r ❛t•❛s❡s✳ ✹✳✷✳✶ ❆ ✜rst ♠♦❞❡• ❋♦r♠✉•❛t✐♦♥ ✇✐t❤ ▼❆P ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ ✇❡ ❛r❡ ❣✐✈❡♥ N ❢✉♥❝t✐♦♥❛• s❤❛♣❡s S i = (X i , f i ) ❛s ✐♥ ❝❤❛♣t❡r ✷✱ X i ❜❡✐♥❣ d✲❞✐♠❡♥s✐♦♥❛• ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ f i ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X i ✇✐t❤ ✈❛•✉❡s ✐♥ ❛ ❣✐✈❡♥ s✐❣♥❛• s♣❛❝❡ M ✳ ❲❡ ✐♥t❡♥❞ t♦ ✜♥❞✱ •✉st ❛s ✐♥ s❡❝t✐♦♥ ✹✳✶✱ ❛ t❡♠♣•❛t❡ t❤❛t ✇♦✉•❞ ❜❡ ♥♦✇ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ❛♥❞ ❞❡❢♦r♠❛t✐♦♥s φ i ∈ ❉✐✛(E) s❛t✐s❢②✐♥❣ ✿ C (X i ,f i ) = (φ v i ) * C (X,f ) + ǫ i ✭✹✳✻✮ ❲❡ r❡♠✐♥❞ t❤❛t (φ v i ) * C (X,f ) ✐s t❤❡ ❛❝t✐♦♥ ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ s✐❣♥❛•✳ ❖♥❡ ♥♦t✐❝❡❛❜•❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✉rr❡♥ts ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ✐s t❤❛t t❤❡ ❢♦r♠❛• ▼❆P ❞❡r✐✈❛t✐♦♥ ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ ❡①♣•❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✶ ❝❛♥ ❜❡ ❛❞❛♣t❡❞ ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞•② t♦ ❛•• s♦rts ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ ✇❤❡r❡ t❤❡ t❡♠♣•❛t❡ X ✐s ❛ss✉♠❡❞ t♦ ❝♦♥s✐st ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts ❛♥❞ t❤✉s ❛ ✜♥✐t❡ s❡t ♦❢ s✐❣♥❛• ✈❛•✉❡s f ✱ ✇❡ ❛•s♦ ❦♥♦✇ ❢r♦♠ s❡❝t✐♦♥ ✶✳✷✳✺ ✐♥ ❝❤❛♣t❡r ✶✱ t❤❛t t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ v i ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ❜② ✐♥✐t✐❛• ♠♦♠❡♥t❛ p i 0 ✭✇❤✐❝❤ ❛r❡ t❤❡♠s❡•✈❡s ✜♥✐t❡ s❡ts ♦❢ ❝♦✈❡❝t♦rs ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ X✮ ❛♥❞ t❤✉s t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❜•❡♠ t❛❦❡s t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠ ✿ (X, f , p i 0 ) = arg min (X,f,p i 0 ) N i=1 E(p i 0 ) + γ C (X i ,f i ) -(φ p i 0 ) * C (X,f ) 2 W ′ ✭✹✳✼✮ ✶✸✷ γ ❜❡✐♥❣ ❛ ✇❡✐❣❤t ♣❛r❛♠❡t❡r ❜❡t✇❡❡♥ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❝♦st ❛♥❞ t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ ✭♦r t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ r❡s✐❞✉❛• ♣r✐♦r✬s ✈❛r✐❛♥❝❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♣r✐♦r✬s ✈❛r✐❛♥❝❡✮✱ φ p i 0 ❞❡♥♦t❡s t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ✇✐t❤ ✐♥✐t✐❛• ♠♦♠❡♥t✉♠ p i 0 ✳ ◆♦t❡ t❤❛t✱ ✐♥ t❤✐s ❞✐s❝r❡t❡ s❡tt✐♥❣✱ t❤✐s ♠✐♥✐♠✐③❛t✐♦♥ ♠❛❦❡s s❡♥s❡ ❜❡❝❛✉s❡ ❛•• ✈❛r✐❛❜•❡s X✱f ❛♥❞ p i 0 •✐✈❡ ✐♥ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• s♣❛❝❡s✳ ❋♦r ❛ ❝♦❤❡r✲ ❡♥t ❢♦r♠✉•❛t✐♦♥ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s s❡tt✐♥❣✱ ✇❡ r❡❢❡r t♦ t❤❡ •❛t❡r ❞✐s❝✉ss✐♦♥ ♦❢ ✹✳✷✳✹✳ ❲❡ ❛•s♦ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤✐s ❛t•❛s ❡st✐♠❛t✐♦♥ ❢♦r♠✉•❛t✐♦♥ r❡♠❛✐♥s ❣❡♥❡r❛• ❡♥♦✉❣❤ t♦ ✐♥❝♦r♣♦r❛t❡ ❛•• ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ♣r♦✈✐❞❡❞ t❤❛t ♦♥❡ ❝❛♥ ❡①♣r❡ss t❤❡ ✇❛② ❞✐✛❡♦✲ ♠♦r♣❤✐s♠s ❛❝t ♦♥ t❤♦s❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ t❤❛t ❢♦r♠❛•✐s♠✱ ✐❢ t❤❡ t❡♠♣•❛t❡ ❢s❤❛♣❡ (X, f ) ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ s♦♠❡❤♦✇ ❛s ❛ ♣r♦t♦t②♣❡ ♦r ♠❡❛♥ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s✱ ♥♦t❡ t❤❛t ✐t ✐s ♥♦t t❤❡ ❛❝t✉❛• ♠❡❛♥ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❢❝✉rr❡♥ts✱ ✐✳❡ 1 N N i=1 C (X i ,f i ) ✳ ■♥❞❡❡❞✱ t❤✐s ❛r✐t❤♠❡t✐❝ ♠❡❛♥ ✐s ♥♦t r❡❝t✐✜❛❜•❡ ✐♥ ❣❡♥❡r❛•✱ s♦ t❤❛t t❤❡r❡ ✐s ♥♦ ✇❛② t♦ t❤✐♥❦ ♦❢ ✐t ❛s ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡✳ ■♥st❡❛❞✱ t❤❡ ❛♣♣r♦❛❝❤ ❢♦r♠❛•✐③❡❞ ❜② ❡q✳✭✹✳✼✮ ♦♣t✐♠✐③❡s ♦♥ ❢❝✉rr❡♥ts t❤❛t ❛r❡ ❛•r❡❛❞② ✇r✐tt❡♥ ✉♥❞❡r t❤❡ ❢♦r♠ C (X,f ) ✳ ◆♦✇✱ ✐❢ ✇❡ s♣❡❝✐❢② t❤❡ ♣r❡✈✐♦✉s t♦ t❤❡ ❝❛s❡ ♦❢ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s f i : X i → R✱ ✇❤✐❝❤ s❤❛•• ❜❡ t❤❡ s✐t✉❛t✐♦♥ ❝♦♥s✐❞❡r❡❞ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✱ ❛s ❞✐s❝✉ss❡❞ ✐♥ ✷✳✷✱ ✇❡ ❤❛✈❡ (φ v i 0 ) * C (X,f ) = C φ v i 0 .(X,f ) ✇✐t❤ φ p i 0 .(X, f ) = (φ p i 0 (X), f • (φ p i 0 ) -1 ) ❛♥❞ t❤❡r❡❢♦r❡ ❡q✳✭✹✳✼✮ ✇r✐t❡s ✿ min (X,f,p i 0 ) J(X, f, (p i 0 )) . = N i=1 E(p i 0 ) + γ C (X i ,f i ) -C (φ p i 0 (X),f •(φ p i 0 ) -1 ) 2 W ′ ✭✹✳✽✮ ❆ss✉♠✐♥❣ t❤❛t ♦♥❡ ❝❛♥ s♦•✈❡ t❤✐s ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ✭✇❤✐❝❤ ✐s t❤❡ s✉❜•❡❝t ♦❢ ♥❡①t s❡❝t✐♦♥✮✱ t❤❡ ♦✉t♣✉t ✐s ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) t❤❛t ❝❛♣t✉r❡s t❤❡ ❝♦♠♠♦♥ ❢❡❛t✉r❡s ♦❢ t❤❡ s❡t ♦❢ ♦❜s❡r✈❛t✐♦♥s ❜♦t❤ ✐♥ s❤❛♣❡ ❛♥❞ ✐♥ s✐❣♥❛• ❛s ✇❡•• ❛s ❛ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥s t❤❛t s✉♣♣♦s❡❞•② ❡①♣•❛✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ✈❛r✐❛t✐♦♥ ❛r♦✉♥❞ t❤✐s ❝♦♠♠♦♥ t❡♠♣•❛t❡✳ ❚❤❡ r❡st ♦❢ t❤❡ ✈❛r✐❛❜✐•✐t② ✐s ❛•• ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❢❝✉rr❡♥t r❡s✐❞✉❡s C (X i ,f i ) -C (φ p i 0 (X),f •(φ p i 0 ) -1 ) t❤❛t ❝♦♠❜✐♥❡ ❤✐❣❤❡r✲❢r❡q✉❡♥❝② ❣❡♦♠❡tr✐❝❛• ✈❛r✐❛t✐♦♥s ♥♦t ❝❛♣t✉r❡❞ ❜② t❤❡ φ i ✬s ❛s ✇❡•• ❛s t❤❡ ✈❛r✐❛❜✐•✐t② ♦❢ s✐❣♥❛•s f i ❛r♦✉♥❞ f ✳ ❆t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ❲❡ ♥♦✇ ❝♦♠❡ ✐♥t♦ ♠♦r❡ ❞❡t❛✐•s ❛❜♦✉t t❤❡ ❡✛❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ❡q✳✭✹✳✽✮✳ ■♥ ❬✷✽(✱ ✇❛s ♣r♦♣♦s❡❞ ❛♥ ❛•t❡r♥❛t❡❞ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥s✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❤♦✇❡✈❡r t❤❡ ✐♥❝♦♥✈❡♥✐❡♥t ♦❢ ✉♣❞❛t✐♥❣ t❤❡ t❡♠♣•❛t❡ ♣❛rt ❜② ❛ ❣r❛❞✐❡♥t ❝♦♠♣✉t❡❞ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts✱ ✇❤✐❝❤ ✜rst r❡q✉✐r❡s ❞❡❝♦♥✈♦•✉t✐♦♥ s❝❤❡♠❡s •✐❦❡ ▼❛t❝❤✐♥❣ P✉rs✉✐t ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❛♥❞ ❛•s♦ •♦♦s❡s t❤❡ str✉❝t✉r❡ ♦❢ r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ t❤❡ t❡♠♣•❛t❡✳ ■♥st❡❛❞✱ t❤❡ ❛•❣♦r✐t❤♠ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❞❡✈❡•♦♣ ✐s ❛ s✐♥❣•❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t s❝❤❡♠❡ t❤❛t s✐♠✉•t❛♥❡♦✉s•② ✉♣❞❛t❡s X✱ f ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♠♦♠❡♥t❛ p i 0 ✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s✱ ✇❡ s✐♠♣•② ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ ♦❢ t❤❡s❡ ✈❛r✐❛❜•❡s✳ ❆s ✇❡ s❛✐❞✱ t❤❡ t❡♠♣•❛t❡ X ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♠♣♦s❡❞ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts q = (q k ) k=1,..n ❛♥❞ ❛•• ✐♥✐t✐❛• ♠♦♠❡♥t❛ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ t♦ ✈❡❝t♦rs ♦❢ E ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t q k ✱ ✐✳❡ p i 0 = (p i 0,k ) k=1,..n ❣❡♥❡r❛t✐♥❣ ✐♥✐t✐❛• ✈❡❝t♦r ✜❡•❞s v i 0 ❜② ✿ v i 0 = n k=1 K V (q k , .)p i 0,k . ✶✸✸ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❡♥❡r❣② ✐s t❤❡♥ ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞ s✐♥❝❡ ✿ E = N i=1 H r (q, p i ) = N i=1 n k,l=1 K V (q k , q l ). p i k , p i l ❛♥❞ t❤✉s ✿ ∇ q k E = N i=1 ∂ q k H r (q, p i ) = N i=1 n l=1 (∂ 1 K V (q k , q l ) + ∂ 2 K V (q l , q k )) p i k , p i l ∇ p i k E = ∂ p k H r (q, p i ) = n l=1 K V (q k , q l )p i l ✭✹✳✾✮ ◆♦t❡ t❤❛t ✐♥ ♠♦st ❝❛s❡s✱ K V ✐s s②♠♠❡tr✐❝ ✐♥ ❜♦t❤ ✐ts ✈❛r✐❛❜•❡s ✭❛s ❢♦r ❛•• r❛❞✐❛• ❦❡r♥❡•s ❢♦r ✐♥st❛♥❝❡✮ s♦ t❤❡ t✇♦ ♣❛rt✐❛• ❞❡r✐✈❛t✐✈❡s ✐♥ t❤❡ ✜rst ❡q✉❛t✐♦♥ ❛❜♦✈❡ ❛r❡ ❛❝t✉❛••② ❡q✉❛• ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ❜❡❝♦♠❡s ♣❛rt✐❝✉•❛r•② s✐♠♣•❡✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠s g i ✐s ❛❣❛✐♥ s•✐❣❤t•② ♠♦r❡ t❡❝❤♥✐❝❛• ❜✉t ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ✐♥ t❤❡ ❡①❛❝t s❛♠❡ ✇❛② ❛s ❢♦r ❛ r❡❣✐str❛t✐♦♥ ♣r♦❜•❡♠✳ ■♥❞❡❡❞✱ g i ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣♦✐♥ts q i k,1 ✱ t❤❡ ✜♥❛• ♣♦s✐t✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥ts s❤♦t ❜② ♠♦♠❡♥t✉♠ p i 0 ❛♥❞ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✜♥❛• ♠♦♠❡♥t❛ p i k,1 ✳ ❚❤❡r❡❢♦r❡✱ ∇ q i 1 g i = ∇ q 1 g i (q i 1 ) ✇❤❡r❡ ∇ q 1 g i ✇❛s ❛•r❡❛❞② ❡①♣r❡ss❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✸✳✷✱ ❛♥❞ ∇ p i 1 g i = 0✳ ❚❤❡s❡ ❣r❛❞✐❡♥ts ❜❡✐♥❣ ❝♦♠♣✉t❡❞ ❛t t❤❡ ✜♥❛• st❛t❡ ✈❛r✐❛❜•❡s q i 1 ❛♥❞ p i 1 ✱ ♦♥❡ ♥❡❡❞ t♦ ♣✉•• t❤❡♠ ❜❛❝❦ ♦♥ t❤❡ ✐♥✐t✐❛• t❡♠♣•❛t❡ ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t❛✱ ✇❤✐❝❤✱ ❛s ❛•r❡❛❞② ❡①♣•❛✐♥❡❞ ✐♥ ❝❤❛♣t❡r ✷ s❡❝t✐♦♥ ✷✳✸✳✷✱ ✐s t❤❡ r♦•❡ ♦❢ t❤❡ ❜❛❝❦✇❛r❞ ❡✈♦•✉t✐♦♥ ❡q✉❛t✐♦♥✱ ✐✳❡ ❢♦r ❡❛❝❤ i ∈ {1, , , N }✱ ✇❡ ❤❛✈❡ ∇ q g i = Q i 0 , ∇ p i g i = P i 0 ✇✐t❤ ✿      Qi t = -(∂ 2 p,q H r (q i t , p i t )) * Q i t + (∂ 2 q,q H r (q i t , p i t )) * P i t Ṗ i t = -(∂ 2 p,p H r (q i t , p i t )) * Q i t + (∂ 2 q,p H r (q i t , p i t )) * P i t Q i 1 = -∇ q 1 g i (q i 1 ), P i 1 = 0 ✭✹✳✶✵✮ ❆♥❞ ✐t r❡s✉•ts t❤❛t t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• J ✇✐t❤ r❡s♣❡❝t t♦ ✈❛r✐❛❜•❡s q ❛♥❞ p i ✐s ❣✐✈❡♥ ❜② ✿ ∇ q J = N i=1 (∂ q H r (q, p i 0 ) -Q i 0 ) ∇ p i 0 J = ∂ p H r (q, p i 0 ) -P i 0 ✭✹✳✶✶✮ ❚❤❡r❡ st✐•• r❡♠❛✐♥s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ♦♥ t❤❡ t❡♠♣•❛t❡✱ ✇❤✐❝❤ ✇❡ s❤❛•• ❞❡♥♦t❡ ✐♥ s❤♦rt ❜② f k = f (q k )✳ ❲❡ s❡❡ t❤❛t ♦♥•② t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠s g i ❞❡♣❡♥❞ ♦♥ f ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❡①❛❝t•② ❛s ✐♥ ❝❤❛♣t❡r ✷✳ ■♥❞❡❡❞✱ ✇❡ r❡♠✐♥❞ t❤❛t✱ ✐♥ t❤❡ ♠♦❞❡• ♦❢ tr❛♥s♣♦rt ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣✱ t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ❛r❡ ♥♦t ♠♦❞✐✜❡❞ ❜② t❤❡ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ s♦ t❤❡r❡ ✐s ♥♦ ❜❛❝❦✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ♥❡❡❞❡❞✳ ◆♦✇✱ t❛❦✐♥❣ s✐♠✐•❛r ♥♦t❛t✐♦♥s ❛♥❞ ❝♦♥✈❡♥t✐♦♥s✱ •❡t✬s ❞❡♥♦t❡ C (X i ,f i ) = m i k=1 δ ξ i k (x i k , f i k ) ❛♥❞ C φ p i 0 .(X,f ) = m k=1 δ ξ k (q k , fk ) ✇❤❡r❡ ❡❛❝❤ ❞✐r❛❝ ❡♥❝♦❞❡s ♦♥❡ ❢❛❝❡ ♦❢ t❤❡ t❡♠♣•❛t❡ ♦r t❤❡ s✉❜•❡❝ts✳ ❈♦♥s✐❞❡r❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✶✸✹ ❛✉①✐•✐❛r② ✈❛r✐❛❜•❡ f ✱ ✇❡ ❤❛✈❡ ✿ g i = C (X,f ) , C (X,f ) W ′ -2 C (X,f ) , C (X i ,f i ) W ′ + C (X i ,f i ) , C (X i ,f i ) W ′ = m k,l=1 k g (q k , ql )k f ( fk , fl ) ξ k , ξ l -2 m k=1 m i l=1 k g (q k , xi l )k f ( fk , f i l ) ξ k , ξ i l + m i k,l=1 k g (x i k , xi l )k f ( f i k , f i l ) ξ i k , ξ i l ❆♥❞ s♦✱ t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ f ✇r✐t❡s ✿ ∇ fk J = N i=1 ∇ fk g i , ✇✐t❤ ∇ fk g i = m l=1 k g (q k , ql )[∂ 1 k f ( fk , fl ) + ∂ 2 k f ( fl , fk )] ξ k , ξ l -2 m i l=1 k g (q k , xi l )∂ 1 k f ( fk , f i l ) ξ k , ξ i l ✭✹✳✶✷✮ ❙✐♥❝❡ f ✐s ❜❛s✐❝❛••② t❤❡ ✐♥t❡r♣♦•❛t❡❞ ✈❛•✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t t❤❡ ❝❡♥t❡r ♦❢ ❡❛❝❤ ❝❡•• ♦❢ t❤❡ ♠❡s❤ ♦♥ X✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ J ✇✐t❤ r❡s♣❡❝t t♦ f ❝❛♥ ❜❡ s✐♠♣•② ♦❜t❛✐♥❡❞ ❜② ❞✐str✐❜✉t✐♥❣ ∇ f J ♦♥ t❤❡ ♣♦✐♥ts ♦❢ X✳ ◆♦✇✱ t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ✐s r♦✉❣❤•② ❛ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ♦♥ ❛•• ✈❛r✐❛❜•❡s (q, f, p i ) t♦✲ ❣❡t❤❡r✱ ✇❤✐❝❤ ✇❡ r❡❝❛♣ ✐♥ ❛•❣♦r✐t❤♠ ✷✳ Pr❛❝t✐❝❛••②✱ ❛♥② t②♣❡ ♦❢ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ❝♦✉•❞ ❜❡ ❛♣♣•✐❡❞✱ t❤❡ s✐♠♣•❡st ♦♥❡ ❜❡✐♥❣ ❛♥ ❛❞❛♣t✐✈❡ st❡♣ s❝❤❡♠❡✳ ❍♦✇❡✈❡r✱ ✐♥ ♦✉r s✐t✉❛✲ t✐♦♥✱ s✐♥❝❡ ✇❡ ❛r❡ ♦♣t✐♠✐③✐♥❣ ♦♥ s❡✈❡r❛• s♦rts ♦❢ ✈❛r✐❛❜•❡s ✭♣♦✐♥t ♣♦s✐t✐♦♥s✱ ♠♦♠❡♥t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s✮ ✇❤✐❝❤ ❛r❡ ♥♦t ♥❡❝❡ss❛r✐•② ❤♦♠♦❣❡♥❡♦✉s✱ ❛ ❜❡tt❡r ❛♣♣r♦❛❝❤ ✐s t♦ ✉s❡ ❞✐✛❡r❡♥t st❡♣s ❢♦r q✱ f ❛♥❞ t❤❡ p i ✬s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❛❞❛♣t❡❞ s❡♣❛r❛t❡•② ❜② ❞♦✐♥❣ ❛ ❜❛s✐❝ •✐♥❡ s❡❛r❝❤ ✐♥ ♦r❞❡r t♦ ✜♥❞ ❛ ❜❡tt❡r ❞❡s❝❡♥t ❞✐r❡❝t✐♦♥ ❛❢t❡r ❡❛❝❤ ❣r❛❞✐❡♥t ❝♦♠✲ ♣✉t❛t✐♦♥✳ ❚❤✐s ❛••♦✇s s♦♠❡ ✢❡①✐❜✐•✐t② ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ❛♥❞ ♣r❡✈❡♥ts ✐t ❢r♦♠ ✉♣❞❛t✐♥❣ t♦♦ ❢❛st ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜•❡s ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦t❤❡rs✳ ❲❡ ❛•s♦ ✇❛♥t t♦ ❡♠♣❤❛s✐③❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❡♥❡r❣② ❛♥❞ ❣r❛❞✐❡♥t ✇❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡ ❛t ❡❛❝❤ st❡♣ ❛r❡ t❤❡ ❝♦♠❜✐✲ ♥❛t✐♦♥ ♦❢ t❡r♠s r❡•❛t❡❞ t♦ ♦♥•② ♦♥❡ ♦❜s❡r✈❛t✐♦♥ ❛t ❛ t✐♠❡✳ ❚❤✐s ❤❛s t❤❡ ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❞♦♥❡ ✇✐t❤✐♥ t❤❡ ❋❖| •♦♦♣ ✐♥ ❛•❣♦r✐t❤♠ ✷ ❝❛♥ ❜❡ ♣❛r❛••❡•✐③❡❞ str❛✐❣❤t❢♦r✇❛r❞•②✱ ✐♠♣r♦✈✐♥❣ s✐❣♥✐✜❝❛♥t•② t✐♠❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❲♦r❦✐♥❣ ❢♦r ✐♥st❛♥❝❡ ♦♥ ❛ ♠❛❝❤✐♥❡ ✇✐t❤ ❛s ♠❛♥② ♣r♦❝❡ss♦rs ❛s t❤❡ ♥✉♠❜❡r N ♦❢ s✉❜•❡❝ts ❞♦ ♥♦t ♠❛❦❡ ❛t•❛s ❡st✐♠❛t✐♦♥ •♦♥❣❡r ✐♥ t✐♠❡ t❤❛♥ ❛ s✐♥❣•❡ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ s✉❜•❡❝ts✳ ❆t t❤✐s st❛❣❡✱ ❛•❣♦r✐t❤♠ ✷ s✉✛❡rs ❢r♦♠ s❡✈❡r❛• ❞r❛✇❜❛❝❦s ❡ss❡♥t✐❛••② ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❞❡♣❡♥❞❡♥❝② ♦❢ ❡♥❡r❣✐❡s ❛♥❞ ❣r❛❞✐❡♥ts ✐♥ ❡✐t❤❡r t❤❡ s❝❛•❡ ♦r t❤❡ s❛♠♣•✐♥❣ ♦❢ t❡♠♣•❛t❡ ❛♥❞ ❞❛t❛✳ ❚❤❡ ♦t❤❡r ✐ss✉❡ ❝♦♠❡s ❢r♦♠ ❜♦✉♥❞❛r② ❜❡❤❛✈✐♦r t❤❛t ✇❡ ❤❛✈❡ ❛•r❡❛❞② ♦✉t•✐♥❡❞ ❢r♦♠ ❛ t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇ ✐♥ ❝❤❛♣t❡r ✷ t❤❡♦r❡♠ ✷✳✷✳✹✳ ❆•• t❤❡s❡ ♠♦t✐✈❛t❡ t❤❡ ❡✛♦rts t❤❛t ✇❡ ♣r❡s❡♥t ✐♥ t❤❡ ❢♦••♦✇✐♥❣ t♦ ♣r♦✈✐❞❡ ❛ ♠♦r❡ r♦❜✉st ❛♥❞ st❛❜•❡ ♥✉♠❡r✐❝❛• s❝❤❡♠❡✳ ◆♦r♠❛•✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ s❝❛•❡s ❉❡s♣✐t❡ t❤❡ ♦❜✈✐♦✉s ✐♥✈❛r✐❛♥❝❡ ♦❢ ♦✉r ❛•❣♦r✐t❤♠ ✐❢ ✇❡ ❛♣♣•② ❛ ❝♦♠♠♦♥ ✐s♦♠❡tr② ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛• s♣❛❝❡ t♦ ❛•• ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ✐♥✐t✐❛• t❡♠♣•❛t❡✱ ✐t ♥♦ •♦♥❣❡r r❡♠❛✐♥s tr✉❡ ✶✸✺ ❆•❣♦r✐t❤♠ ✷ ❆t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s |❡q✉✐r❡✿ ◆ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s (X i , f i )✱ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❢s❤❛♣❡ (X, f ) = (q k , f k ) k=1,..,n ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣ δ✳ ✶✿ s❡t p i = 0 ❢♦r ❛•• i ∈ {1, .., N }✳ ✷✿ ✇❤✐•❡ ❈♦♥✈❡r❣❡♥❝❡ ❞♦ ✸✿ ❢♦r ✐❂✶✱✳✳✱◆ ❞♦ ✹✿ ❋•♦✇ (X, f ) ❜② ❢♦r✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ p i ❛♥❞ st♦r❡ t❤❡ ❞❡❢♦r♠❡❞ t❡♠♣•❛t❡ ❛t t✐♠❡ ✶ φ v i 0 .(X, f ) = (q i k,1 , f k )✳ ✺✿ ❈♦♠♣✉t❡ t❤❡ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ (X i , f i ) ❛♥❞ φ v i 0 .(X, f )✳ ✻✿ ❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥ts ♦❢ g i ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜♥❛• ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ q i k,1 ✳ ✼✿ ❆♣♣•② t❤❡ ❜❛❝❦✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❡q✳✭✹✳✶✵✮ ❛♥❞ ❞❡❞✉❝❡ ∇ q J ❛♥❞ ∇ p i J ❛s ✐♥ ❡q✳✭✹✳✶✶✮✳ ✽✿ ❈♦♠♣✉t❡ ∇ f J ✇✐t❤ ❡q✳✭✹✳✶✷✮✳ ✾✿ ❡♥❞ ❢♦r ✶✵✿ ❯♣❞❛t❡ ✈❛r✐❛❜•❡s ✿ q ← q + δ∇ q J f ← f + δ∇ f J p i ← p i + δ∇ p i J ✶✶✿ ❈♦♠♣✉t❡ ♥❡✇ ❢✉♥❝t✐♦♥❛• J(q, f, (p i )) ❛♥❞ ♣♦ss✐❜•② ❛❞❛♣t st❡♣ δ ❜② •✐♥❡ s❡❛r❝❤✳ ✶✷✿ ❡♥❞ ✇❤✐•❡ ✶✸✿ r❡t✉r♥ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ❛♥❞ ◆ ✐♥✐t✐❛• ♠♦♠❡♥t❛ (p i )✳ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❡s❝❛•✐♥❣✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r❡❝✐s❡•② ❡①❛♠✐♥❡ t❤❡ ❡✛❡❝t ♦❢ s❝❛•✐♥❣ ❜♦t❤ ✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ❛♥❞ t❤❡ s✐❣♥❛• s♣❛❝❡ ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛• J ❛♥❞ ✐ts ❣r❛❞✐❡♥ts✳ ❚❤❡ ♦❜•❡❝t✐✈❡ ✐s t♦ ♠❛❦❡ t❤❡ ❡♥❡r❣✐❡s ❛♥❞ ✈❛r✐❛❜•❡s✬ ✉♣❞❛t❡s ✐♥ ❛•❣♦r✐t❤♠ ✷ ✐♥✈❛r✐❛♥t ✐❢ ❢♦r ✐♥st❛♥❝❡ ✇❡ ❛♣♣•② ❛♥② ❤♦♠♦t❤❡t✐❝ tr❛♥s❢♦r♠❛t✐♦♥ t♦ ❛•• t❤❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rts ♦❢ t❤❡ ✐♥✐t✐❛• t❡♠♣•❛t❡ ❛♥❞ t❤❡ ❞❛t❛ t♦❣❡t❤❡r ♦r ✐❢ ✇❡ ♠✉•t✐♣•② ❛•• s✐❣♥❛•s ❜② ❛ ❝❡rt❛✐♥ ❝♦♥st❛♥t✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛ ✜rst ❤♦♠♦t❤❡t② ♦♥ t❤❡ ❣❡♦♠❡tr✐❝❛• s♣❛❝❡✱ x → x = λ g x ❛♥❞ ❤♦♠♦✲ t❤❡t② ♦♥ t❤❡ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s f → f = λ s f ✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛r❛♠❡t❡rs ♦❢ ❦❡r♥❡•s K V ❛♥❞ k g ❛r❡ ❛•s♦ r❡s❝❛•❡❞ ❜② λ g ❛♥❞ t❤❛t t❤❡ ♦♥❡ ♦❢ ❦❡r♥❡• k f ✐s r❡s❝❛•❡❞ ❜② λ s s♦ t❤❛t ✇❡ ❤❛✈❡ ✿ KV (λ g x, λ g y) = K V (x, y)✱ kg (λ g x, λ g y) = k g (x, y) ❛♥❞ kf (λ s f, λ s g) = k f (f, g)✳ ❚❤❛t ❜❡✐♥❣ s❡t ✉♣✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ ♥❡✇ ♠♦♠❡♥t❛ ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❤❛✈❡ t♦ ❜❡ ♠✉•t✐♣•✐❡❞ ❜② t❤❡ ❢❛❝t♦r λ g ✱ ✐✳❡ p = λ g p s♦ t❤❛t t❤❡ ❞✐s♣•❛❝❡♠❡♥t ❣❡♥❡r❛t❡❞ ❜② p ✐s ❛•s♦ r❡s❝❛•❡❞ ❜② λ g ✳ ❚❤❡ ♥❡✇ ❡♥❡r❣② ♦❢ ❞❡❢♦r♠❛t✐♦♥ ✐s t❤❡♥ Ẽ = i,j KV (x i , xj ) pi , pj = λ 2 g E✳ ❖♥❡ ❝❛♥ t❤✉s ♥♦r♠❛•✐③❡ t❤❡ ❡♥❡r❣② ❜② t❤❡ ❢❛❝t♦r 1/λ 2 g ✱ ✇❤✐❝❤ ♠❛❦❡s E ✐♥❞❡♣❡♥❞❡♥t ♦❢ s❝❛•❡ ✐♥ t❤❡ ❛•❣♦r✐t❤♠✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t t❡r♠✱ t❤❡ ❞❡♣❡♥❞❡♥❝② ✐♥ λ g ✐s ❣✐✈❡♥ ❜② ❞♦t ♣r♦❞✉❝ts ❜❡t✇❡❡♥ d✲✈❡❝t♦rs ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ d✲❞✐♠❡♥s✐♦♥❛• ❛r❡❛ ♦❢ t❤❡ s❤❛♣❡s ❛♥❞ t❤❡r❡❢♦r❡ t♦ λ d g ✳ ■t r❡s✉•ts t❤❛t t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ ✐s ♣r♦♣♦rt✐♦♥❛• t♦ λ 2d g ❛♥❞ ❤❛s t♦ ❜❡ ♥♦r♠❛•✐③❡❞ ❜② 1/λ 2d g t♦ ❜❡ s❝❛•❡✲✐♥✈❛r✐❛♥t✳ ◆♦t❡ ✐♥❝✐❞❡♥t❛••② t❤❛t✱ ✇✐t❤♦✉t t❤❡s❡ ♥♦r♠❛•✐③❛t✐♦♥s✱ t❤❡ ❜❛•❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ r❡❣✉•❛r✐t② ❛♥❞ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠s ✐s ♠♦❞✐✜❡❞ ❜② s❝❛•❡ ✐❢ d = 1✱ ✇❤✐❝❤ ✐s ♥♦t ❞❡s✐r❛❜•❡ ✐♥ ♣r❛❝t✐❝❡✳ ❆s ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• ❞❡♣❡♥❞❡♥❝②✱ ✇❡ s❡❡ ✶✸✻ t❤❛t t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ ✐s ♥♦t ❛✛❡❝t❡❞ ❜② t❤❡ s❝❛•❡ λ s ✳ ◆♦✇✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❣r❛❞✐❡♥ts✬ ❜❡❤❛✈✐♦r✳ ❆s ✇❡ •✉st s❛✐❞✱ ✇❡ ❤❛✈❡✱ ❢♦r t❤❡ ❢✉♥❝✲ t✐♦♥❛• ❞❡♣❡♥❞❡♥❝② ♦❢ t❤❡ ❝r✐t❡r✐♦♥ J ♣r♦♣❡r•② r❡♥♦r♠❛•✐③❡❞✱ J( f ) = J(f ) = J( f λs ) ❛♥❞ ∇ f J = 1 λs ∇ f J✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛❞✐❡♥t ♦♥ f ♠✉st ❜❡ ♠✉•t✐♣•✐❡❞ ❜② λ 2 s ✐♥ ♦r❞❡r t♦ ✉♣❞❛t❡ t❤❡ ❢✉♥❝t✐♦♥ f ❜② ❛ t❡r♠ ✇❤✐❝❤ ✐s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ✈❛•✉❡s ♦❢ ❢✉♥❝t✐♦♥❛• s✐❣♥❛•s✳ ■♥ t❤❡ s❛♠❡ s♣✐r✐t✱ ❢♦❝✉s✐♥❣ ♦♥ t❤❡ ❞❡♣❡♥❞❡♥❝② ✐♥ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥t ♣♦s✐✲ t✐♦♥s x✱ J(x) = J( x λg ) ⇒ ∇ x J = 1 λg ∇ xJ ✱ s♦ t❤❡ ❣r❛❞✐❡♥t ♦❢ J ✇✐t❤ r❡s♣❡❝t t♦ x ❤❛s t♦ ❜❡ r❡♥♦r♠❛•✐③❡❞ ❜② ❛ ❢❛❝t♦r λ 2 g ✳ ❋✐♥❛••②✱ ✇❡ ❤❛✈❡ ❛❣❛✐♥ J(p) = J( 1 λg .p) ❛♥❞ •✉st ❛s ♣r❡✈✐♦✉s•②✱ ✇❡ ✜♥❞ t❤❛t t❤❡ ♥♦r♠❛•✐③❛t✐♦♥ ❢❛❝t♦r ✐♥ ❢r♦♥t ♦❢ ∇ p J ✐s ❛•s♦ λ 2 g ✳ ■♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s ♦❢ ❛•❣♦r✐t❤♠ ✷✱ ♦♥❡ ❝❛♥ ❤❛✈❡ ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ t②♣✐❝❛• s❝❛•❡s λ g ❛♥❞ λ f ❜② s✐♠♣•② •♦♦❦✐♥❣ ❛t t❤❡ ❛✈❡r❛❣❡ s♣❛t✐❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s✳ ❚❤❡ ♥♦r♠❛•✐③❛t✐♦♥s t❤❛t ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ t❤❡♥ ❣✉❛r❛♥t❡❡ t❤❛t ✉s✐♥❣ s✐♠✐•❛r ❜✉t r❡s❝❛•❡❞ ❞❛t❛ ✇✐•• ♣r♦✈✐❞❡ ❝♦♠♣❛r❛❜•❡ ❡♥❡r❣✐❡s ❛♥❞ ❡♥❡r❣② ❞❡❝r❡❛s❡ ❞✉r✐♥❣ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♠✉st ✐♥s✐st t❤❛t s✉❝❤ ❛ ♥♦r♠❛•✐③❛t✐♦♥ ✐s ♥♦t ❛❜s♦•✉t❡•② ❢✉♥❞❛♠❡♥t❛• ✐♥ t❤❡♦r② ❜✉t ✐t ❤❛s t❤❡ ❣r❡❛t ❛❞✈❛♥t❛❣❡ ♦❢ ♥♦r♠❛•✐③✐♥❣ t❤❡ ❝❤♦✐❝❡ ♦❢ s♦♠❡ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❛•❣♦r✐t❤♠✱ ♥♦t❛❜•② t❤❡ ❜❛•❛♥❝❡ ❜❡t✇❡❡♥ ❞❡❢♦r♠❛t✐♦♥ ❝♦st ❛♥❞ ❞❛t❛ ❛tt❛❝❤♠❡♥t ❛s ✇❡•• ❛s t❤❡ ✐♥✐t✐❛• st❡♣s ❢♦r t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✱ ✇❤✐❝❤ ❝♦✉•❞ ♦t❤❡r✇✐s❡ ✈❛r② ❛ •♦t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s✳ ◆♦r♠❛•✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❡♠♣•❛t❡ s❛♠♣•✐♥❣ ❯s✐♥❣ ❛ s✐♠✐•❛r ♠❡t❤♦❞✱ ✇❡ ♥♦✇ ❢♦❝✉s ♦♥ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛•❣♦r✐t❤♠ ✷ ✇✐t❤ r❡s♣❡❝t t♦ s❛♠♣•✐♥❣ ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ t❡♠♣•❛t❡✳ ◆❛♠❡•②✱ ✇♦r❦✐♥❣ ❢♦r ✐♥st❛♥❝❡ ✇✐t❤ s✉r❢❛❝❡s✱ ✇❡ ✇✐s❤ t♦ ✉♥❞❡rst❛♥❞ q✉❛•✐t❛t✐✈❡•② t❤❡ ❛s②♠♣t♦t✐❝❛• ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• J ❛♥❞ t❤❡ ❞✐✛❡r❡♥t ❣r❛❞✐❡♥ts ✇❤❡♥ t❤❡ ♠❡s❤ ♦♥ t❤❡ t❡♠♣•❛t❡ ✐s r❡✜♥❡❞✳ ▲❡t (X (n) , f ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❞✐s❝r❡t✐③❡❞ t❡♠♣•❛t❡s s❛♠♣•❡❞ ❢r♦♠ ❛ ❝♦♥t✐♥✉♦✉s t❡♠✲ ♣•❛t❡ ❢s❤❛♣❡ (X, f )✱ ✇✐t❤ n t❤❡ t♦t❛• ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ X (n) s✉❝❤ t❤❛t C (X (n) ,f ) W ′ --→ C (X,f ) ❛♥❞ v (n) 0 V -→ v 0 ✳ ❙✐♥❝❡ t❤❡ ✐♥✐t✐❛• ✈❡❝t♦r ✜❡•❞ ♦♥ X (n) ✐s r❡•❛t❡❞ t♦ t❤❡ ♠♦♠❡♥t❛ ❜② t❤❡ r❡•❛t✐♦♥ v (n) 0 (x) = n k=1 K V (q (n) k , x)p (n) k ✱ ✇❡ s❡❡ t❤❛t p (n) ✐s ♦❢ ❛s②♠♣t♦t✐❝❛• ♦r❞❡r 1 n ✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥❛•s J(X (n) , p (n) 0 ) = v (n) 0 2 V + C (φ v (n) 0 (X (n) ),f ) -C (Y,f Y ) 2 W ′ t❤❡♥ ❝♦♥✈❡r❣❡s t♦ ✐ts ❝♦♥t✐♥✉♦✉s ❝♦✉♥t❡r♣❛rt v 0 2 V + C (φ v 0 (X),f ) -C (Y,f Y ) 2 W ′ ✳ ❚❤❡ ✈❛•✉❡s ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❡♥❡r❣✐❡s ❛♥❞ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❛•❣♦✲ r✐t❤♠ ❛r❡ t❤❡r❡❢♦r❡ st❛❜•❡ ✇✐t❤ t❤❡ s❛♠♣•✐♥❣ ♦❢ t❤❡ t❡♠♣•❛t❡✳ ▲❡t✬s ♥♦✇ ❡①❛♠✐♥❡ t❤❡ ❣r❛❞✐❡♥ts ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❡♥❡r❣② E✳ ❚❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ p ✐s ❣✐✈❡♥ ❜② ✿ ∇ p (n) l E = 2 n k=1 K V (q (n) k , q (n) l )p (n) k = 2v (n) 0 (q (n) k ) ✇❤✐❝❤ ✐s ❛s②♠♣t♦t✐❝❛••② ♦❢ ♦r❞❡r 1 ✭❜❡❝❛✉s❡ ✐t ❝♦♥✈❡r❣❡s t♦✇❛r❞ t❤❡ ✈❡❝t♦r ✜❡•❞ v 0 ✮✳ ❆s ♥♦t✐❝❡❞ ♣r❡✈✐♦✉s•②✱ t❤❡ ♠♦♠❡♥t❛ p (n) l ❛r❡ ♦❢ ♦r❞❡r 1 n s♦ t❤❛t ❤♦♠♦❣❡♥❡✐t② r❡q✉✐r❡s t♦ r❡s❝❛•❡ t❤❡ ❣r❛❞✐❡♥t ∇ p (n) E ❜② ❛ ❢❛❝t♦r 1 n ✳ ■♥ ❛ s✐♠✐•❛r ✇❛②✱ ✇❡ ❤❛✈❡ ∇ q (n) l E = 2 n k=1 ∂ 1 K V (q (n) l , q (n) k )p (n) k , p (n) l ✶✸✼ ❋✐❣✉r❡ ✹✳✷✿ •r❛❞✐❡♥t ❞❡♣❡♥❞❡♥❝② ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts n✳ ❚❤❡ ✜❣✉r❡ s❤♦✇s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢❝✉rr❡♥t ♥♦r♠ ❜❡t✇❡❡♥ t✇♦ s✉r❢❛❝❡s ✿ t❤❡ t❡♠♣•❛t❡ ✭t❤❡ ✉♥❞❡r♥❡❛t❤ sq✉❛r❡✮ ❛♥❞ ❛ t❛r❣❡t ✭t❤❡ ❛❜♦✈❡ ❜✐❣❣❡r sq✉❛r❡✮✳ ❚❤❡ ❣r❛❞✐❡♥t ✐s ❝♦♠♣✉t❡❞ ♥✉♠❡r✐❝❛••② ❢♦r t❤r❡❡ ❞✐✛❡r❡♥t s❛♠♣•✐♥❣ ♦❢ t❤❡ t❡♠♣•❛t❡ s✉r❢❛❝❡ ✿ ✾✵✵ ♣♦✐♥ts ✭•❡❢t✮✱ ✸✻✵✵ ♣♦✐♥ts ✭❝❡♥t❡r✮ ❛♥❞ ✶✵✵✵✵ ♣♦✐♥ts ✭r✐❣❤t✮✳ ❲❡ ♦♥•② ❞✐s♣•❛② ❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣♦✐♥t ♣♦s✐t✐♦♥s r❡♣r❡s❡♥t❡❞ ❛s ❛ ✈❡❝t♦r ❛t ❡❛❝❤ ♦❢ t❤❡s❡ ♣♦✐♥ts✳ ❖❜s❡r✈❡ t❤❡ ❣•♦❜❛• ❞❡❝r❡❛s❡ ✇❤❡♥ t❤❡ s❛♠♣•✐♥❣ ✐s ❤✐❣❤❡r ❛s ✇❡•• ❛s t❤❡ ❜♦✉♥❞❛r② ❡✛❡❝t ✇❤✐❝❤ ♠❛❦❡s t❤❡ ❣r❛❞✐❡♥t ❤✐❣❤❡r ♦♥ t❤❡ ❜♦✉♥❞❛r②✱ ♣❛rt✐❝✉•❛r•② ❢♦r ❤✐❣❤ s❛♠♣•✐♥❣✳ ❛♥❞ n k=1 ∂ 1 K V (q (n) l , q (n) k )p (n) k ❝♦♥✈❡r❣❡s t♦✇❛r❞ ∂ 1 K V (q, .)p✱ p (n) l ✐s ♦❢ ♦r❞❡r 1 n s♦ ∇ q (n) E ✐s ❛s②♠♣t♦t✐❝❛••② ♣r♦♣♦rt✐♦♥❛• t♦ 1 n ❛♥❞ ♠✉st ❜❡ t❤❡r❡❢♦r❡ ♥♦r♠❛•✐③❡❞ ❜② ❛ ❢❛❝t♦r n✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ g✱ t❤❡ ❣r❛❞✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ ✐♥✐t✐❛• ♣♦s✐t✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ ✐s ❝♦♠♣✉t❡❞ ❜❛s✐❝❛••② ❜② ✢♦✇✐♥❣ ❜❛❝❦✇❛r❞✱ ✇✐t❤ ❡q✳✭✹✳✶✵✮✱ t❤❡ ❣r❛❞✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ ✜♥❛• ✈❛r✐❛❜•❡s q 1 ❛♥❞ p 1 ✳ ❆s st❛t❡❞ ♣r❡✈✐♦✉s•②✱ ✇❡ ❤❛✈❡ ∇ p 1 g = 0 ❛♥❞ ∇ q 1 g ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ C (φ p 0 (X),f ) -C (Y,f Y ) 2 W ′ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ s❤♦♦t❡❞ s❤❛♣❡ φ v 0 (X)✳ ■♥ t❤❡♦r❡♠ ✷✳✷✳✹ ♦❢ ❝❤❛♣t❡r ✷✱ ✇❡ ❞❡r✐✈❡❞ ❛ ❣❡♥❡r❛• ❢♦r♠✉•❛ ❢♦r t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s✉❝❤ ❢❝✉rr❡♥t ♥♦r♠s r❡•②✐♥❣ ♦♥ t❤❡ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ ❈❛rt❛♥✬s ❢♦r♠✉•❛✳ ❆s ✇❡ ❡①♣•❛✐♥❡❞✱ ♣✉tt✐♥❣ ❛s✐❞❡ t❤❡ ❜♦✉♥❞❛r② t❡r♠s t❤❛t ✇❡ s❤❛•• tr❡❛t ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞✐s♣•❛❝❡♠❡♥t ♦❢ ❛♥ ✐♥t❡r✐♦r ♠❡s❤ ❝❡•• ✐s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ d✲✈♦•✉♠❡ ♦❢ t❤✐s ❝❡••✳ ❚❤✉s ❢♦r ❛ s❤❛♣❡ s❛♠♣•❡❞ ✇✐t❤ n ♣♦✐♥ts✱ ✐t ❞❡❝r❡❛s❡s ❛s 1 n ✳ ■t r❡s✉•ts t❤❛t ❛s②♠♣t♦t✐❝❛••② ∇ q 1 g ∼ 1 n ❛♥❞ s♦ ✐s ∇ q g t❤❛♥❦s t♦ t❤❡ •✐♥❡❛r✐t② ♦❢ t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥ ✿ t❤✐s ❣✐✈❡s ❛ r❡♥♦r♠❛•✐③❛t✐♦♥ ❜② ❢❛❝t♦r n ❢♦r ∇ q g✳ ❲✐t❤ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❣❡♥❡r❛•✐③❡❞ t♦ ❢❝✉rr❡♥ts ✭t❤❡♦r❡♠ ✷✳✷✳✹✮✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ ❡①❛❝t s❛♠❡ r❡❛s♦♥ t❤❛t ∇ f g ✐s ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ d✲✈♦•✉♠❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❡•• ❛♥❞ t❤✉s t❤❡ ♥♦r♠❛•✐③❛t✐♦♥ ❢❛❝t♦r ✐s ❛❣❛✐♥ n✳ ❚♦ s✉♠ ✉♣ t❤❡ ✇❤♦•❡ ❞✐s❝✉ss✐♦♥✱ ✇❡ ❝♦♥❝•✉❞❡ t❤❛t ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ✇✐t❤ r❡s♣❡❝t t♦ s❛♠♣•✐♥❣ ♦❢ t❤❡ t❡♠♣•❛t❡ ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❞♦✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ r❡♥♦r♠❛•✐③❛t✐♦♥ ♦❢ ❣r❛❞✐❡♥ts ✿ ❢♦r ❛ t❡♠♣•❛t❡ ✇✐t❤ n ♣♦✐♥ts✱ ♠✉•t✐♣•② ∇ q J✱ ∇ f J ❜② n ❛♥❞ ❞✐✈✐❞❡ ∇ p J ❜② n✳ ❇♦✉♥❞❛r② t❡r♠s ❛♥❞ r❡❣✉•❛r✐③❛t✐♦♥ ♦❢ ❣r❛❞✐❡♥ts ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts r❡✈❡❛•s t❤❛t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ W ′ ✲♥♦r♠s ✐♥✲ ✈♦•✈❡s ✐♥ ❛❞❞✐t✐♦♥ ✐♥t❡❣r❛•s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ s✉❜♠❛♥✐❢♦•❞ X✳ ❚❤❡s❡ ✐♥❞✉❝❡ ♣❛rt✐❝✉•❛r ❡✛❡❝ts ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ X ✐♥ t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♣r♦❝❡ss t❤❛t ♠✉st ❜❡ ❛❝❝♦✉♥t❡❞ ❢♦r ✇❤❡♥ r❡♥♦r♠❛•✐③✐♥❣ ❣r❛❞✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❡♠♣•❛t❡ ♠❡s❤ ❛❝❝✉r❛❝②✳ ❆ss✉♠✐♥❣ t❤❛t t❤❡ X (n) ❛r❡ r❡❣✉•❛r•② ❞✐s❝r❡t✐③❡❞ ✇✐t❤ n ♣♦✐♥ts✱ ✇❡ ♠❛② ❛ss✉♠❡ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r✐❡s ∂X (n) ✐s ♦❢ ♦r❞❡r n 1-1/d ❛♥❞ •♦❝❛• (d -1)✲✈♦•✉♠❡ ❡•❡♠❡♥ts ♦♥ ∂X (n) ❛r❡ ♣r♦♣♦rt✐♦♥❛• t♦ n 1/d-1 ✳ ■t r❡s✉•ts t❤❛t✱ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♣♦✐♥ts✱ ∇ q 1 g ❞❡♣❡♥❞s ♦♥ t❤❡ s✐③❡ ♦❢ t❤❡ ♠❡s❤ ❛s②♠♣t♦t✐❝❛••② ✐♥ n 1/d-1 ✶✸✽ ❛s ♦♣♣♦s❡❞ t♦ ✐♥t❡r✐♦r ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❞❡♣❡♥❞❡♥❝② ✐s ✐♥ n -1 ✳ ❚❤❡r❡ ✐s t❤❡r❡❢♦r❡ ❛ ♠✉•t✐♣•✐❝❛t✐✈❡ ❢❛❝t♦r n 1/d ❜❡t✇❡❡♥ t❤❡ ❣r❛❞✐❡♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ❛♥❞ ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ X (n) ✱ ✇❤✐❝❤ ✐s ❝♦rr♦❜♦r❛t❡❞ ♥✉♠❡r✐❝❛••② ❜② t❤❡ s✐♠♣•❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✹✳✷✳ ◆♦✇✱ ✐♥ ❛•❣♦r✐t❤♠ ✷✱ ∇ q 1 g s❡r✈❡s ❛s t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ❢♦r t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥ t❤❛t ❝♦♠♣✉t❡s ∇ q g✱ ❛♥❞ t❤✉s ✇❡ ❣❡t ❛ s✐♠✐•❛r ❜❡❤❛✈✐♦r ✇❤❡♥ ✇❡ ✉♣❞❛t❡ t❤❡ t❡♠♣•❛t❡ X ❜② t❤❡ ❣r❛❞✐❡♥t ∇ q J✳ ■♥❞❡❡❞✱ t❤❡ ❣r❛❞✐❡♥t ✐s ❝♦♠♣✉t❡❞ ❢♦r t❤❡ L 2 ♠❡tr✐❝ ♦♥ t❤❡ s❡t ♦❢ ♣♦✐♥ts q✱ ✇❤✐❝❤ r❡s✉•ts ✐♥ ✉♣❞❛t✐♥❣ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥t ♣♦s✐t✐♦♥s ❜② s✐♠♣•② ❛❞❞✐♥❣ t❤❡ ❣r❛❞✐❡♥t✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡ ❤✐❣❤ ❜♦✉♥❞❛r② t❡r♠s ♠❛② ❝❛✉s❡✱ ❛t ❝❡rt❛✐♥ st❡♣s ♦❢ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✱ ✉♥❞❡s✐r❡❞ ❡✛❡❝ts •✐❦❡ s❡•❢✲❝r♦ss✐♥❣ ♦r ❝❤❛♥❣❡s ✐♥ t❤❡ t♦♣♦•♦❣② ♦❢ X✳ ❖♥❡ ❝♦♥✈❡♥✐❡♥t s♦•✉t✐♦♥ t♦ ❛❞❞r❡ss s✉❝❤ ✐ss✉❡s ✐s t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ X ✇✐t❤ r❡s♣❡❝t t♦ ❛ ♠♦r❡ r❡❣✉•❛r ♠❡tr✐❝ t❤❛♥ L 2 ✳ ◆❛♠❡•②✱ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ❛ •❛✉ss✐❛♥ ❦❡r♥❡• K reg ♦❢ s❝❛•❡ λ reg > 0 ❛♥❞ ❝♦♠♣✉t❡ ∇q J = K reg (q, q)∇ q J✳ ❚❤❡♥✱ -K reg (q, q)∇ q J ✐s st✐•• ❛ ❞❡s❝❡♥t ❞✐r❡❝t✐♦♥ ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• J✱ ❛s t❤❡ ♠❛tr✐① K reg (q, q) ✐s ♣♦s✐t✐✈❡ ❞❡✜✲ ♥✐t❡✳ ❍♦✇❡✈❡r✱ t❤✐s ♥❡✇ ❣r❛❞✐❡♥t ✐s r❡❣✉•❛r✐③❡❞ ❛t ❛ s❝❛•❡ ❣✐✈❡♥ ❜② t❤❡ r❡❣✉•❛r✐③❛t✐♦♥ ❦❡r♥❡• ❛♥❞ ✐t ❝❛♥ ❜❡ t❤✉s ❝♦♥s✐❞❡r❡❞ ❛s ❛ s❛♠♣•✐♥❣ ♦❢ ❛ ❞❡♥s❡ ✈❡❝t♦r ✜❡•❞ ❜❡•♦♥❣✐♥❣ t♦ t❤❡ |❑❍❙ V reg ✳ ❚❤❡ ❞✐s♣•❛❝❡♠❡♥t ♦❢ t❤❡ t❡♠♣•❛t❡ ❢r♦♠ ✐ts ✐♥✐t✐❛• ♣♦s✐t✐♦♥ ✐♥ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ✐s t❤❡♥ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s •✐✈✐♥❣ ✐♥ V reg s♦ t❤❛t✱ ✇✐t❤ s✉✣❝✐❡♥t•② s♠❛•• st❡♣s δ ✐♥ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✱ ✐t ❡♥s✉r❡s t❤❛t t❤❡ ✜♥❛• ❡st✐♠❛t✐♦♥ ♦❢ X ✐s ❛♥ ❡•❡♠❡♥t ♦❢ t❤❡ ❞✐✛❡♦♠♦r♣❤✐❝ ♦r❜✐t ♦❢ ✐ts ✐♥✐t✐❛• ❡st✐♠❛t❡✳ ■♥ ♠♦r❡ ♣r❡❝✐s❡ t❡r♠s✱ ♦♥❡ ❝❛♥ ❛❝t✉❛••② s❤♦✇ t❤❛t t❤❡ ✢♦✇ ♦❢ t❤❡ r❡❣✉•❛r✐③❡❞ ❣r❛❞✐❡♥ts ✐s ✐♥❞❡❡❞ ❞❡✜♥❡❞ ❛t ❛•• t✐♠❡s✱ ❛s st❛t❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✿ Pr♦♣❡rt② ✹✳✷✳✶✳ ▲❡t q s = (q k,s ) k=1,..n ❜❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥ts ❛t ✐t❡r❛t✐♦♥ s ✭✇❤✐❝❤ ✇❡ ❝♦♥s✐❞❡r ❛s ❛ ❝♦♥t✐♥✉✉♠✮ ❛♥❞ u s (x) = - n k=1 K reg (q s,k , x)∇ q s,k J t❤❡ ✈❡❝t♦r ✜❡•❞ ♦❜t❛✐♥❡❞ ❜② t❤❡ r❡❣✉•❛r✐③❡❞ ❣r❛❞✐❡♥t ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❚❤❡♥ t❤❡ ✢♦✇ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ẏ(s) = u s (y) ❡①✐sts ❢♦r ❛•• s ∈ R + ✳ Pr♦♦❢✳ ❉✉❡ t♦ t❤❡ r❡❣✉•❛r✐t② ♦❢ t❤❡ r❡❣✉•❛r✐③❛t✐♦♥ ❦❡r♥❡• K reg ✱ ✇❡ s❡❡ t❤❛t t❤❡ ✈❡❝t♦r ✜❡•❞s u s ❛r❡ s♠♦♦t❤ ❢♦r ❛•• s s♦ t❤❛t ❈❛✉❝❤②✲▲✐♣s❝❤✐t③ t❤❡♦r❡♠ ❣✐✈❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠❛①✐♠❛• s♦•✉t✐♦♥s✳ ❋♦r ❛ ❣✐✈❡♥ ✐♥✐t✐❛• ❝♦♥❞✐t✐♦♥ x✱ t❤❡ s♦•✉t✐♦♥ ✐s ❞❡✜♥❡❞ ♦♥ ❛♥ ✐♥t❡r✈❛• [0, s f [ ❛♥❞ ✇❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t s f = +∞✳ ▲❡t✬s ❞❡♥♦t❡ ✐♥ s❤♦rt J(s) ❢♦r J(q s , p i 0,s )✳ ❙✐♥❝❡ ✇❡ ❦♥♦✇ t❤❛t |u s | ∞ ≤ ❈t❡.|u s | Vreg ✱ t❤✐s ✇✐•• ❜❡ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ✿ s f 0 |u s | 2 Vreg ds ≤ J(0) . ✭✹✳✶✸✮ ❲❡ ❤❛✈❡ ✐♥❞❡❡❞ ✿ |u s | 2 Vreg = n k=1 (∇ q s,k J) T n l=1 K reg (q s,k , q s,l )∇ q s,l J = - n k=1 (∇ q s,k J) T u s (q s,k ) ≤ - n k=1 (∇ q s,k J) T dq s,k ds - N i=1 (∇ p i 0,s J) T dp i 0,s ds ≥0 = - dJ ds . ❚❤❡ ♣♦s✐t✐✈❡♥❡ss ♦❢ t❤❡ s❡❝♦♥❞ t❡r♠ ✐s s✐♠♣•② ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t ❜② ❞❡✜♥✐t✐♦♥ dp i 0,s ds = -∇ p i 0,s J✳ ■t r❡s✉•ts t❤❛t s f 0 |u s | 2 Vreg ≤ J(0) -J(s f ) ≤ J(0) ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢✳ ❚❤❡ ❝♦♥s❡q✉❡♥❝❡ ✐s t❤❛t r❡❣✉•❛r✐③❛t✐♦♥ ♣r♦✈✐❞❡s ♥✉♠❡r✐❝❛• st❛❜✐•✐t② ♦❢ t❤❡ t❡♠♣•❛t❡ s✉♣♣♦rt ❜② t❤❡ ❛•❣♦r✐t❤♠✳ ❇✉t ✐♥❝✐❞❡♥t❛••②✱ ✐t ❝♦♥str❛✐♥s t❤❡ t❡♠♣•❛t❡ t♦ •✐✈❡ ✐♥ ❛ ♣❛rt✐❝✉•❛r ♦r❜✐t ❛♥❞ ♠❛❦❡s t❤❡ ❛•❣♦r✐t❤♠ ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥✐t✐❛•✐③❛✲ t✐♦♥✳ ❊✈❡♥ t❤♦✉❣❤ s♠❛••❡r ✈❛•✉❡s ♦❢ λ reg ❛••♦✇ ♠♦r❡ ✢❡①✐❜✐•✐t② ✭✇❤✐•❡ ❞✐♠✐♥✐s❤✐♥❣ t❤❡ st❛❜✐•✐t②✮✱ ❝❤❛♥❣❡s ♦❢ t♦♣♦•♦❣② ❛s ❝r❡❛t✐♦♥ ♦❢ ❤♦•❡s ♦r ❞✐s❝♦♥♥❡❝t✐♦♥s ♦♥ X ❝❛♥♥♦t ❜❡ ♦❜t❛✐♥❡❞ ❜② ♦✉r ❛♣♣r♦❛❝❤ ❛♥❞ t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ✐s ❝♦♥s❡q✉❡♥t•② ❝r✉✲ ❝✐❛•✳ ❲❡ ❝❛♥ ❛r❣✉❡ ❤♦✇❡✈❡r t❤❛t t♦♣♦•♦❣✐❝❛• ❝❤❛♥❣❡s ✐♥ ❛♣♣•✐❝❛t✐♦♥s t♦ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❛r❡ ❣❡♥❡r❛••② ♥♦t ❞❡s✐r❛❜•❡✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ♠♦♠❡♥t❛✱ t❤❡ ❜❡❤❛✈✐♦r ✐s ❡①❛❝t•② ❞✉❛•✳ ■♥❞❡❡❞✱ p✱ ❛s ❛ ❝♦✈❛r✐❛❜•❡✱ •✐✈❡s ✐♥ t❤❡ ❞✉❛• ♦❢ t❤❡ s♣❛❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s V ❛♥❞ ♥❡❡❞ ♥♦t ❜❡ r❡❣✉•❛r ❛t ❛••✳ ■♥ ❢❛❝t✱ ♦♥❝❡ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞✱ ♦♥❡ ❤❛s t❤❡ tr❛♥s✈❡rs❛•✐t② ❡q✉❛t✐♦♥ ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ✭s❡❝t✐♦♥ ✶✳✸✳✷✮ ✇❤✐❝❤ ✐♠♣♦s❡s t❤❛t ✿ p 1 = -∇ q 1 g ❚❤✐s s✉❣❣❡sts t❤❛t t❤❡ ♠♦♠❡♥t❛ ♠✉st ❜❡ s✐♥❣✉•❛r ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ X ✭❛s②♠♣✲ t♦t✐❝❛••② ✐♥ n 1-1/d ✐♥ t❤❡ ❞✐s❝r❡t❡ s❡tt✐♥❣✮✱ ❞✉❡ ♦♥❝❡ ♠♦r❡ t♦ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❙✉❝❤ s✐♥❣✉•❛r✐t✐❡s ❛r❡ ❞✐✣❝✉•t t♦ ❝r❡❛t❡ ❜② t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ♣r♦❝❡ss ❜❡❝❛✉s❡ ∇ p J ✐s ♦❜t❛✐♥❡❞ ♥✉♠❡r✐❝❛••② ❛s ❛ r❡❣✉•❛r✐③❡❞ ✈❡❝t♦r ✜❡•❞ t❤r♦✉❣❤ t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥✳ ■t r❡s✉•ts t❤❛t t❤❡ ❢r❛♠❡✇♦r❦ ✇✐•• r❡q✉✐r❡ ♥✉♠❡r♦✉s ❞❡s❝❡♥t st❡♣s ❢♦r s✉r❢❛❝❡s ✇✐t❤ ❜♦✉♥❞❛r✐❡s ✐♥ ♦r❞❡r t♦ r❡♣r♦❞✉❝❡ t❤✐s ♠♦♠❡♥t✉♠✬s s✐♥❣✉•❛r✲ ✐t✐❡s✱ ❛s ✇❡ ❤❛✈❡ ❛•s♦ ♦❜s❡r✈❡❞ ✐♥ ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✳ ❚❤✉s✱ ❛ ✇❛② t♦ ✐♥❝r❡❛s❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ s♣❡❡❞ ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ✐s t♦ s✐♠♣•② ❡①tr❛❝t t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ✭✇❤✐❝❤ ❝❛♥ ❜❡ ❞♦♥❡ ♦♥•② ♦♥❝❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ s✐♥❝❡ t❤❡ ♣r❡✈✐♦✉s r❡❣✉•❛r✐③❛t✐♦♥ ♦♥ t❤❡ t❡♠♣•❛t❡ ♣♦s✐t✐♦♥✬s ❡✈♦•✉t✐♦♥ ❡♥s✉r❡s t❤❛t ❜♦✉♥❞❛r② ♣♦✐♥ts ✇✐•• r❡♠❛✐♥ ❜♦✉♥❞❛r② ♣♦✐♥ts ❞✉r✐♥❣ t❤❡ ✇❤♦•❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✮ ❛♥❞✱ ❛❢t❡r ❡❛❝❤ ❣r❛❞✐❡♥t ❝♦♠♣✉t❛t✐♦♥✱ ♠✉•t✐♣•② ∇ p J ♦♥ t❤❡ ❜♦✉♥❞❛r② ❜② ❛ ❢❛❝t♦r n 1/d ✳ ❆s ✇❡ ❤❛✈❡ ❝♦♥✜r♠❡❞ ♦♥ ❛ ❢❡✇ ♥✉♠❡r✐❝❛• ❡①❛♠♣•❡s✱ t❤✐s str❛✐❣❤t❢♦r✇❛r❞ ♣r♦❝❡❞✉r❡ ❛••♦✇s t♦ s✐❣♥✐✜❝❛♥t•② ❞❡❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ❢♦r ❛ ♣r❡❞❡✜♥❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤r❡s❤♦•❞ ✉♣ t♦ ❛ ❢❛❝t♦r ✷ ♦r ✸✳ ❋✐rst r❡s✉•ts ❛♥❞ ❞✐s❝✉ss✐♦♥ ❲❡ ♥♦✇ ❛♣♣•② ❛•❣♦r✐t❤♠ ✷✱ ✇✐t❤ t❤❡ ❛❞❞✐t✐♦♥❛• ♥♦r♠❛•✐③❛t✐♦♥s ❛♥❞ r❡❣✉•❛r✐③❛t✐♦♥s ♦❢ ❣r❛❞✐❡♥ts ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥s✱ ❢♦r s❡✈❡r❛• s✐♠✉•❛t❡❞ ❛♥❞ r❡❛• ❞❛t❛s❡ts ✐♥✈♦•✈✐♥❣ s❡ts ♦❢ s❤❛♣❡s ❝❛rr②✐♥❣ ❞✐✛❡r❡♥t r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s✳ ❆s ♠❡♥t✐♦♥❡❞✱ t❤❡ ❛•❣♦r✐t❤♠ ❡st✐♠❛t❡s ❛ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) t❤❛t ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❜❛s❡•✐♥❡ ♦❢ t❤❡ ❞❛t❛ ❡♥s❡♠❜•❡ ❛♥❞ ❛ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♠❛♣♣✐♥❣ t❤❡ t❡♠♣•❛t❡ ♦♥ ❡❛❝❤ s✉❜•❡❝t r❡♣r❡s❡♥t❡❞ ❜② t❤❡✐r ✐♥✐t✐❛• ♠♦♠❡♥t❛ (p i )✳ ❲❡ ✜rst s❤♦✇✱ ♦♥ t❤❡ s✐♠♣•❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡s ✹✳✸ ❛♥❞ ✹✳✹✱ t❤❡ ✐♥t❡r❡st ♦❢ ❞♦✐♥❣ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❡st✐♠❛t✐♦♥ t♦❣❡t❤❡r ✐♥st❡❛❞ ♦❢ s❡♣❛r❛t✐♥❣ ❜♦t❤✳ ❚❤✐s ✐s ♣r❡❝✐s❡•② ♠❛❞❡ ♣♦ss✐❜•❡ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❆s ✇❡ s❡❡✱ t❤♦s❡ ♥♦r♠s ❡♥❢♦r❝❡ ♠♦♠❡♥t❛ t♦ ♠❛t❝❤ r❡❣✐♦♥s ✇✐t❤ s✐♠✐•❛r s✐❣♥❛•s ♦♥ ❡❛❝❤ ♦t❤❡r ✇❤✐❝❤ ❤❛s ❛•s♦ ✐♠♣♦rt❛♥t ✐♠♣❛❝t ♦♥ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ❢s❤❛♣❡ ✐ts❡•❢✳ ■♥❞❡❡❞✱ t❤❡ s✐❣♥❛• ✇❡ ♦❜t❛✐♥ ♦♥ t❤❡ t❡♠♣•❛t❡ ✐s ♠✉❝❤ ❝•♦s❡r t♦ t❤❡ ❡①♣❡❝t❡❞ ♣❛tt❡r♥ t❤❛♥ t❤❡ ♦♥❡ ❣✐✈❡♥ ❜② s❡♣❛r❛t✐♥❣ ❣❡♦♠❡tr② ❛♥❞ s✐❣♥❛• ✭✜❣✉r❡ ✹✳✹✮ ✇❤✐❝❤ ❜❛s✐❝❛••② ❛✈❡r❛❣❡s t❤❡ s✐❣♥❛•s •♦❝❛t❡❞ ❛t ❝♦rr❡s♣♦♥❞✐♥❣ ✶✹✵ ❋✐❣✉r❡ ✹✳✸✿ ❚❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♦♥ ❛ s❡t ♦❢ t❤r❡❡ ❝♦•♦r❡❞ ❝✉r✈❡s ❜❛s❡❞ ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❚❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥s ❛r❡ ❡st✐♠❛t❡❞ ✉s✐♥❣ ❜♦t❤ ❣❡♦♠❡tr✐❝❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡ t❤r♦✉❣❤ t❤❡ ❢❝✉rr❡♥t ♥♦r♠✳ ❖♥ t❤❡ ✜rst r♦✇✱ ✇❡ s❤♦✇ t❤❡ t❤r❡❡ s✉❜•❡❝ts ❛♥❞ t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ✭♦♥ t❤❡ •❡❢t ✜❣✉r❡✮ ❛♥❞ t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t❡❞ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ✭r✐❣❤t ✜❣✉r❡✮✳ ❖♥ t❤❡ s❡❝♦♥❞ r♦✇ ❛r❡ ❞✐s♣•❛②❡❞ t❤❡ tr❛•❡❝t♦r✐❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❞❡❢♦r♠❛t✐♦♥s ❢r♦♠ t❤❡ t❡♠♣•❛t❡ t♦ ❡❛❝❤ ♦♥❡ ♦❢ t❤❡ s✉❜•❡❝ts✳ ✶✹✶ ❋✐❣✉r❡ ✹✳✹✿ ❚❤❡ s❛♠❡ ❡①❛♠♣•❡ ❢♦r ✇❤✐❝❤ t❤❡ ❣❡♦♠❡tr② ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❛r❡ ❝♦♠♣✉t❡❞ s❡♣❛r❛t❡•②✱ ❜② ✜rst ❡st✐♠❛t✐♥❣ ❛ t❡♠♣•❛t❡ s❤❛♣❡ ❜② ❝✉rr❡♥ts ❛♥❞ t❤❡♥ ♣✉••✐♥❣ ❜❛❝❦ s✐❣♥❛•s ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t s✉❜•❡❝ts ♦♥ t❤❡ t❡♠♣•❛t❡✳ ❖❜s❡r✈❡ t❤❛t t❤❡ r❡s✉•t✐♥❣ s✐❣♥❛• ♦♥ t❤❡ t❡♠♣•❛t❡ ✐s ❛ ♠♦r❡ s♣r❡❛❞ ❛♥❞ ❛✈❡r❛❣❡❞ ❝r❡♥❡•✳ ♣♦s✐t✐♦♥s ❜✉t ♣r♦✈✐❞❡❞ ❜② ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• ♠❛t❝❤✐♥❣s✳ ❚❤✐s ♦❜s❡r✈❛t✐♦♥ ❛r❣✉❡s ✐♥ ❢❛✈♦r ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ✇❡ ♣r♦♣♦s❡ ❜❡❝❛✉s❡✱ ✐♥ ♠❛♥② ❝❛s❡s✱ ✐t ♠❛❦❡s ♣♦ss✐❜•❡ t♦ •❡❛r♥ s✐❣♥❛•s ✇✐t❤ ❛ ❜❡tt❡r ❛❝❝✉r❛❝② ❛♥❞ r❡s♦•✉t✐♦♥ t❤❛♥ ❛•• ♠❡t❤♦❞s t❤❛t ✐♥t❡♥❞ t♦ ❞❡❝♦✉♣•❡ t❤❡ s❤❛♣❡ ❢r♦♠ t❤❡ ❢✉♥❝t✐♦♥❛• ✐♥❢♦r♠❛t✐♦♥✳ ❖♥ t❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✹✳✺✱ ✇❡ s❤♦✇ t❤❡ ✐♥t❡r❡st ♦❢ t❤❡ ❣r❛❞✐❡♥t r❡❣✉•❛r✐③❛t✐♦♥ ❢♦r t❤❡ t❡♠♣•❛t❡ ❞✐s♣•❛❝❡♠❡♥t ∇ q J ❞❡✈❡•♦♣❡❞ ✐♥ ✹✳✷✳✶✳ ❖❜s❡r✈❡ ✐♥ ♣❛rt✐❝✉•❛r✱ ❡✈❡♥ ♦♥ s✉❝❤ ❛ s✐♠♣•❡ ❡①❛♠♣•❡✱ t❤❡ ❝❤❛♥❣❡ ♦❢ t♦♣♦•♦❣② ♦❢ t❤❡ t❡♠♣•❛t❡ ✐♥ t❤❡ ♥♦♥✲r❡❣✉•❛r✐③❡❞ ❝❛s❡ ✇❤✐❝❤ ✐s ♠❛✐♥•② ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❤✐❣❤❡r ❣r❛❞✐❡♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r✐❡s ✇❤✐❝❤ ❝❛♥ ♣r♦✈♦❦❡ ✉♥❞❡s✐r❡❞ s❡•❢✲❝r♦ss✐♥❣s✳ ❚❤❡ t❤✐r❞ ❡①❛♠♣•❡ ✐s ❛ t❡♠♣•❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❛ s❡t ♦❢ ✬❝✉♣✬ s✉r❢❛❝❡s ✇✐t❤ •❛✉ss✐❛♥ ❞❡❝r❡❛s✐♥❣ s✐❣♥❛•s ❝❡♥t❡r❡❞ ❛t t❤❡ ❜❛s❡✳ ◆♦t✐❝❡ t❤❡ ❡✈♦•✉t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❢s❤❛♣❡ t❤r♦✉❣❤♦✉t t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣s ✹✳✷✳✷ ❊st✐♠❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ❆ r✐❝❤❡r ♠♦❞❡• ❢♦r ❢✉♥❝t✐♦♥❛• ✈❛r✐❛❜✐•✐t② ❚❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ♦❢ ✹✳✷✳✶ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♣r♦✈✐❞✐♥❣ ❛ ❣❡♦♠❡tr✐❝♦✲ ❢✉♥❝t✐♦♥❛• ❜❛s❡•✐♥❡ ❢♦r ❛ ❣✐✈❡♥ ❞❛t❛s❡t✱ ❛♥❞ ❞❡❢♦r♠❛t✐♦♥s ✇❤♦s❡ ✈❛r✐❛❜✐•✐t② ❝❛♥ ❜❡ t❤❡♥ st❛t✐st✐❝❛••② ❛♥❛•②s❡❞ ❜❛s❡❞ ♦♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② t❤❡ ♠♦♠❡♥t❛✱ ❛s ♣✐♦♥❡❡r❡❞ ✐♥ ❬✼✾(✳ ❚❤✐s s❤❛•• ❜❡ ❞❡t❛✐•❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✸✳ ❍♦✇❡✈❡r✱ ✐t ♦♥•② ❛❝❝♦✉♥ts ❢♦r t❤❡ ❣❡♦♠❡tr✐❝❛• ✈❛r✐❛❜✐•✐t② ♦❢ t❤❡ ❞❛t❛s❡t ❛r♦✉♥❞ t❤❡ t❡♠♣•❛t❡✳ ■♥ ❣❡♥❡r✐❝ s✐t✉❛t✐♦♥s✱ s✐❣♥❛•s ❛r❡ ❛•s♦ •✐❦❡•② t♦ ✈❛r② ❛❝r♦ss ✐♥❞✐✈✐❞✉❛•s ❛♥❞ t❤✐s ✈❛r✐❛❜✐•✐t② ❝❛♥ ❤❛✈❡ ❛ st❛t✐st✐❝❛• s✐❣♥✐✜❝❛♥❝❡✳ ❋♦r ❡①❛♠♣•❡✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❖❈❚ ❞❛t❛s❡t ♣r❡s❡♥t❡❞ ♣r❡✈✐♦✉s•②✱ t❤❡ t❤✐❝❦♥❡ss ✐s ❡①♣❡❝t❡❞ t♦ ❞❡❝r❡❛s❡ ✐♥ s❡✈❡r❛• •♦❝❛t✐♦♥s ❛s ❛ ✶✹✷ ■♥✐t✐❛• t❡♠♣•❛t❡ ◆♦♥✲r❡❣✉•❛r✐③❡❞ ■t ✷ ■t ✺ ■t ✸✵ |❡❣✉•❛r✐③❡❞ ❋✐❣✉r❡ ✹✳✺✿ ❙❡✈❡r❛• ✐♥t❡r♠❡❞✐❛t❡ ✐t❡r❛t✐♦♥s ✐♥ t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ❜♦t❤ ❢♦r ♥♦♥✲r❡❣✉•❛r✐③❡❞ ❛♥❞ r❡❣✉•❛r✐③❡❞ ❣r❛❞✐❡♥ts✳ ❋✐❣✉r❡s s❤♦✇ t❤❡ ❞❛t❛s❡t ♦❢ t❡♥ ❝•♦s❡ ♣❛r❛❜♦•❛s ✭✇❤✐❝❤ ❛r❡ ♣✉r❡ ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s ✐♥ t❤❛t ❝❛s❡✮ ❛♥❞ t❤❡ ❡✈♦•✉t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ✭✐♥ ❜•✉❡✮ ❛t ❞✐✛❡r❡♥t st❡♣s ♦❢ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✳ ✶✹✸ ■♥✐t✐❛• ❙t❡♣ ✺ ❙t❡♣ ✶✺ ❙t❡♣ ✷✺ ❙t❡♣ ✺✵ ❋✐❣✉r❡ ✹✳✻✿ ❚❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♦♥ ❛ s❡t ♦❢ ✻ ✬❝✉♣✬ s❤❛♣❡s ✇✐t❤ ❛rt✐✜❝✐❛• •❛✉ss✐❛♥ s✐❣♥❛•s✳ ❖♥ t♦♣✱ t❤❡ ❡✈♦•✉t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❞✉r✐♥❣ ❣r❛❞✐❡♥t ❞❡s❝❡♥t✳ ❖♥ t❤❡ ❜♦tt♦♠✱ ✇❡ s❤♦✇ t❤❡ ✜♥❛• r❡s✉•ts ♦❢ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ ❛♥❞ ❡❛❝❤ ♦❢ t❤❡ s✉❜•❡❝ts ✿ t❤❡ ♠♦r❡ tr❛♥s♣❛r❡♥t s✉r❢❛❝❡ ✐s t❤❡ s✉❜•❡❝t✱ t❤❡ s♦•✐❞ ♦♥❡ ✐s t❤❡ ❞❡❢♦r♠❡❞ t❡♠♣•❛t❡✳ ✶✹✹ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣•❛✉❝♦♠❛ s♦ t❤❛t ✐♥t❡r✲✐♥❞✐✈✐❞✉❛• t❤✐❝❦♥❡ss ✈❛r✐❛❜✐•✐t② ❝❛♥ ❜❡ ❛ ❤✐❣❤•② r❡•❡✈❛♥t ❜✐♦♠❛r❦❡r t♦ ♣❡r❢♦r♠ ❝•❛ss✐✜❝❛t✐♦♥✳ ❲✐t❤ ♦✉r ♣r❡✈✐♦✉s ♠♦❞❡•✱ t❤✐s ✈❛r✐❛❜✐•✐t② ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ r❡s✐❞✉❛•s ♦❢ ♠❛t❝❤✐♥❣ t♦ ❡❛❝❤ s✉❜•❡❝t✱ ♥❛♠❡•② t❤❡ ❢❝✉rr❡♥ts ❛ss♦❝✐❛t❡❞ t♦ S i -(φ v i 0 ) * S✱ ✇❤✐❝❤ ❝♦✉•❞ ❜❡ ❛•s♦ ❛♥❛•②s❡❞ ❢r♦♠ ❛ st❛t✐st✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❛s ❡•❡♠❡♥ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• s♣❛❝❡ ♦❢ ❢❝✉rr❡♥ts✳ ❚❤❡ ❡ss❡♥t✐❛• ❞r❛✇❜❛❝❦ ✐s t❤❛t t❤❡s❡ t❡r♠s ❞♦ ♥♦t ♦♥•② r❡♣r❡s❡♥t t❤❡ ❢✉♥❝t✐♦♥❛• ✈❛r✐❛❜✐•✐t② ❜✉t ❛•s♦ ✐♥❝•✉❞❡ ❤✐❣❤✲❢r❡q✉❡♥❝② ❣❡♦♠❡tr✐❝❛• ✈❛r✐❛t✐♦♥s ♥♦t ❝❛♣t✉r❡❞ ❜② t❤❡ ❞❡❢♦r♠❛t✐♦♥s ❛s ✇❡•• ❛s ♥♦✐s❡ ❛♥❞ ❛rt✐❢❛❝ts ♦❢ t❤❡ ❞❛t❛s❡ts✳ ■♥ ❛❞❞✐t✐♦♥✱ s✉❝❤ r❡s✐❞✉❛•s ❛r❡ ♥❡❛r•② ✐♠♣♦ss✐❜•❡ t♦ ✈✐s✉❛•✐③❡ ❡①❝❡♣t ❛s •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❉✐r❛❝ ❢❝✉rr❡♥ts ❛♥❞ t❤✉s ♠❛❦✐♥❣ t❤❡ ✐♥t❡r♣r❡t❛❜✐•✐t② ♦❢ ❞✐s❝r✐♠✐♥❛t✐✈❡ ❞✐r❡❝t✐♦♥s ❢♦r P❈❆ ♦r ▲❉❆ ♣r❛❝t✐❝❛••② ✐♥tr❛❝t❛❜•❡✳ ❚❤✐s ♠♦t✐✈❛t❡s ❛♥ ❛❞❛♣t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢r❛♠❡✇♦r❦ t❤❛t ✇♦✉•❞ ❡st✐♠❛t❡✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ t❡♠♣•❛t❡ ❛♥❞ ❞❡❢♦r♠❛t✐♦♥s✱ ❛ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛• ❢♦r ❡❛❝❤ s✉❜•❡❝t✱ ♥❛♠❡•② ❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ t❡♠♣•❛t❡ t❤❛t ♠❡❛s✉r❡s t❤❡ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ ❜❛s❡✲ •✐♥❡ ❛♥❞ s✉❜•❡❝t✬s ❢✉♥❝t✐♦♥❛•✳ ❚❤✐s ✐s ♣r❡❝✐s❡•② t❤❡ ♠♦❞❡• ✐♥tr♦❞✉❝❡❞ ✐♥ ✶✳✹✳✶ ❛♥❞ ❛•r❡❛❞② ✐♠♣•❡♠❡♥t❡❞ ❢♦r r❡❣✐str❛t✐♦♥ ✐♥ ✷✳✸✳✷✳ ■♥ t❤❛t ❝❛s❡✱ ❛ ❣❡♥❡r❛t✐✈❡ ♠♦❞❡• t❤❛t ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ ✐s t❤❡ ❢♦••♦✇✐♥❣ ✿ (X i , f i ) = φ i (X), (f + ζ i ) • φ i,-1 + ǫ i ✭✹✳✶✹✮ ❚❤❡ ❞❛t❛ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ❣❡♥❡r❛t❡❞ ❜② ❛ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✇✐t❤ ❛ s✉❜♠❛♥✲ ✐❢♦•❞ X ❛♥❞ ❛ s✐❣♥❛• f ♦♥ ✇❤✐❝❤ ✐s ❛❞❞❡❞ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥s ζ i : X → R ❛♥❞ ❞❡❢♦r♠❡❞ ❜② ❞✐✛❡♦♠♦r♣❤✐s♠ φ i ✳ ◆♦t❡ t❤❛t ✇❡ ❝♦♥s✐❞❡r t❤❡ r❡s✐❞✉❛•s (ζ i ) t♦ ❜❡ ❞❡✜♥❡❞ ♦♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ❛♥❞ t❤✉s ❛❞❞❡❞ ❜❡❢♦r❡ ❞❡❢♦r♠❛t✐♦♥ ✐♥st❡❛❞ ♦❢ ❛❢t❡r✳ ❚❤❡ r❡❛s♦♥ ❢♦r t❤✐s ❝❤♦✐❝❡ ✐s ❡ss❡♥t✐❛••② t❤❛t r❡s✐❞✉❛•s ✇✐•• t❤❡♥ s❤❛r❡ ❛ ❝♦♠♠♦♥ ❣❡♦♠❡tr✐❝❛• s✉♣✲ ♣♦rt✱ ❡♥❛❜•✐♥❣ ❡❛s✐❡r ♥✉♠❡r✐❝❛• ❡st✐♠❛t✐♦♥ ❛♥❞ ❛ ♠✉❝❤ ♥✐❝❡r ❢r❛♠❡✇♦r❦ ❢♦r st❛t✐st✐❝s ✭❝❢ s❡❝t✐♦♥ ✹✳✸✮✳ ❏✉st ❛s ✇❡ ❞✐❞ ❢♦r t❤❡ ♣r❡✈✐♦✉s ♠♦❞❡•✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ ❛ ❢♦r♠❛• ▼❆P ❡st✐♠❛t♦r ❛ss♦❝✐❛t❡❞ t♦ ❡q✳✭✹✳✶✹✮✳ ❲❡ r❡♣r❡s❡♥t ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❜② ❢❝✉rr❡♥ts ❛♥❞ ❛ss✉♠❡ •❛✉ss✐❛♥ ❞❡♥s✐t② ♣r✐♦rs ♦♥ t❤❡ r❡s✐❞✉❛• ❢❝✉rr❡♥ts ǫ i ✳ ❲❡ ❤❛✈❡ ♥♦✇ ❤✐❞❞❡♥ ✈❛r✐❛❜•❡s t❤❛t ❛r❡ ❜♦t❤ t❤❡ ❞❡❢♦r♠❛t✐♦♥s φ i ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ ✐♥✐t✐❛• ♠♦♠❡♥t❛ p i 0 ♦♥ ✇❤✐❝❤ ✇❡ ❛ss✉♠❡ ❛ ♣r✐♦r ♣r♦❜❛❜✐•✐t② ❞✐str✐❜✉t✐♦♥ ❛s ❣✐✈❡♥ ❜② ❡q✳✭✹✳✸✮✱ ❛♥❞ t❤❡ r❡s✐❞✉❛•s ζ i ✇❤✐❝❤ ✇❡ s❤❛•• ❝♦♥s✐❞❡r ❛s L 2 ❢✉♥❝t✐♦♥s ♦♥ X ✇✐t❤ ❛ ♣r✐♦r ❞✐str✐❜✉t✐♦♥ ✿ p res (ζ|X) = C res . exp(-ζ 2 L 2 (X) /(2σ 2 res )) ✭✹✳✶✺✮ ✇❤❡r❡ ✇❡ r❡♠✐♥❞ t❤❛t ζ L 2 (X) st❛♥❞s ❢♦r t❤❡ L 2 ♥♦r♠ ♦♥ t❤❡ s✉❜♠❛♥✐❢♦•❞ X✳ ❚❤❡♥✱ ♦♥❡ ❝❛♥ ✇r✐t❡ ❛♥ ❡q✉✐✈❛•❡♥t ▼❆P ❡st✐♠❛t♦r ✐♥ t❤❛t ❝♦♥t❡①t ❛♥❞ ✉s✐♥❣ t❤❡ ❋❛st ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ▼♦❞❡s✱ ✐t •❡❛❞s t♦ ❛ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜•❡♠ ♦❢ t❤❡ ❢♦r♠ ✿ min (X,f,p i 0 ,ζ i ) N i=1 E(p i 0 ) + γ res ζ i 2 L 2 (X) + γ W C (X i ,f i ) -C φ p i 0 (X),(f +ζ i )•(φ p i 0 ) -1 2 W ′ ✭✹✳✶✻✮ ✇❤❡r❡ γ res , γ W > 0 ❛r❡ t❤❡ ✇❡✐❣❤t ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ t❡r♠s✳ ❈♦♠♣❛r❡❞ t♦ t❤❡ ♣r❡✈✐♦✉s ♠♦❞❡• ♦❢ ❡q✳✭✹✳✽✮✱ ✇❡ ❤❛✈❡ ❛❞❞✐t✐♦♥❛• ✈❛r✐❛❜•❡s ζ i t♦ ❡st✐♠❛t❡ ❛♥❞ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛♥ ❛❞❞✐t✐♦♥❛• t❡r♠ ✐♥ t❤❡ ❢✉♥❝t✐♦♥❛• t❤❛t ❞❡♣❡♥❞s ♦♥ t❤❡ ζ i ✬s ❛♥❞ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ X✳ ◆♦t❡ t❤❛t ✐❢ X ❛♥❞ f ❛r❡ ✜①❡❞✱ t❤❡♥ t❤❡ ♣r♦❜•❡♠ ❣✐✈❡♥ ✐♥ ❡q✳✭✹✳✽✮ r❡❞✉❝❡s t♦ s♦•✈❡ t❤❡ r❡❣✐str❛t✐♦♥ ✇✐t❤ r❡s✐❞✉❛• ♦❢ (X, f ) ♦♥ ❡❛❝❤ (X i , f i )✱ ✇❤✐❝❤ ✐s ❡①❛❝t•② ✇❤❛t ✇❛s st✉❞✐❡❞ ✐♥ ❝❤❛♣t❡r ✷ s❡❝t✐♦♥ ✷✳✸✳✷✳ ❆•❣♦r✐t❤♠✐❝❛••②✱ t❤❡ ❢r❛♠❡✇♦r❦ ❞♦❡s♥✬t ❝❤❛♥❣❡ ♠✉❝❤ ❡①❝❡♣t ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛• ❣r❛❞✐❡♥ts✱ ✇❤✐❝❤ ✇❡ ❞❡t❛✐• ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✳ ✶✹✺ ◆❡✇ ❛•❣♦r✐t❤♠ ❆s ✐♥ ❛•❣♦r✐t❤♠ ✷✱ ✇❡ ♣r♦♣♦s❡ ❛ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ❛•❣♦r✐t❤♠ t❤❛t s✐♠✉•t❛♥❡♦✉s•② ♦♣t✐♠✐③❡s t❤❡ ❢✉♥❝t✐♦♥❛• ✇✐t❤ r❡s♣❡❝t t♦ ❛•• ✈❛r✐❛❜•❡s t♦❣❡t❤❡r✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s✱ ✇❡ ♦♥•② ♥❡❡❞ t♦ ♠❛❦❡ ❡①♣•✐❝✐t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♥❡✇ t❡r♠s E f (ζ i ) . = ζ i 2 L 2 (X) ❛♥❞ ❡①♣r❡ss t❤❡ ❣r❛❞✐❡♥ts ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥❡✇ ✈❛r✐❛❜•❡s ζ i ✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ t❤❡ L 2 ✲♥♦r♠s E f (ζ i ) ❛r❡ ❛•♠♦st str❛✐❣❤t❢♦r✇❛r❞ ❜❡❝❛✉s❡✱ ❢♦r ❛ ❞✐s❝r❡t✐③❡❞ s❤❛♣❡ X ❛♥❞ s✐❣♥❛•s ζ i ✱ t❤❡s❡ ❛r❡ ❜❛s✐❝❛••② t❤❡ s✉♠ ♦❢ t❤❡ ✈♦•✉♠❡s ♦❢ t❤❡ ♠❡s❤ ❝❡••s ✇❡✐❣❤t❡❞ ❜② t❤❡ ✐♥t❡r♣♦•❛t❡❞ ✈❛•✉❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ζ i t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝❡••s✳ ◆❛♠❡•②✱ ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥s ♦❢ s❡❝t✐♦♥ ✹✳✷✳✶✱ ✇❡ ❤❛✈❡ ✿ E f (ζ) = m l=1 ζ2 l .|ξ l | ✭✹✳✶✼✮ ✇❤❡r❡ ❢♦r ❡❛❝❤ k✱ qk ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ l th ❝❡••✱ |ξ l | ❡q✉❛•s ✐ts d✲ ❞✐♠❡♥s✐♦♥❛• ✈♦•✉♠❡ ❛♥❞ ζl ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❝❡•• ✈❡rt✐❝❡s✬ s✐❣♥❛• ✈❛•✉❡s✳ ◆♦✇✱ ❞✐✛❡r❡♥t✐❛t✐♥❣ ❡q✳✭✹✳✶✼✮ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥t ♣♦s✐t✐♦♥s s✐♠♣•② r❡q✉✐r❡s t♦ ❝♦♠♣✉t❡ ❣r❛❞✐❡♥ts ♦❢ t❤❡ •♦❝❛• ✈♦•✉♠❡s |ξ l | ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ q k ✬s✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❝♦♠♣✉t❛t✐♦♥ ❛s ✇❡ ❞✐❞ ❢♦r t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❢❝✉rr❡♥t ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s✳ ❚❤❡ ❣r❛❞✐❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡s✐❞✉❛• ❢✉♥❝t✐♦♥s✬ ✈❛•✉❡s (ζ k ) ✐s ✿ ∇ ζ k E f = m l=1 2 ζl ∇ ζ k ζl .|ξ l | ❛♥❞ ❣r❛❞✐❡♥ts ∇ ζ k ζl ❛r❡ s✐♠♣•② ❝♦♠♣✉t❡❞ ❜② r❡❞✐str✐❜✉t✐♦♥ ❢r♦♠ ❝❡♥t❡rs t♦ ✈❡r✲ t✐❝❡s✳ ❋✐♥❛••②✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡ t❡r♠s g i = C (X i ,f i ) - C (φ i (X),(f +ζ i )•φ i,-1 ) 2 W ′ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡s✐❞✉❛•s ζ i ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❡①❛❝t s❛♠❡ ✇❛② ❛s ✇✐t❤ r❡s♣❡❝t t♦ f ✱ ❝❢ ❡q✳✭✹✳✶✷✮✳ ■♥ ❢❛❝t✱ ✇❡ s❡❡ t❤❛t ∇ f g = N i=1 ∇ ζ i g i ✳ ❚❤❡ •❛st ✐♠♣♦rt❛♥t ✐ss✉❡ ✐s t❤❡ ♦♥❡ ♦❢ ❣r❛❞✐❡♥t ♥♦r♠❛•✐③❛t✐♦♥s ❛♥❞ r❡❣✉•❛r✐③❛t✐♦♥s t❤❛t ❤❛s ❜❡❡♥ t❤♦r♦✉❣❤•② ❞✐s❝✉ss❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✷✳✶✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❞✐✛❡r❡♥t ❝♦♥❝•✉✲ s✐♦♥s ❤♦•❞ ✇✐t❤ t❤✐s ♥❡✇ ♠♦❞❡• ❛♥❞ ✇❡ ♦♥•② ♥❡❡❞ t♦ ❡①❛♠✐♥❡ t❤❡ ♥❡✇ ♣❡♥❛•✐③❛t✐♦♥ t❡r♠s ❛♥❞ t❤❡ ❜❡❤❛✈✐♦r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥❡✇ ✈❛r✐❛❜•❡s ζ i ✳ ❆s ❢❛r ❛s s❝❛•❡s ❛r❡ ❝♦♥❝❡r♥❡❞✱ t❤❡ ❛tt❛❝❤♠❡♥t t❡r♠ g ❛♥❞ ∇ ζ i g ❝❛♥ ❜❡ tr❡❛t❡❞ ❡①❛❝t•② t❤❡ s❛♠❡ ❛s ✇✐t❤ r❡s♣❡❝t t♦ f ✭❝❢ ✹✳✷✳✶✮✳ E f (ζ i ) ❜❡✐♥❣ t❤❡ sq✉❛r❡❞ L 2 ♥♦r♠ ♦❢ ζ i ♦♥ X✱ r❡s❝❛•✐♥❣ ❜② λ g ♦♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ♣❛rt ❛♥❞ ❜② λ f ♦♥ t❤❡ s✐❣♥❛• ♣❛rt ♠✉•t✐♣•✐❡s s✉❝❤ t❡r♠s ❜② λ d g λ 2 f ❛♥❞ t❤✉s ✇❡ ❛❝❤✐❡✈❡ s❝❛•❡ ✐♥✈❛r✐❛♥❝❡ ❜② ❛♣♣•②✐♥❣ r❡♥♦r♠❛•✐③❛t✐♦♥ ❢❛❝t♦rs 1/(λ d g λ 2 f ) ♦♥ t❤❡ E f ✳ ❋♦r s✐♠✐•❛r r❡❛s♦♥s✱ t❤❡ ❣r❛❞✐❡♥t ∇ ζ i E f ✐s t♦ ❜❡ r❡♥♦r♠❛•✐③❡❞ ❜② 1/λ d g ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ∇ q E f ❜② λ 2 g /λ 2 f ✳ ❖♥ t❤❡ ❞❡♣❡♥❞❡♥❝② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❡♠♣•❛t❡ s❛♠♣•✐♥❣✱ ✉s✐♥❣ t❤❡ ♥♦t❛t✐♦♥s ❛♥❞ r❡❛s♦♥✐♥❣ ♦❢ s✉❜s❡❝t✐♦♥ ✹✳✷✳✶✱ ✇❡ ❡❛s✐•② s❡❡ t❤❛t ∇ ζ k E f ✐s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ •♦❝❛• ✈♦•✉♠❡ ❡•❡♠❡♥t ♦♥ X (n) ✇❤✐❝❤ ✐s ❛s②♠♣t♦t✐❝❛••② ♦❢ ♦r❞❡r 1/n ❛♥❞ t❤✉s t❤❡s❡ t❡r♠s ♠✉st ❜❡ r❡♥♦r♠❛•✐③❡❞ ❜② ❛ ❢❛❝t♦r n✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❡r♠ E f ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ ♣♦✐♥ts q ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ♣❛rt✐❝✉•❛r ✐♥st❛♥❝❡ ♦❢ ❈❛rt❛♥✬s ❢♦r♠✉•❛ ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✭t❛❦✐♥❣ ω(x, m) = ζ(x)✮✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐t ✐♥✈♦•✈❡s s♦♠❡ ❜♦✉♥❞❛r② ❡✛❡❝ts ❛s ✐♥ ✹✳✷✳✶✳ ❆ ❣♦♦❞ s♦•✉t✐♦♥ ✐s ❛❣❛✐♥ t♦ r❡❣✉•❛r✐③❡ t❤❡ ❣r❛❞✐❡♥t ❜② t❤❡ ❦❡r♥❡• K reg (q, q)✳ ❲❡ s✉♠ ✉♣ ❛•• t❤❡ ❞✐✛❡r❡♥t st❡♣s ♦❢ t❤❡ ❢✉•• ♣r♦❝❡ss ✐♥ ❛•❣♦r✐t❤♠ ✸✳ ◆♦t❡ t❤❛t t❤✐s ♥❡✇ ❛•❣♦r✐t❤♠ ✐s ❛ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ✿ t❛❦✐♥❣ t❤❡ ♣❛r❛♠❡t❡r γ res → +∞ ❛♥❞ ❛ ♥✉•• ❣r❛❞✐❡♥t st❡♣ ✐♥ t❤❡ r❡s✐❞✉❛•s ζ i ✱ ✇❡ ❡①❛❝t•② ❣❡t ❜❛❝❦ t♦ ❛•❣♦r✐t❤♠ ✷✳ ✶✹✻ ❆•❣♦r✐t❤♠ ✸ ❆t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s |❡q✉✐r❡✿ ◆ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s (X i , f i )✱ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❢s❤❛♣❡ (X, f ) = (q k , f k ) k=1,..,n ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣ δ✳ ✶✿ s❡t p i = 0 ❛♥❞ ζ i = 0 ❢♦r ❛•• i ∈ {1, .., N }✳ ✷✿ ✇❤✐•❡ ❈♦♥✈❡r❣❡♥❝❡ ❞♦ ✸✿ ❢♦r ✐❂✶✱✳✳✱◆ ❞♦ ✹✿ ❋•♦✇ (X, f ) ❜② ❢♦r✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ✇✐t❤ p i ❛♥❞ st♦r❡ t❤❡ ❞❡❢♦r♠❡❞ t❡♠♣•❛t❡ ❛t t✐♠❡ ✶ φ v i 0 .(X, f ) = (q i k,1 , f k + ζ i k )✳ ✺✿ ❈♦♠♣✉t❡ t❤❡ ❢❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ (X i , f i ) ❛♥❞ φ v i 0 .(X, f + ζ i )✳ ✻✿ ❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥ts ♦❢ g i ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜♥❛• ♣♦✐♥t ❝♦♥✜❣✉r❛t✐♦♥ q i k,1 ✳ ✼✿ ❆♣♣•② t❤❡ ❜❛❝❦✇❛r❞ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❡q✳✭✹✳✶✵✮ ❛♥❞ ❞❡❞✉❝❡ ∇ q g i ❛♥❞ ∇ p i g i ✳ ✽✿ ❈♦♠♣✉t❡ ∇ q E ❛♥❞ ∇ p i E ✇✐t❤ ❡q✳✭✹✳✾✮✳ ✾✿ ❈♦♠♣✉t❡ ∇ ζ i J = ∇ ζ i g i + ∇ ζ i E f ✶✵✿ ❡♥❞ ❢♦r ✶✶✿ ❉❡❞✉❝❡ ∇ f J = N i=1 ∇ ζ i J✳ ✶✷✿ ◆♦r♠❛•✐③❡ ✇✐t❤ r❡s♣❡❝t t♦ s❝❛•❡ ❛♥❞ s❛♠♣•✐♥❣ ❛•• t❤❡ ❣r❛❞✐❡♥ts✳ ✶✸✿ ❆♣♣•② r❡❣✉•❛r✐③❛t✐♦♥ ❦❡r♥❡• t♦ ∇ q J✳ ✶✹✿ ❯♣❞❛t❡ ✈❛r✐❛❜•❡s ✿ q ← q + δ∇ q J f ← f + δ∇ f J p i ← p i + δ∇ p i J ζ i ← ζ i + δ∇ ζ i J ✶✺✿ ❈♦♠♣✉t❡ ♥❡✇ ❢✉♥❝t✐♦♥❛• J(q, f, (p i ), (ζ i )) ❛♥❞ ❛❞❛♣t st❡♣ δ ❜② •✐♥❡ s❡❛r❝❤✳ ✶✻✿ ❡♥❞ ✇❤✐•❡ ✶✼✿ r❡t✉r♥ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f )✱ N ✐♥✐t✐❛• ♠♦♠❡♥t❛ (p i ) ❛♥❞ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s (ζ i )✳ ✹✳✷✳✸ |❡s✉•ts ♦♥ t❤❡ r❡t✐♥❛ ❞❛t❛s❡t ❲❡ ❤❛✈❡ ❛♣♣•✐❡❞ ❛•❣♦r✐t❤♠s ✷ ❛♥❞ ✸ t♦ ❛ ❞❛t❛s❡t ♦❢ r❡t✐♥❛ ♠❡♠❜r❛♥❡s✳ ❚❤✐s ✐s ❛ ❝✉rr❡♥t •♦✐♥t ♣r♦•❡❝t ✇✐t❤ ❉r ❋❛✐s❛• ❇❡❣✱ ❉r ❊✈❣❡♥✐② ▲❡❜❡❞✱ ❉r ▼❛r✐♥❦♦ ❙❛r✉♥✐❝ ❛♥❞ ❙✐❡✉♥ ▲❡❡ ❢r♦♠ ❇✐♦♠❡❞✐❝❛• |❡s❡❛r❝❤ ❣r♦✉♣ ✐♥ ❙✐♠♦♥ ❋r❛s❡r ❯♥✐✈❡rs✐t②✳ ❚❤❡ ❞❛t❛ ✇❛s ❛❝q✉✐r❡❞ ✉s✐♥❣ ❖♣t✐♠❛• ❈♦❤❡r❡♥❝❡ ❚♦♠♦❣r❛♣❤② ✭❖❈❚✮ ❝♦✉♣•❡❞ ✇✐t❤ ❝♦♠♣r❡ss✐✈❡ s❛♠♣•✐♥❣ ❢♦r r❛♣✐❞ ✐♠❛❣❡ ❛❝q✉✐s✐t✐♦♥✱ ❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✽✼(✳ ❚❤❡ r❛✇ ❞❛t❛ ❝♦♥s✐st ✐♥ ✸❉ ✐♠❛❣❡s ♦❢ ❛ ♣♦rt✐♦♥ ♦❢ t❤❡ r❡t✐♥❛ ❢♦r ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ s✉❜•❡❝ts✱ ✇❤✐❝❤ ❛r❡ t❤❡♥ r❡❣✉•❛r✐③❡❞✳ ❚❤❡ ♥❡①t ♣r❡✲♣r♦❝❡ss✐♥❣ st❡♣ ✐s t❤❡ s❡❣♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t •❛②❡rs ❛♥❞ ♠❡♠❜r❛♥❡s ✐♥ t❤❡ r❡t✐♥❛ ✇❤✐❝❤ ❛r❡ ❜❛s✐❝❛••② ✭❢r♦♠ t❤❡ ♠♦st s✉♣❡r✜❝✐❛• t♦ t❤❡ ❞❡❡♣❡st✮ ✿ t❤❡ ✐♥♥❡r •❛②❡r ♠❡♠❜r❛♥❡ ✭■▲▼✮✱
♣❡ ♦❢ t❤❡ r❡t✐♥❛ •❛②❡rs✳ ■♥ ♦✉r ♣❛rt✐❝✉•❛r ❝❛s❡✱ ✇❡ ✇♦✉•❞ ✇✐s❤ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❡✛❡❝ts ♦❢ ❣•❛✉❝♦♠❛ ❜♦t❤ ♦♥ t❤❡ s❤❛♣❡ ❛♥❞ t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ◆❋▲✳ ❖♥❡ ✉s✉❛• ❡♠♣✐r✐❝❛• ❤②♣♦t❤❡s✐s ❢♦r♠✉•❛t❡❞ ❜② ❝•✐♥✐❝✐❛♥s ✐s t❤❛t ❣•❛✉❝♦♠❛ ✐♥❞✉❝❡s ❛ ❞❡❣❡♥❡r❛t✐♦♥ ♦❢ t✐ss✉❡s ♣r♦✈♦❦✐♥❣ s♦♠❡ •♦❝❛•✐③❡❞ t❤✐❝❦♥❡ss ❞❡❝r❡❛s❡✱ ♣♦ss✐❜•② ❝♦♠✐♥❣ ❛•♦♥❣ ✇✐t❤ ♠❡♠❜r❛♥❡s✬ ❞❡❢♦r♠❛t✐♦♥s✳ ■♥ ♦r❞❡r t♦ ❛ss❡ss ✐t q✉❛•✐t❛t✐✈❡•② ✐♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ ❛♣♣•✐❡❞ ❛•❣♦r✐t❤♠ ✷ t♦ ❡st✐♠❛t❡ ❛ t❡♠♣•❛t❡ ❢♦r t❤❡ ❝♦♥tr♦• ♣♦♣✉•❛t✐♦♥ ❛♥❞ ❛ t❡♠♣•❛t❡ ❢♦r t❤❡ s✉❜•❡❝ts ✇✐t❤ ❝♦♥✜r♠❡❞ ❣•❛✉❝♦♠❛✳ ❇♦t❤ t❤❡s❡ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✹✳✾✳ ■t s❤♦✇s ✐♥❞❡❡❞ q✉✐t❡ ❝•❡❛r•② ❛♥ ❛✈❡r❛❣❡ •♦ss ♦❢ t❤✐❝❦♥❡ss ❢♦r ❣•❛✉❝♦♠❛t♦✉s s✉❜•❡❝ts ❝♦♠♣❛r❡❞ t♦ ❝♦♥tr♦•s ♣❛rt✐❝✉•❛r•② ✐♥ s♦♠❡ ❛r❡❛s ❛r♦✉♥❞ t❤❡ ♦♣❡♥✐♥❣ ❛s ✇❡•• ❛s ❛ r❛t❤❡r s✐❣♥✐✜❝❛♥t ❝❤❛♥❣❡ ✐♥ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❡♠❜r❛♥❡ ❛s ✇❡ s❡❡ ♦♥ t❤❡ ✐♠❛❣❡s ♦❢ t❤❡ ✜rst r♦✇ ✐♥ ✜❣✉r❡ ✹✳✾ ❛♥❞ s♦♠❡ s♠❛•• ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ♦♣❡♥✐♥❣✳ ❚❤✐s✱ ❤♦✇❡✈❡r✱ ❝♦♥st✐t✉t❡ ♦♥•② ❛ q✉❛•✐t❛t✐✈❡ st✉❞② ❛t t❤✐s ♣♦✐♥t✳ ❋♦r t❤❡ t❤✐❝❦♥❡ss ♠❛♣s✱ ✐t ✐s ♦♥•② ❣✐✈✐♥❣ s♦♠❡ ♠❡❛♥ t❤✐❝❦♥❡ss ❢♦r ❜♦t❤ ❝•❛ss❡s ❜✉t ♥♦t t❤❡ ✐♥tr❛✲❝•❛ss ❛♥❞ ✐♥t❡r✲❝•❛ss ❝♦✈❛r✐❛♥❝❡s✳ ❆s ❢♦r t❤❡ s❤❛♣❡s ♦❢ t❤❡ ◆❋▲ s✉r❢❛❝❡s✱ s✐♥❝❡ ✇❡ ❤❛✈❡ t✇♦ t❡♠♣•❛t❡s✱ t❤❡ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥s✬ ♠♦♠❡♥t❛ ❛r❡ •②✐♥❣ ♦♥ t✇♦ ❞✐✛❡r❡♥t s✉r❢❛❝❡s ❛❝❝♦r❞✐♥❣•② t♦ t❤❡ ❝•❛ss✳ ❉♦✐♥❣ ❢✉rt❤❡r st❛t✐st✐❝s ✇♦✉•❞ t❤❡♥ r❡q✉✐r❡ t♦ ❡st✐♠❛t❡ •✐❦❡•✐❤♦♦❞s ♦❢ ❞❡❢♦
❛•❣♦r✐t❤♠ ✷✳ ❚❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ✇❛s✱ ✐♥ ❜♦t❤ ❝❛s❡✱ ❛ sq✉❛r❡ ✇✐t❤ ❛ ❝❡♥t❡r❡❞ ♦♣❡♥✐♥❣✱ ✇✐t❤ ❛•• ❢✉♥❝t✐♦♥❛• ✈❛•✉❡s ❛t ③❡r♦✳ ❋✐❣✉r❡ ✹✳✶✵✿ ❉❡❢♦r♠❛t✐♦♥ ❛♥❞ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ♦❜t❛✐♥❡❞ ❜② ❛•❣♦r✐t❤♠ ✸ ❢♦r t✇♦ s✉❜•❡❝ts✳ ✶✺✵ ❬✻(✳ ■♥st❡❛❞✱ t❤❡ ❛♣♣r♦❛❝❤ ✇❡ ✐♠♣•❡♠❡♥t❡❞ ✇✐t❤ ❛•❣♦r✐t❤♠ ✸ ✐s ♠❡❛♥t t♦ s✐♠♣•✐❢② t❤❡ ❢♦••♦✇✐♥❣ st❛t✐st✐❝❛• ❛♥❛•②s✐s ❛ •✐tt•❡ ❜✐t✳ ■t ❡st✐♠❛t❡s ♦♥•② ♦♥❡ s✐♥❣•❡ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❛♥❞ ❢♦r ❡❛❝❤ s✉❜•❡❝t✱ ❛ ♠♦♠❡♥t✉♠ p i 0 ❢♦r t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ t♦ t❤❡ s✉❜•❡❝t ❛♥❞ ❛ r❡s✐❞✉❛• t❤✐❝❦♥❡ss ζ i s♦ t❤❛t✱ ♥♦✇✱ ❛•• ♠♦♠❡♥t❛ ❛♥❞ r❡s✐❞✉❛•s ❢♦r ❜♦t❤ ❝•❛ss❡s ❛r❡ ❞❡✜♥❡❞ ♦♥ t❤✐s ❝♦♠♠♦♥ t❡♠♣•❛t❡ s✉♣♣♦rt ❛♥❞ ❝❛♥ ❜❡ st✉❞✐❡❞ ❛•• t♦❣❡t❤❡r✳ ❚❤✐s ❡♥❛❜•❡s ❛ ❞❡❡♣❡r st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ t❤❡ ✈❛r✐❛❜✐•✐t② ❛♥❞ t❤❡ ❞❡s✐❣♥ ♦❢ ❝•❛ss✐✜❝❛t✐♦♥ ❛•❣♦r✐t❤♠s ✇❤✐❝❤ ✇❡ ♣♦st♣♦♥❡ t❤❡ ❡①♣♦s✐t✐♦♥✱ t♦❣❡t❤❡r ✇✐t❤ r❡s✉•ts ♦♥ t❤✐s ❞❛t❛s❡t✱ t♦ s❡❝t✐♦♥ ✹✳✸✳ ✹✳✷✳✹ ❆ ✇❡••✲♣♦s❡❞ ❝♦♥t✐♥✉♦✉s ❢r❛♠❡✇♦r❦ ❢♦r ❛t•❛s ❡st✐♠❛t✐♦♥ ❆s ✇❡ ❤❛✈❡ ♠❡♥t✐♦♥❡❞✱ t❤❡ t✇♦ ♣r❡✈✐♦✉s ♠♦❞❡•s ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ t❤❛t ✇❡ ❞❡r✐✈❡❞ ❛r❡ ❡ss❡♥t✐❛••② ✈❛•✐❞ ✐♥ t❤❡ ❞✐s❝r❡t❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ ❛•• ✈❛r✐❛❜•❡s ❛r❡ ❛ss✉♠❡❞ t♦ •✐✈❡ ✐♥ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• s♣❛❝❡s✳ ❋r♦♠ t❤❡ ❛•❣♦r✐t❤♠✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤✐s ✐s ♦❢ ❝♦✉rs❡ ❛ ♥❛t✉r❛• ❛ss✉♠♣t✐♦♥ ❛♥❞ ✇❡ ❤❛✈❡ ❛ss❡ss❡❞ ♦♥ s❡✈❡r❛• ❡①❛♠♣•❡s t❤❛t t❤❡ ❛•❣♦r✐t❤♠s s❡❡♠ t♦ ♣r♦✈✐❞❡ ❡①♣❡❝t❡❞ ❛♥❞ st❛❜•❡ r❡s✉•ts✳ (❡t✱ t❤❡ q✉❡st✐♦♥ ♦❢ ❤❛✈✐♥❣ ❛♥ ✉♥❞❡r✲ •②✐♥❣ ❝♦♥t✐♥✉♦✉s ♠♦❞❡• t❤❛t✱ ❛♠♦♥❣ ♦t❤❡r ♣r♦♣❡rt✐❡s✱ ❣✉❛r❛♥t❡❡s t❤❡♦r❡t✐❝❛••② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r ❢♦r t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❜•❡♠ r❡♠❛✐♥s ❛ ✈❡r② ✐♠♣♦rt❛♥t ♣♦✐♥t ❢♦r ✐♥q✉✐r②✳ ■t ✐s q✉✐t❡ ❝•❡❛r t❤❛t t❤❡ ❢♦r♠✉•❛t✐♦♥ ♦❢ ❡q✳✭✹✳✶✻✮ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❛ ✇❡••✲♣♦s❡❞ ♣r♦❜•❡♠ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s s✐t✉❛t✐♦♥✱ ✜rst ❜❡❝❛✉s❡ ♦♥❡ ♥❡❡❞s t♦ ❣✐✈❡ ❛ ♠❡❛♥✐♥❣ t♦ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦✈❡r t❤❡ ✬s❤❛♣❡✬ X ✐♥ t❤❛t ❝❛s❡✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥t❡♥❞ t♦ ❛❞❞r❡ss t❤❡s❡ ✐ss✉❡s ❢r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• ♣❡rs♣❡❝t✐✈❡ ✿ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♠♣•❡♠❡♥t❛t✐♦♥s ♦❢ s✉❝❤ ❢r❛♠❡✇♦r❦s✱ ❛•t❤♦✉❣❤ ♥♦t ✈❡r② ❢❛r ❢r♦♠ t❤❡ ♦♥❡s ♣r❡✈✐♦✉s•② ♣r❡s❡♥t❡❞✱ ✇✐•• ❜❡ •❡❢t ❢♦r ❢✉t✉r❡ ✇♦r❦✳ ❍❡♥❝❡ t❤❡ ✜rst ♣♦✐♥t ✐s t♦ ❣✐✈❡ ❛ ♣r♦♣❡r s❡tt✐♥❣ ❢♦r t❤❡ s❡❛r❝❤ ♦❢ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ X✱ s✐♥❝❡ t❤❡ s❡t ♦❢ ❛•• ♣♦ss✐❜•❡ s❤❛♣❡s ✭❢♦r ✐♥st❛♥❝❡ ♦❢ ❛•• r❡❝t✐✜❛❜•❡ s✉❜s❡ts✮ •❛❝❦s str✉❝t✉r❡ t♦ ❝♦♥s✐❞❡r t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r t❤✐s ✇❤♦•❡ s❡t ❛♥❞ ✐t ✐s ✈❡r② ✉♥•✐❦❡•② t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❢♦r ❛t•❛s ❡st✐♠❛t✐♦♥ ❝♦✉•❞ ❜❡ ♣r♦✈❡❞ ✐♥ t❤✐s •❛r❣❡ s♣❛❝❡✳ ■♥st❡❛❞✱ ✇❡ ❝❛♥ ❢♦••♦✇ t❤❡ ✐❞❡❛ ♦❢ ❤②♣❡rt❡♠♣•❛t❡ ✐♥tr♦❞✉❝❡❞ ❢♦r ✐♠❛❣❡s ✐♥ ❬✺✼( ❛♥❞ •❛t❡r ❛❞❛♣t❡❞ t♦ s✉r❢❛❝❡ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ✐♥ ❬✺✽(✳ ❋✐①✐♥❣ ❛ ❤②♣❡rt❡♠♣•❛t❡ s❤❛♣❡ X 0 ✱ ✇❡ s❤❛•• •♦♦❦ ❢♦r t❡♠♣•❛t❡s t❤❛t ❛r❡ ❝♦♥str❛✐♥❡❞ t♦ •✐✈❡ ✐♥ t❤❡ ♦r❜✐t ♦❢ X 0 ❢♦r ❛ ❝❡rt❛✐♥ ❣r♦✉♣ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s✳ ◆❛♠❡•②✱ •❡t✬s ❝♦♥s✐❞❡r ❛ s♣❛❝❡ ♦❢ ❛❞♠✐ss✐❜•❡ ✈❡❝t♦r ✜❡•❞s V 0 ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ ❣r♦✉♣ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s G V 0 ✳ ▲❡t X 0 . = G V 0 .X 0 ❜❡ t❤❡ ♦r❜✐t ♦❢ t❤❡ ❤②♣❡rt❡♠♣•❛t❡ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ G V 0 ✳ ❚❤❡♥ ✭❝❢ ✶✳✷✳✸✮ ✇❡ ❤❛✈❡ t❤❡ |✐❡♠❛♥♥✐❛♥ ❞✐st❛♥❝❡ d X 0 ♦♥ X 0 ❞❡✜♥❡❞ ❜② ✿ d X 0 (X 1 , X 2 ) = inf v 0 ∈L 2 V 0 1 0 v 0 t 2 V 0 dt 1 2 / φ v 0 1 .X 1 = X 2 . ■♥ t❤❛t s❡tt✐♥❣✱ ✇❡ ❝❛♥ t❤✉s r❡♣•❛❝❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r X ❜② ❛♥ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r ❛ ✈❡❝t♦r ✜❡•❞ v 0 ✐♥ L 2 V 0 t❤❛t ❞✐s♣•❛❝❡s t❤❡ ✐♥✐t✐❛• ❤②♣❡rt❡♠♣•❛t❡ ✐♥ t❤❡ ♦r❜✐t✳ ◆♦t❡ t❤❛t t❤❡ ❣r❛❞✐❡♥t r❡❣✉•❛r✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡ t❤❛t ✇❡ ❛♣♣•✐❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠s ❜② ❝♦♥✈♦•✈✐♥❣ ❣r❛❞✐❡♥ts t❤r♦✉❣❤ ❛ ❦❡r♥❡• ❛t ❡❛❝❤ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣ ❝❛♥ ❜❡ s❡❡♥ ❛s s♦♠❡ ❦✐♥❞ ♦❢ ❣r❡❡❞② ✇❛② t♦ ❛•s♦ r❡str✐❝t t❤❡ ✉♣❞❛t❡s ♦❢ t❤❡ t❡♠♣•❛t❡ ✐♥ t❤❡ ♦r❜✐t ♦❢ t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ✭t❤✐s ✐s ❢♦r♠✉•❛t❡❞ ❜② ♣r♦♣❡rt② ✹✳✷✳✶✮✳ ❍❡r❡✱ ✇❡ ❤❛✈❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ ♠❡tr✐❝ d X 0 t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣❡♥❛•✐③❡ t❤❡ ❞✐s♣•❛❝❡♠❡♥ts ♦❢ t❤❡ t❡♠♣•❛t❡ ❢r♦♠ ✐ts ✐♥✐t✐❛• ♣♦s✐t✐♦♥ ✿ s✉❝❤ ♣❡♥❛•✐③❛t✐♦♥ s❤♦✉•❞ ❤♦♣❡❢✉••② ❣✉❛r❛♥t❡❡ ❛ ❝♦♠♣❛❝t♥❡ss ♣r♦♣❡rt② ❢♦r X ♥❡❡❞❡❞ ❢♦r ♣r♦✈✐♥❣ ❡①✐st❡♥❝❡ ♦❢ s♦•✉t✐♦♥s t♦ t❤❡ ❛t•❛s ♣r♦❜•❡♠✳ ✶✺✶ ❚❤✉s✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ ❢✉♥❝t✐♦♥❛• ✿ J(X, f, (v i ), (ζ i )) = γ 0 2 d X 0 (X 0 , X) 2 + γ f 2 f L 2 (X) + 1 2 N i=1 1 0 v i t 2 V + γ ζ ζ i 2 L 2 (X) + γ W C (X i ,f i ) -C (φ v i (X),(f +ζ i )•(φ v i ) -1 ) 2 W ′ ✭✹✳✶✽✮ ❈♦♠♣❛r❡❞ t♦ t❤❡ ♦♥❡ ♦❢ ❡q✳✭✹✳✶✻✮✱ ✇❡ ❡ss❡♥t✐❛••② ❛❞❞ ♣❡♥❛•t② t❡r♠s ❢♦r t❤❡ t❡♠♣•❛t❡ ❞✐s♣•❛❝❡♠❡♥t ❢r♦♠ ✐ts ✐♥✐t✐❛•✐③❛t✐♦♥ ❛♥❞ ❢♦r t❤❡ ❜❛s❡•✐♥❡ s✐❣♥❛• f ♦♥ X✳ ❚❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ❢♦r t❤❡ ❣r♦✉♣ ♦❢ ❢s❤❛♣❡s (X i , f i ) ✇r✐t❡s ✿ inf{J(X, f, (v i ), (ζ i )) | X ∈ X 0 , f ∈ L 2 (X), (v i ) ∈ (L 2 V ) N , (ζ i ) ∈ (L 2 (X)) N } ✭✹✳✶✾✮ ◆♦t❡ t❤❛t✱ ❤❡r❡✱ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦✈❡r X ∈ X 0 ♠❛❦❡s s❡♥s❡ ❜❡❝❛✉s❡ ✐t ✐s ❡q✉✐✈❛•❡♥t t♦ ♠✐♥✐♠✐③✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❛ t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞ v 0 ∈ L 2 V 0 ❜② s❡tt✐♥❣ X = φ v 0 (X 0 )✳ Pr♦✈✐♥❣ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r t♦ ❡q✳✭✹✳✶✾✮ ✐s ❤♦✇❡✈❡r ♥♦t ✐♠♠❡❞✐❛t❡✱ t❤❡ ❡ss❡♥t✐❛• r❡❛s♦♥ ❜❡✐♥❣ t❤❛t ✇❡ ❤❛✈❡ ♥♦ ✇❡❛❦ s❡♠✐✲❝♦♥t✐♥✉✐t② ✐♥ L 2 (X) ♦❢ t❤❡ ❛♣♣•✐❝❛t✐♦♥ f → C (X i ,f i ) -C (φ v i (X),(f +ζ i )•(φ v i ) -1 ) 2 W ′ ✳ (❡t t❤❡ r❡s✉•t ✇❡ ❤❛✈❡ ✐s s✉♠♠❡❞ ✉♣ ❜② t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠ ✿ ❚❤❡♦r❡♠ ✹✳✷✳✶✳ ❆ss✉♠❡ t❤❛t W ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥ C 2 0 (E × R, Λ d E * )✱ t❤❛t X 0 ❛♥❞ (X i ) 1≤i≤N ❛r❡ ✜♥✐t❡ ✈♦•✉♠❡ ❜♦✉♥❞❡❞ ♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ t❤❛t f i ∈ L 2 (X i ) ❢♦r 1 ≤ i ≤ N ✳ ❆ss✉♠❡ t❤❛t γ f /γ W ❛♥❞ γ ζ /γ W ❛r❡ •❛r❣❡ ❡♥♦✉❣❤ ✇✐t❤ γ f , γ W , γ ζ ≥ 0 ❛♥❞ γ 0 > 0✳ ❚❤❡♥ J(X, f, (ζ i ), (v i )) . = γ 0 2 d X 0 (X 0 , X) 2 + γ f 2 X |f (x)| 2 dH d (x) + 1 2 N i=1 v i 2 L 2 ([0,1],V ) + γ ζ X |ζ i (x)| 2 dH d (x) + γ W C X i ,f i -C φ v i (X),f +ζ i 2 W ′ ✭✹✳✷✵✮ ❛❝❤✐❡✈❡s ✐ts ♠✐♥✐♠✉♠ ♦♥ {(X, f, ζ i , (v i )) | X ∈ X 0 , f ∈ L 2 (X), ζ = (ζ i ) ∈ L 2 (X) N , (v i ) ∈ L 2 ([0, 1], V ) N } ❛♥❞ ❛♥② ♠✐♥✐♠✐③❡r (X * , f * , (ζ i ❚❤❡ ♣r♦❜•❡♠ ♦❢ ❞✐♠❡♥s✐♦♥❛•✐t② r❡❞✉❝t✐♦♥ ✐s ❛♥ ❡①tr❡♠❡•② ✇✐❞❡ t♦♣✐❝ ✐♥ st❛t✐st✐❝❛• •❡❛r♥✐♥❣ ❛♥❞ ✐t ✐s ❝•❡❛r•② ♦✉t ♦❢ t❤❡ s❝♦♣❡ ♦❢ t❤✐s t❤❡s✐s t♦ ♣r❡t❡♥❞ t♦ ❜❡ ❡①❤❛✉st✐✈❡ ✐♥ ❛♥② ✇❛②✳ ❖✉r ♠♦r❡ ❤✉♠❜•❡ ♣✉r♣♦s❡ ✐s t♦ r❡✈✐❡✇ ❜r✐❡✢② t❤❡ •❡❛r♥✐♥❣ ♠❡t❤♦❞s t❤❛t ✇❡ ❛❝t✉❛••② ❡①♣❡r✐♠❡♥t❡❞ ✇✐t❤ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✳ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✜rst ❝♦♥s✐❞❡r t❤❡ ❣❡♥❡r✐❝ ❢r❛♠❡✇♦r❦ ♦❢ ❞❛t❛ r❡❞✉❝t✐♦♥ ❛♥❞ ♣•❛❝❡ ♦✉rs❡•✈❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❞❛t❛ •②✐♥❣ ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ❡✉❝•✐❞❡❛♥ s♣❛❝❡ R p ✳ ❚❤❡ ❝♦••❡❝t✐♦♥ ♦❢ ✬s✉❜•❡❝ts✬ ✐♥ t❤❛t s❡tt✐♥❣ ✐s t❤✉s ❛ s❡t ♦❢ ♣♦✐♥ts ✐♥ R p t❤❛t ✇❡ ✇✐•• ❞❡♥♦t❡ (x i ) i=1,..,N ❛♥❞ x = 1 N N i=1 x i t❤❡ ♠❡❛♥ ♦❢ t❤❡ ❞❛t❛s❡t✳ ❲❡ ✇✐•• ❛•s♦ ❞❡♥♦t❡ X t❤❡ ❝❡♥t❡r❡❞ ❞❛t❛ ♠❛tr✐① X = (x i -x) ∈ R p×N ✳ ❯♣ t♦ t❤✐s ❣•♦❜❛• tr❛♥s•❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛✱ ✇❡ s❤❛•• ❛ss✉♠❡ ❢r♦♠ ♥♦✇ ♦♥ t❤❛t t❤❡ ❞❛t❛s❡t ✐s ❝❡♥t❡r❡❞✳ Pr✐♥❝✐♣❛• ❈♦♠♣♦♥❡♥t ❆♥❛•②s✐s ❚❤❡ ♠♦st ❝♦♠♠♦♥ ❛♥❞ s✐♠♣•❡ t❡❝❤♥✐q✉❡ ❢♦r ❛♥❛•②③✐♥❣ s✉❝❤ ♠✉•t✐✈❛r✐❛t❡ •✐♥❡❛r ❞❛t❛ ✐s t❤❡ P❈❆ ❛♥❞ ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡•② ✉s❡❞ ✐♥ ♠❛♥② s✐t✉❛t✐♦♥s✳ ❚❤❡ ✐❞❡❛ ❜❡❤✐♥❞ t❤❡ ♠❡t❤♦❞ ✐s t♦ ✜♥❞ ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ ❛ •♦✇❡r ❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ♦♥ ✇❤✐❝❤ t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ♣♦✐♥ts ❤❛s ❛ ♠❛①✐♠✉♠ ✈❛r✐❛♥❝❡✳ ❚❤❡ P❈❆ t❤✉s ❝♦♥s✐sts ✐♥ ❝♦♠♣✉t✐♥❣ ✐t❡r❛t✐✈❡•② t❤❡ ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts w 1 , .., w k , .. ∈ R p ✳ ❚❤❡ ✜rst ♦♥❡ ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ✉♥✐t ✈❡❝t♦r ♠❛①✐♠✐③✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣r♦•❡❝t✐♦♥s x T i w 1 ✐✳❡ w 1 = arg max w =1 X T w 2 = arg max w =1 w T (XX T )w ❚❤❡ k th ❝♦♠♣♦♥❡♥t ✐s ❣✐✈❡♥ ❜② s♦•✈✐♥❣ t❤❡ ♣r❡✈✐♦✉s ♣r♦❜•❡♠ ♦♥ t❤❡ ♥❡✇ ❞❛t❛ ♠❛tr✐① X -k-1 s=1 w s w T s X ♦❜t❛✐♥❡❞ ❜② •✉st r❡♠♦✈✐♥❣ t♦ ❡❛❝❤ ❞❛t❛ ♣♦✐♥t ✐ts ♣r♦•❡❝t✐♦♥ ♦♥ t❤❡ s✉❜s♣❛❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ (k -1) ♣r❡✈✐♦✉s ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts✳ ❲❡ ✇✐•• ❛•s♦ ❝❛•• t❤❡s❡ ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts ♣r✐♥❝✐♣❛• ♠♦❞❡s ♦❢ ✈❛r✐❛t✐♦♥ s✐♥❝❡ t❤❡② ❡♥❝♦❞❡ t❤❡ ✶✺✸ ❞✐r❡❝t✐♦♥s ♦❢ ♠❛✐♥ ✈❛r✐❛t✐♦♥s ✐♥ t❤❡ ❞❛t❛s❡t✳ ❖♥❡ ♦❢ t❤❡ ❣r❡❛t ❛❞✈❛♥t❛❣❡ ♦❢ P❈❆ ✐s t❤❛t ✐t ❝❛♥ ❜❡ ❛•s♦ ❝♦♠♣✉t❡❞ ❞✐r❡❝t•② ❛s t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ❡♠♣✐r✐❝❛• ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① XX T ✐♥ t❤❡ ❞❡❝r❡❛s✐♥❣ ♦r❞❡r ♦❢ t❤❡ ❡✐❣❡♥✈❛•✉❡s ✭❬✹✼(✮✳ ❚❤✉s t❤❡ ♠❛tr✐① W ♦❢ ❛•• t❤❡ ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ ❡✐❣❡♥✈❛•✉❡s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s②♠♠❡tr✐❝ ♠❛tr✐① XX T ✱ ✇❤✐❝❤ ✇r✐t❡s ✿ XX T = W T ΛW ✇❤❡r❡ Λ ✐s t❤❡ ❞✐❛❣♦♥❛• ♠❛tr✐① ♦❢ t❤❡ ✭♣♦s✐t✐✈❡✮ ❡✐❣❡♥✈❛•✉❡s✳ ❖❜s❡r✈❡✱ s✐♥❝❡ t❤❡ r❛♥❦ ♦❢ XX T ✐s •❡ss t❤❛♥ min(N, p)✱ t❤❛t t❤❡r❡ ❛r❡ ❛t ♠♦st min(N, p) ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛•✉❡s✳ ❍♦✇❡✈❡r✱ t❤❡ ❣❡♥❡r❛• ♣❛r❛❞✐❣♠ ✐s t❤❛t t❤❡ ❞❛t❛ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✇✐t❤ ♠✉❝❤ •❡ss ❝♦♠♣♦♥❡♥ts ❛♥❞ t❤❛t ♦♥•② ❛ ❢❡✇ ♦♥❡s s❤♦✉•❞ ❜❡ ❛❜•❡ t♦ ❝❛♣t✉r❡ t❤❡ ❡ss❡♥t✐❛• ♣❛rt ♦❢ t❤❡ st❛t✐st✐❝❛• ✈❛r✐❛❜✐•✐t②✳ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ t❤❡ ❞❡❣❡♥❡r❛t❡ ❝❛s❡ ✇❤❡r❡ ❛•• t❤❡ ❞❛t❛ •✐❡ ✐♥ ❛ •♦✇✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ♦❢ R p ✱ t❤❡ ✜rst ❡✐❣❡♥✈❡❝t♦rs ✇✐•• ♣r❡❝✐s❡•② ❜❡ ✈❡❝t♦rs ♦❢ t❤❛t s✉❜s♣❛❝❡✳ ❚❤❡ ✐♥t❡r❡st ♦❢ P❈❆ ✐s t✇♦❢♦•❞ ✿ ✐t ✐s ✜rst ❛ ✇❛② t♦ ❡①tr❛❝t t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ ♠❛✐♥ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❞❛t❛s❡t ✐♥ t❤❡ ♦r✐❣✐♥❛• s♣❛❝❡ ❜✉t ✐t ♣r♦✈✐❞❡s ✐♥ ❛❞❞✐t✐♦♥ ❛ •♦✇❡r ❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ♦♥ ✇❤✐❝❤ t♦ ♣r♦•❡❝t ❛♥❞ ✈✐s✉❛•✐③❡ t❤❡ ❞❛t❛✳ ◆♦✇✱ ✐♥ s✐t✉❛t✐♦♥s ✇❤❡♥ ♦♥❡ ❤❛s ❛ •❡❛r♥✐♥❣ s❡t ♦❢ ❞❛t❛ ❝♦♥s✐st✐♥❣ ✐♥ s❡✈❡r❛• s✉❜•❡❝ts ❜❡•♦♥❣✐♥❣ t♦ ❞✐✛❡r❡♥t ❦♥♦✇♥ ❣r♦✉♣s ♦r ❝•❛ss❡s✱ t❤✐s r❡❞✉❝❡❞ s✉❜s♣❛❝❡ ❝❛♥ ❜❡ t❤❡♥ ✉s❡❞ t②♣✐❝❛••② t♦ •❡❛r♥ ❛ ❝•❛ss✐✜❡r ❢r♦♠ t❤❡ ❞❛t❛✱ ❢♦r ✐♥st❛♥❝❡ ✉s✐♥❣ ✈❛r✐♦✉s ❝•✉st❡r✐♥❣ ♠❡t❤♦❞s ✐♥ t❤❡ P❈❆ s✉❜s♣❛❝❡s✳ |❡♠❛r❦ ✹✳✸✳✶✳ ❆♥ ✐♠♣♦rt❛♥t t❤✐♥❣ t♦ ♥♦t❡ ❛•s♦ ✐s t❤❛t P❈❆ ✐s ❜❛s❡❞ ❡①❝•✉s✐✈❡•② ♦♥ t❤❡ ❝♦✈❛r✐❛♥❝❡ str✉❝t✉r❡ ✇✐t❤✐♥ t❤❡ ❞❛t❛s❡t✳ ❚❤❡ ❝♦♠♣♦♥❡♥t ✈❡❝t♦rs ♥❡❝❡ss❛r✐•② ❜❡•♦♥❣ t♦ t❤❡ ✈❡❝t♦r s♣❛❝❡ s♣❛♥♥❡❞ ❜② t❤❡ ❞❛t❛ ♣♦✐♥ts✳ ■♥ t❤✐s r❡s♣❡❝t✱ ✐t ✐s ❡q✉✐✈❛•❡♥t t♦ ♣❡r❢♦r♠ P❈❆ ❜② ✇♦r❦✐♥❣ ♦♥ t❤❡ •✐♥❡❛r s✉❜s♣❛❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❞❛t❛ ❛♥❞ r❡♣•❛❝❡ t❤❡ ❞❛t❛ ♠❛tr✐① ❜② G 1 2 ✇❤❡r❡ G = X T X ∈ R N ×N ✐s t❤❡ •r❛♠ ♠❛tr✐① ♦❢ t❤❡ ❞❛t❛✳ ■♥ t❤❛t ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡ ✈❡❝t♦rs ♣r♦✈✐❞❡❞ ❜② P❈❆ ❛r❡ t♦ ❜❡ t❤♦✉❣❤t ❛s ✇❡✐❣❤ts ♦❢ ❛ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛s❡t✳ ▼♦r❡♦✈❡r✱ ✇❡ s❡❡ t❤❛t ♦t❤❡r ♠❡tr✐❝s ❝♦✉•❞ ❜❡ ✉s❡❞ ✐♥st❡❛❞ ♦❢ t❤❡ ❡✉❝•✐❞❡❛♥ ♦♥❡ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ •r❛♠ ♠❛tr✐①✱ ✇❤✐❝❤ ✐s ❢♦r ✐♥st❛♥❝❡ t❤❡ ❜❛s✐s ♦❢ ❦❡r♥❡• P❈❆ ♠❡t❤♦❞s✳ ▲✐♥❡❛r ❉✐s❝r✐♠✐♥❛♥t ❆♥❛•②s✐s ◆❡✈❡rt❤❡•❡ss✱ ✐t✬s ✐♠♣♦rt❛♥t t♦ ♣♦✐♥t ♦✉t t❤❛t t❤❡ ✈❡❝t♦rs ❣✐✈❡♥ ❜② t❤❡ P❈❆ ❛r❡ ♦♥•② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❣•♦❜❛• ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ✇❤♦•❡ ❞❛t❛s❡t✳ ■♥ t❤❡ s✐t✉❛t✐♦♥s t❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✱ ♦✉r •❡❛r♥✐♥❣ s❡ts ❝♦♥s✐st ❣❡♥❡r❛••② ♦❢ s✉❜•❡❝ts ❜❡•♦♥❣✐♥❣ t♦ ❦♥♦✇♥ ❝•❛ss❡s ✭❝♦♥tr♦•s ❛♥❞ ❣•❛✉❝♦♠❛t♦✉s ✐♥ t❤❡ ❡①❛♠♣•❡ ♦❢ t❤❡ r❡t✐♥❛ ❞❛t❛❜❛s❡✮✳ ❙✐♥❝❡ P❈❆ ✐s ♥♦t s✉♣❡r✈✐s❡❞ ❜② t❤❡ ❝•❛ss❡s✱ ✐t ❞♦❡s ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r✲❝•❛ss ✈❛r✐❛♥❝❡ ❛♥❞ ❢♦r t❤❛t r❡❛s♦♥✱ t❤❡ r❡s✉•t✐♥❣ s✉❜s♣❛❝❡s ♠✐❣❤t ❜❡ ✈❡r② ♣♦♦r•② r❡•❡✈❛♥t t♦ ❞✐s❝r✐♠✐♥❛t❡ ❜❡t✇❡❡♥ ❣r♦✉♣s✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ✐♥t❡♥❞ t♦ s❤♦✇ ♦♥ ✜❣✉r❡ ✹✳✶✶ ✿ ♥♦♥❡ ♦❢ t❤❡ t✇♦ ✜rst P❈❆ ❛①✐s ✐s ❛❜•❡ t♦ s❡♣❛r❛t❡ ❝•❛ss❡s ❜❡❝❛✉s❡ t❤❡ ✐♥t❡r✲❝•❛ss ✈❛r✐❛t✐♦♥ ✐s ❜❛s✐❝❛••② ♦✈❡r✇❤❡•♠❡❞ ❜② t❤❡ ❣r❡❛t❡r ✐♥♥❡r ✈❛r✐❛♥❝❡ ♦❢ t❤❡ t✇♦ ❝•❛ss ❝•✉st❡rs✳ ▲✐♥❡❛r ❉✐s❝r✐♠✐♥❛♥t ❆♥❛•②s✐s ✭♦r ▲❉❆✮✱ ✇❤✐❝❤ ✐s ❛•s♦ ✈❡r② ❝•♦s❡ t♦ ❋✐s❝❤❡r✬s ❞✐s❝r✐♠✲ ✐♥❛♥t ❛♥❛•②s✐s✱ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ st❛t✐st✐❝❛• •❡❛r♥✐♥❣ ♣r❡❝✐s❡•② t♦ ♦✈❡r❝♦♠❡ t❤✐s ❦✐♥❞ ♦❢ ✐ss✉❡s✳ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ ✇❡••✲❞❡t❛✐•❡❞ ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✹✼✱ ✹✽(✳ ❲❡ •✉st ❜r✐❡✢② r❡♠✐♥❞ ✐ts ❡ss❡♥t✐❛• •✐♥❡s✳ ▲❡t✬s ❢♦❝✉s ✜rst ♦♥ t❤❡ ❝❛s❡ ♦❢ t✇♦ ❝•❛ss❡s✳ ❲❡ ❝♦♥s✐❞❡r ❛❣❛✐♥ t❤❡ s❡t ♦❢ ♣♦✐♥ts (x i ) ∈ R p×N ✇✐t❤ ❣•♦❜❛• ❝❡♥t❡r x ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡s❡ ♣♦✐♥ts ❜❡•♦♥❣ t♦ t✇♦ ❝•❛ss❡s 0 ❛♥❞ 1 ❣✐✈❡♥ ❜② t❤❡ s❡ts C 0 ⊂ {1, ..., N } ❛♥❞ C 1 = {1, ..., N }\C 0 ♦❢ r❡s♣❡❝t✐✈❡ s✐③❡s N 0 ❛♥❞ N 1 = N -N 0 ✳ ❲❡ ❛•s♦ ❝❛•• x 0 ❛♥❞ x 1 t❤❡ ❝❡♥t❡rs ♦❢ ❡❛❝❤ ❝•❛ss✳ ❚❤❡ ✶✺✹ ❋✐❣✉r❡ ✹✳✶✶✿ ❆ s✐♠✉•❛t❡❞ ❡①❛♠♣•❡ ♦❢ P❈❆ ♦♥ ❛ s❡t ♦❢ ♣♦✐♥ts ✐♥ R 3 ✳ ❖♥ t❤❡ •❡❢t ✜❣✉r❡✱ t❤❡ ♣♦✐♥ts ❞r❛✇♥ ✉s✐♥❣ t✇♦ ❡•♦♥❣❛t❡❞ ❛♥❞ tr❛♥s•❛t❡❞ ♥♦r♠❛• ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ❝♦•♦rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ t✇♦ ❞✐✛❡r❡♥t ❝•❛ss❡s✳ ■♥ r❡❞✱ ✇❡ s❤♦✇ t❤❡ t✇♦ ✜rst ♣r✐♥❝✐♣❛• ❞✐r❡❝t✐♦♥s ❢♦✉♥❞ ❜② P❈❆✳ ❖♥ t❤❡ r✐❣❤t ✜❣✉r❡ ❛r❡ r❡♣r❡s❡♥t❡❞ t❤❡ ♣♦✐♥ts ♣r♦•❡❝t❡❞ ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜s♣❛❝❡✳ ❣♦❛• ♦❢ ▲❉❆ ✐s t♦ ✜♥❞ ♦♥❡ ❞✐s❝r✐♠✐♥❛♥t ❛①✐s ❜❡t✇❡❡♥ X C 0 ❛♥❞ X C 1 ✳ ■t ✐♥tr♦❞✉❝❡s t✇♦ ♠❛tr✐❝❡s ♦❢ R p×p ✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ✇✐t❤✐♥✲❝•❛ss s❝❛tt❡r ♠❛tr✐① ❛♥❞ ✐s ❞❡✜♥❡❞ ❜② ✿ S w = i∈C 0 (x i -x 0 )(x i -x 0 ) T + i∈C 1 (x i -x 1 )(x i -x 1 ) T . ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s t❤❡ ❜❡t✇❡❡♥✲❝•❛ss s❝❛tt❡r ♠❛tr✐① ✿ S b = N 0 (x 0 -x)(x 0 -x) T + N 1 (x 1 -x)(x 1 -x) T . |❡♠❛r❦ ✹✳✸✳✷✳ ❙✐♠♣•❡ ❝♦♠♣✉t❛t✐♦♥s s❤♦✇ t❤❛t S b ∝ (x 1 -x 0 )(x 1 -x 0 ) T ❛♥❞ t❤✉s S b ✐s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥ ♠❛tr✐① ♦♥ t❤❡ •✐♥❡ s♣❛♥♥❡❞ ❜② (x 1 -x 0 )✳ ■♥ ♦r❞❡r t♦ s❡♣❛r❛t❡ t❤❡ t✇♦ ❝•❛ss❡s ❛s ♠✉❝❤ ❛s ♣♦ss✐❜•❡✱ t❤❡ ♣r✐♥❝✐♣•❡ ♦❢ ▲❉❆ ✐s t♦ ✜♥❞ ❛ ❞✐r❡❝t✐♦♥ v t❤❛t ♠❛①✐♠✐③❡s t❤❡ ❜❡t✇❡❡♥✲❝•❛ss ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣r♦•❡❝t✐♦♥s ♦♥ ❙♣❛♥(v) ✇❤✐•❡ ♠✐♥✐♠✐③✐♥❣ t❤❡ ✇✐t❤✐♥✲❝•❛ss ✈❛r✐❛♥❝❡✳ ❚❤✉s✱ ✐t ✐♥t❡♥❞s t♦ s♦•✈❡ ✿ v = arg max u =1 u T S b u u T S w u . ✭✹✳✷✶✮ ❲❡ s❡❡ t❤❛t t❤✐s ♣r♦❜•❡♠ ♠❛❦❡s s❡♥s❡ ♦♥•② ✐❢ S w ✐s ❛ ♥♦♥✲s✐♥❣✉•❛r ♠❛tr✐①✳ ❋r♦♠ t❤❡ ✈❡r② ❞❡✜♥✐t✐♦♥ ♦❢ S w ✱ t❤✐s ✐s ❝•❡❛r•② ♥♦t t❤❡ ❝❛s❡ ✐❢ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜•❡❝ts ✐s s♠❛••❡r t❤❛♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❡♠❜❡❞❞✐♥❣ s♣❛❝❡✱ ✐✳❡ ✐❢ N < p✳ ❲❡ s❤❛•• ❜❡ ❞❡❛•✐♥❣ ✇✐t❤ t❤❛t ❝❛s❡ ✐♥ ❛ ❢♦••♦✇✐♥❣ ♣❛r❛❣r❛♣❤✳ ■❢ p ≪ N ✱ ✉♥•❡ss t❤❡ ❞❛t❛ str✐❝t•② •✐❡ ✐♥ ❛ •♦✇❡r ❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ♦❢ R N ✱ S w ✐s ❣❡♥❡r❛••② ♥♦♥✲s✐♥❣✉•❛r ❛♥❞✱ ❝❢ ❬✶✷(✱ v ✐s t❤❡ ❡✐❣❡♥✈❡❝t♦r ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ •❛r❣❡st ❡✐❣❡♥✈❛•✉❡ λ ♦❢ S b S -1 w ✱ t❤❛t ✐s ✿ S b v = λS w v ✇❤✐❝❤ ❛♠♦✉♥ts ✐♥ s♦•✈✐♥❣ ❛ ❣❡♥❡r❛•✐③❡❞ ❡✐❣❡♥✈❛•✉❡ ♣r♦❜•❡♠ ❜❡t✇❡❡♥ S b ❛♥❞ S w ✳ ❙✐♥❝❡ S b ✐s ❛ r❛♥❦ ♦♥❡ ♠❛tr✐①✱ t❤❡r❡ ✐s ♦♥•② ♦♥❡ ♥♦♥✲③❡r♦ ❡✐❣❡♥✈❛•✉❡✳ ■♥ t❤✐s s✐♠♣•❡ ❝❛s❡✱ ♦♥❡ ❝❛♥ ❛❝t✉❛••② ❡①♣r❡ss t❤❡ tr✉❡ s♦•✉t✐♦♥ ✿ ✶✺✺ ❋✐❣✉r❡ ✹✳✶✷✿ ❚❤❡ ▲❉❆ ❞✐r❡❝t✐♦♥ ♦❜t❛✐♥❡❞ ♦♥ t❤❡ ❡①❛♠♣•❡ ♦❢ ✜❣✉r❡ ✹✳✶✶ ❛♥❞ t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ t❤✐s s✉❜s♣❛❝❡✳ Pr♦♣❡rt② ✹✳✸✳✶✳ ❚❤❡ ▲❉❆ ✈❡❝t♦r v s♦•✈✐♥❣ ✭✹✳✷✶✮ ✐s ❣✐✈❡♥ ❡①♣•✐❝✐t•② ❜② ✿ v = ± S -1 w (x 1 -x 0 ) S -1 w (x 1 -x 0 ) Pr♦♦❢✳ ❙✐♥❝❡ S w v = 1 λ S b v ❛♥❞ S b v ∝ (x 1 -x 0 ) t❤❛♥❦s t♦ t❤❡ ♣r❡✈✐♦✉s r❡♠❛r❦✱ ✇❡ s❡❡ t❤❛t S w v ❜❡•♦♥❣s t♦ t❤❡ •✐♥❡ s♣❛♥♥❡❞ ❜② t❤❡ ✈❡❝t♦r (x 1 -x 0 ) ❛♥❞ t❤✉s v ✐s ❝♦••✐♥❡❛r t♦ S -1 w (x 1 -x 0 )✳ ■♥ ✜❣✉r❡ ✹✳✶✷✱ ✇❡ s❤♦✇ t❤❡ r❡s✉•t ♦❢ ▲❉❆ ♦♥ t❤❡ ❡①❛♠♣•❡ ♦❢ t❤❡ t✇♦ tr❛♥s•❛t❡❞ •❛✉s✲ s✐❛♥ ❞✐str✐❜✉t✐♦♥s ♦❢ ✜❣✉r❡ ✹✳✶✶✳ ❆s ✇❡ ❡①♣❡❝t❡❞✱ ❝♦♠♣❛r❡❞ t♦ P❈❆✱ t❤❡ ♣r♦•❡❝t✐♦♥s ♦❢ t❤❡ ❞❛t❛ ♣♦✐♥ts ♦♥ t❤❡ ▲❉❆ ❛①✐s ❛r❡ ❢❛r ❜❡tt❡r s❡♣❛r❛t❡❞✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ ❛♣♣•✐❝❛t✐♦♥s ✇❡ s❤♦✇ ✐♥ t❤✐s t❤❡s✐s ❛r❡ ❢♦❝✉s❡❞ ♦♥ t❤❡ ❝❛s❡ ♦❢ t✇♦ ❝•❛ss❡s✱ •❡t✬s ♠❡♥t✐♦♥ t❤❛t ▲❉❆ ❣❡♥❡r❛•✐③❡s t♦ ❛♥ ❛r❜✐tr❛r② ♥✉♠❜❡r ♦❢ ❝•❛ss❡s (C j ) j=1,..,k ✐♥ ❛ q✉✐t❡ ♥❛t✉r❛• ✇❛②✳ ❲✐t❤ ❛♥❛•♦❣♦✉s ❝♦♥✈❡♥t✐♦♥s✱ ♦♥❡ ❞❡✜♥❡s t❤❡ ♠❛tr✐❝❡s S w ❛♥❞ S b ❜② ✿ S w = k j=1 i∈C j (x i -x j )(x i -x j ) T S b = k j=1 N j (x j -x)(x j -x) T ◆♦t❡ t❤❛t S b ✐s ♥♦✇ ❛ ♠❛tr✐① ✇❤♦s❡ r❛♥❦ ✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ✈❡❝t♦r s♣❛❝❡ s♣❛♥♥❡❞ ❜② t❤❡ ❝•❛ss ❝❡♥t❡rs x i ❛♥❞ t❤✉s S b ✐s ❛ ♠❛tr✐① ♦❢ r❛♥❦ •❡ss ♦r ❡q✉❛• t❤❛♥ k -1✱ s✐♥❝❡ k i=1 N i (x i -x) = 0✳ ❚❤✐s ❜❡✐♥❣ s❡t✱ s✐♠✐•❛r•② t♦ t❤❡ ❝❛s❡ k = 2✱ t❤❡ ♠✉•t✐✲❝•❛ss ▲❉❆ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ t❤❡ ♥♦♥✲③❡r♦ ❣❡♥❡r❛•✐③❡❞ ❡✐❣❡♥✈❛•✉❡s ❜❡t✇❡❡♥ S b ❛♥❞ S w ✳ ❚❤✐s ♣r♦✈✐❞❡s s❡✈❡r❛• ▲❉❆ ✈❡❝t♦rs✱ t❤❛t ❛r❡ ❛•• ♦❜t❛✐♥❡❞ ❛s •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ♣♦✐♥ts✱ ❛♥❞ t❤✉s ❛ ❞✐s❝r✐♠✐♥❛♥t s✉❜s♣❛❝❡ ✭♦❢ ❞✐♠❡♥s✐♦♥ ≤ k -1✮ ✐♥ ✇❤✐❝❤ ♦♥❡ ❝❛♥ ♣r♦•❡❝t ❛♥❞ ❛♥❛•②③❡ t❤❡ ❞❛t❛ ✇✐t❤ ❞❡✜♥✐t❡•② ❜❡tt❡r ❝•❛ss s❡♣❛r❛t✐♦♥ ♣r♦♣❡rt✐❡s✳ |❡❣✉•❛r✐③❡❞ ▲❉❆ ❆s ✇❡ ♠❡♥t✐♦♥❡❞✱ t❤❡ ♣r❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✈❛•✐❞ ✇❤❡♥ S w ✐s ♥♦♥✲s✐♥❣✉•❛r ❜✉t t❤✐s ❤②♣♦t❤❡s✐s ❜r❡❛❦s ❞♦✇♥ ❛s s♦♦♥ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts ✐s s♠❛••❡r t❤❛♥ ✶✺✻ t❤❡ ❞✐♠❡♥s✐♦♥ N ✳ ■♥ ♦✉r ❝❛s❡s ♦❢ ❛♣♣•✐❝❛t✐♦♥s✱ ✇❡ ❛r❡ t②♣✐❝❛••② ❝♦♥❢r♦♥t❡❞ t♦ t❤❛t s✐♥❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ❛✈❛✐•❛❜•❡ s✉❜•❡❝ts ✐s r❛t❤❡r s♠❛•• ❝♦♠♣❛r❡❞ t♦ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❢❡❛t✉r❡s ✇❡ ✇❛♥t t♦ ❛♥❛•②③❡ ✇❤✐❝❤ ✐s ❣❡♥❡r❛••② ❤✉❣❡ ✭❢♦r ♠♦♠❡♥t❛ ♦r ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s✱ t❤✐s ✐s ♣r♦♣♦rt✐♦♥❛• t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦♥ t❤❡ s❤❛♣❡s✮✳ ❚♦ t❛❝❦•❡ t❤✐s ✐ss✉❡✱ s♦♠❡ ♠❡t❤♦❞s ♣r♦♣♦s❡ t♦ ✜rst ♣r♦•❡❝t t❤❡ ❞❛t❛ ♦♥ ❛ •♦✇❡r ❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ✉s✐♥❣ t②♣✐❝❛••② P❈❆✳ ❍♦✇❡✈❡r✱ ✇❡ ❛r❣✉❡ t❤❛t t❤✐s ❝♦✉•❞ ❜❡ ❞❛♥❣❡r♦✉s ✐♥ t❤❛t ✇❡ ❞♦ ♥♦t r❡❛••② ❝♦♥tr♦• ❤♦✇ ♠✉❝❤ ❜❡t✇❡❡♥✲❝•❛ss ✈❛r✐❛♥❝❡ ✐s •♦st ❜② ❞♦✐♥❣ s♦✳ ❖t❤❡r ❛♣♣r♦❛❝❤❡s •✐❦❡ ✐♥ ❬✹✽( r❡•② ♦♥ ❣❡♥❡r❛•✐③❡❞ ❙❱❉✬s t♦ s♦•✈❡ t❤❡ ❣❡♥❡r❛•✐③❡❞ ❡✐❣❡♥✈❛•✉❡s ♣r♦❜•❡♠✳ (❡t t❤❡ ♠♦st ❝♦♠♠♦♥ ❛♥❞ s✐♠♣•❡ ❛•t❡r♥❛t✐✈❡ ✐s t♦ r❡❣✉•❛r✐③❡ t❤❡ ▲❉❆ ❬✸✽( ❜② r❡♣•❛❝✐♥❣ S w ❜② t❤❡ ♠❛tr✐① ✿ S ε w = S w + ε■❞ ❢♦r ❛ ❝❡rt❛✐♥ ǫ > 0✱ ✇❤✐❝❤ ❛✉t♦♠❛t✐❝❛••② ♠❛❦❡s S ε w ✐♥✈❡rt✐❜•❡ ❛♥❞✱ s✐♠✐•❛r•② t♦ ♣r♦♣✲ ❡rt② ✹✳✸✳✶✱ ❣✐✈❡s v = (S w + ε■❞) -1 (x 1 -x 0 ) ✐♥ t❤❡ t✇♦ ❝•❛ss❡s ❝❛s❡✳ ❚❤❡ ❞r❛✇❜❛❝❦ ✐s t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ❛❞❞✐t✐♦♥❛• r❡❣✉•❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ ε ❝❛♥ ❜❡ ❞♦♥❡ ❜② ♦♣t✐♠✐③✐♥❣ ✐t ✇✐t❤ r❡s♣❡❝t t♦ ❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s ♦♥ ❝r♦ss✲✈❛•✐❞❛t✐♦♥ t❡sts✱ ✇❤✐❝❤ ✇❡ s❤❛•• ❡①♣•❛✐♥ ❛ •✐tt•❡ ♠♦r❡ ❜❡•♦✇✳ ✹✳✸✳✷ •❡♥❡r❛•✐③❡❞ ▲❉❆ ♦♥ ❍✐•❜❡rt s♣❛❝❡s ❚❤❡ ♣r❡✈✐♦✉s ▲❉❆ ✇❛s ♣r❡s❡♥t❡❞ ✐♥ ✐ts ❝•❛ss✐❝❛• s❡tt✐♥❣✱ ♥❛♠❡•② ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ❡✉❝•✐❞❡❛♥ s♣❛❝❡✳ ◆❡✈❡rt❤❡•❡ss✱ t❤❡ t②♣✐❝❛• ❛♣♣•✐❝❛t✐♦♥s t❤❛t ✇❡ ❛✐♠ ❛t ✐♥ t❤✐s ❝❤❛♣t❡r ❤❛✈❡ t♦ ❞❡❛• ✇✐t❤ ♠♦♠❡♥t❛ t❤❛t ❜❡•♦♥❣ t♦ t❤❡ ❞✉❛• V ′ ♦❢ ❛ s♣❛❝❡ ♦❢ ✈❡❝t♦r ✜❡•❞s ♦r r❡s✐❞✉❛• ❢✉♥❝t✐♦♥s ✐♥ t❤❡ s♣❛❝❡ L 2 (X) ❢♦r ✐♥st❛♥❝❡✳ ❚❤♦s❡ ❛r❡ ♠❛♥✐❢❡st•② ❍✐•❜❡rt s♣❛❝❡s ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥s✳ ❊✈❡♥ t❤♦✉❣❤ ♦♥❡ ❝♦✉•❞ ❛r❣✉❡ t❤❛t✱ ✐♥ ♣r❛❝t✐❝❡✱ t❤❡ ❞❛t❛s❡ts ❛r❡ s❛♠♣•❡❞ ✇✐t❤ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ❛♥❞ t❤❛t t❤❡ ♠♦♠❡♥t❛ ❛♥❞ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ •❛r❣❡ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡s✱ t❤❡ ♠❡tr✐❝ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ V ′ ❢♦r ❡①❛♠♣•❡ ✐s ❣❡♥❡r❛••② ♥♦t t❤❡ ❡✉❝•✐❞❡❛♥ ♠❡tr✐❝ ✭✐t ✐♥✈♦•✈❡s ❛ ❦❡r♥❡• ♠❛tr✐①✮✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ s❤♦✇ t❤❛t ▲❉❆ ❝❛♥ ❜❡ ❣❡♥❡r❛•✐③❡❞ ✐♥ ❛ ♥❛t✉r❛• ✇❛② t♦ ❍✐•❜❡rt s♣❛❝❡s ❛♥❞ ❡①♣•❛✐♥ ❤♦✇ ✐t ❝❛♥ ❜❡ ❛❝t✉❛••② r❡❞✉❝❡❞ t♦ ❛ ✉s✉❛• r❡❣✉•❛r✐③❡❞ ▲❉❆ ✐♥ ❛ ❝❡rt❛✐♥ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ❡✉❝•✐❞❡❛♥ s♣❛❝❡✳ ▲❡t✬s ✜① ❛ ❍✐•❜❡rt s♣❛❝❡ H ❛♥❞ ✐ts ✐♥♥❡r ♣r♦❞✉❝t ., . H ✳ ❲❡ t❛❦❡ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥ ✿ (x i ) i=1,..,N ∈ H N ✐s t❤❡ ❞❛t❛s❡t ❛♥❞ x ✐ts ❝❡♥t❡r✱ C j ⊂ {1, .., N } ❢♦r j = 1, .., k ❛r❡ t❤❡ k ❞✐✛❡r❡♥t ❝•❛ss❡s ❛♥❞ x j ∈ H ❛r❡ t❤❡ ❝•❛ss ❝❡♥t❡rs✳ ❲❡ ❝❛♥ t❤❡♥ ❞❡✜♥❡✱ ❜② ❛♥❛•♦❣② t♦ ✉s✉❛• ▲❉❆✱ t❤❡ ✇✐t❤✐♥✲❝•❛ss ❛♥❞ ❜❡t✇❡❡♥✲ ❝•❛ss ♦♣❡r❛t♦rs ✿ S w (x) = k j=1 i∈C j x i -x j , x H (x i -x j ) S b (x) = k j=1 N j x j -x, x H (x j -x) . ◆♦t❡ t❤❛t t❤❡s❡ ❞❡✜♥✐t✐♦♥s ❣❡♥❡r❛•✐③❡ t❤❡ ♣r❡✈✐♦✉s ❡✉❝•✐❞❡❛♥ ❝❛s❡✳ ❲❡ ✐♥tr♦❞✉❝❡ ♥♦✇ t❤❡ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ H 0 = ❙♣❛♥(x i |i = 1, .., N ) s♣❛♥♥❡❞ ❜② t❤❡ ❞❛t❛ ♣♦✐♥ts ❛♥❞ ❞❡♥♦t❡ ❜② S w,H 0 ❛♥❞ S b,H 0 t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs t♦ H 0 ✳ ◆♦t❡ t❤❛t✱ s✐♥❝❡ ❜♦t❤ S w ❛♥❞ S b t❛❦❡ ✈❛•✉❡s ✐♥ H 0 ✱ S w,H 0 ❛♥❞ S b,H 0 ❛r❡ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥s ♦❢ H 0 t♦ ✐ts❡•❢✳ ■♥ ❣❡♥❡r❛•✱ S w,H 0 ✐s ♥♦t ✐♥✈❡rt✐❜•❡ ❛♥❞ ✇❡ ❝♦♥s✐❞❡r ❛s ♣r❡✈✐♦✉s•② t❤❡ r❡❣✉•❛r✐③❡❞ ♠❛tr✐① ✿ S ε w,H 0 = S w,H 0 + ε■❞ H 0 ∈ L(H 0 ) ✶✺✼ ◆♦✇✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt② ✿ Pr♦♣❡rt② ✹✳✸✳✷✳ ▲❡t L : H 0 → R q ✱ q ∈ N * ✱ ❜❡ ❛♥② ✐s♦♠❡tr② ❛♥❞ •❡t✬s ❞❡♥♦t❡ ❜② Sε w , Sb t❤❡ s❝❛tt❡r ♠❛tr✐❝❡s ♦❢ t❤❡ ✐♠❛❣❡ ❞❛t❛ ♣♦✐♥ts L(x i ) ∈ R q ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✱ ❢♦r ❛•• ǫ ≥ 0 ✿ Sε w = LS ε w,H 0 L † , Sb = LS b,H 0 L † ✇❤❡r❡ L † : R q → H 0 ✐s t❤❡ ❛❞•♦✐♥t ♦❢ L✳ ■t r❡s✉•ts t❤❛t✱ ✉♣ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ û → u = L † û✱ t♦ ❛♥② ❡✐❣❡♥✈❛•✉❡ ❛♥❞ ❡✐❣❡♥✈❡❝t♦r (λ j , ûj ) ♦❢ ( Sε w ) -1 S b ❝♦rr❡s♣♦♥❞s t❤❡ ❡✐❣❡♥✈❛•✉❡ ❛♥❞ ❡✐❣❡♥✈❡❝t♦r (λ j , L † ûj ) ♦❢ (S ε w,H 0 ) -1 S b,H 0 ✳ Pr♦♦❢✳ ❇② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ ❢♦r ❛•• a ∈ R q ❛♥❞ x ∈ H ✿ L † a, x H = L(x) T a ❲❡ ❞❡❞✉❝❡ t❤❛t ❢♦r ❛•• a, b ∈ R q ✿ b T Sw (a) = b T   k j=1 i∈C j (L(x i ) -L(x j ))(L(x i ) -L(x j )) T a   = b T   k j=1 i∈C j (L(x i ) -L(x j )) L † a, x i -x j H   = k j=1 i∈C j b T (L(x i ) -L(x j )) L † a, x i -x j H = k j=1 i∈C j L † b, x i -x j H L † a, x i -x j H = L † b, k j=1 i∈C j L † a, x i -x j H (x i -x j ) H = L † b, S w,H 0 (L † a) H = b T (LS w,H 0 L † a) ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t Sw = LS w,H 0 L † ✳ ■♥ ❛❞❞✐t✐♦♥✱ s✐♥❝❡ L ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛♥ ✐s♦♠❡tr②✱ ✇❡ ❤❛✈❡ t❤❛t LL † = ■❞ R q ❛♥❞ ✐t ❢♦••♦✇s t❤❛t ✿ Sε w = LS ε w,H 0 L † . ■♥ t❤❡ s❛♠❡ ✇❛②✱ ♦♥❡ ❝❤❡❝❦s ❡❛s✐•② t❤❛t Sb = LS b,H 0 L † ✳ ◆♦✇✱ ✇❡ ❞❡❞✉❝❡ ✿ ( Sε w ) -1 Sb = (L † ) -1 S ε w,H 0 ) -1 S b,H 0 L † . ■t r❡s✉•ts t❤❛t t❤❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ ( Sε w ) -1 Sb ❛♥❞ (S ε w,H 0 ) -1 S b,H 0 ❛r❡ t❤❡ s❛♠❡ ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ s❡❝♦♥❞ ♦♣❡r❛t♦r ❢r♦♠ t❤❡ ✜rst ♦♥❡ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s û ∈ R q → L † û ∈ H 0 ✳ Pr♦♣❡rt② ✹✳✸✳✷ ♠❡❛♥s ❜❛s✐❝❛••② t❤❛t ▲❉❆ ✐♥ t❤❡ ❍✐•❜❡rt s♣❛❝❡ H ❝❛♥ ❜❡ ❞♦♥❡ ❡q✉✐✈✲ ❛•❡♥t•② ❜② ❛♣♣•②✐♥❣ ✉s✉❛• ✬❡✉❝•✐❞❡❛♥✬ ▲❉❆ ✐♥ R q ♦♥ t❤❡ ❞❛t❛ ♣♦✐♥ts L(x i )✳ ❚❤❡ ♦♥•② r❡♠❛✐♥✐♥❣ ♣♦✐♥t t♦ ❛❞❞r❡ss ✐s t♦ ♠❛❦❡ ❡①♣•✐❝✐t s✉❝❤ ❛♥ ✐s♦♠❡tr② L✳ ❋♦r t❤❛t ♣✉r♣♦s❡✱ •❡t✬s ❛ss✉♠❡ t❤❛t t❤❡ x i ❛r❡ •✐♥❡❛r•② ✐♥❞❡♣❡♥❞❡♥t s♦ t❤❛t t❤❡② ❝♦♥st✐t✉t❡ ❛ ❜❛s✐s ♦❢ H 0 ✳ ✶✺✽ ■❢ H ✐s t②♣✐❝❛••② ❛ ❍✐•❜❡rt s♣❛❝❡ ✇❤♦s❡ ❞✐♠❡♥s✐♦♥ ✐s ✐♥✜♥✐t❡ ♦r ✈❡r② ❜✐❣ ❝♦♠♣❛r❡❞ t♦ N ✱ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ♥♦t ❛❜❡rr❛♥t s✐♥❝❡ N r❛♥❞♦♠ ❡•❡♠❡♥ts ✐♥ H ✇✐•• ❜❡ ❣❡♥❡r✐❝❛••② ✐♥❞❡♣❡♥❞❡♥t✳ ❖t❤❡r✇✐s❡✱ ♦♥❡ ❝❛♥ •✉st r❡str✐❝t t♦ ❛ s✉❜✲❞❛t❛❜❛s❡ ♠❛❞❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t x i ✬s✳ ❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ t❤❡ •r❛♠ ♠❛tr✐① ♦❢ t❤❡ ❞❛t❛s❡t G = ( x i , x j H ) i,j=1,..,N ✱ ✇❤✐❝❤ ✐s s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❛♥❞ ❞❡✜♥❡ ❛♥ ♦♣❡r❛t♦r L : H 0 → R N ❛s ❢♦••♦✇s ✿ ∀x ∈ H 0 , L(x) = G -1 2 γ(x) ✭✹✳✷✷✮ ✇✐t❤ γ(x) = ( x, x i H ) i=1,..,N . ◆♦t❡ t❤❛t t❤❡ ✐♠❛❣❡ L(x i ) i=1,..,N ♦❢ t❤❡ ❞❛t❛ ♠❛tr✐① ❜② L ✐s G 1 2 ✳ ■♥ ❛❞❞✐t✐♦♥ ✿ Pr♦♣❡rt② ✹✳✸✳✸✳ ❚❤❡ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥ L : H 0 → R N ❞❡✜♥❡❞ ❜② ❡q✳✭✹✳✷✷✮ ✐s ❛♥ ✐s♦♠❡tr②✳ ▼♦r❡♦✈❡r✱ t❤❡ ❛❞•♦✐♥t L † ❡q✉❛•s ❢♦r ❛•• a ∈ R q ✿ L † (a) = N i=1 α i x i , ✇✐t❤ α = G -1 2 a ✭✹✳✷✸✮ Pr♦♦❢✳ ▲❡t y, z ∈ H 0 ✳ ❇② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❤❛✈❡ ✿ L(y) T L(z) = γ(y) T G -1 γ(z) . ◆♦✇ ✐❢ ✇❡ ✇r✐t❡ z = N i=1 µ i x i ✱ ✇❡ s❡❡ t❤❛t γ(z) = Gµ ❛♥❞ t❤✉s ✿ L(y) T L(z) = γ(y) T µ = N i=1 y, x i H µ i = y, z H . L ✐s t❤❡r❡❢♦r❡ ❛♥ ✐s♦♠❡tr②✳ ❆ ❝♦♥s❡q✉❡♥❝❡ ✐s t❤❛t ✇❡ ❤❛✈❡ ❛•s♦ LL † = ■❞ R N ✳ ■❢ a ∈ R N ✱ L † (a) ∈ H 0 ✇❤✐❝❤ ✇r✐t❡s L † (a) = N i=1 α i x i ✳ ❚❤✉s✱ a = LL † (a) = G -1 2 γ(L † (a)) = G -1 2 Gα = G 1 2 α ❛♥❞ ✜♥❛••②✱ ✇❡ ❝♦♥❝•✉❞❡ t❤❛t α = G -1 2 a✳ ■♥ ❝♦♥❝r❡t❡ t❡r♠s✱ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣❡rt② s❤♦✇s t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ▲❉❆ ❞✐s❝r✐♠✐♥❛♥t ✈❡❝t♦rs ✐♥ t❤❡ ❍✐•❜❡rt s♣❛❝❡ H ❝❛♥ ❜❡ ❛❝t✉❛••② ❞♦♥❡ ❜② ✜rst ❛♣♣•②✐♥❣ ✉s✉❛• ▲❉❆ ✐♥ t❤❡ ❡✉❝•✐❞❡❛♥ s♣❛❝❡ R N ❢♦r t❤❡ ❞❛t❛ ♠❛tr✐① G 1 2 ✳ ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ❞✐r❡❝t✐♦♥s ✐♥ H ❛r❡ t❤❡♥ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❛❞•♦✐♥t ♦♣❡r❛t♦r L † t❤❛t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❡①♣•✐❝✐t•② ✉s✐♥❣ ❡q✳✭✹✳✷✸✮✳ ❆♥② ♥❡✇ ❞❛t❛ x ∈ H ❝❛♥ ❜❡ ❛•s♦ ♣r♦•❡❝t❡❞ t♦ t❤✐s ❞✐s✲ ❝r✐♠✐♥❛♥t s✉❜s♣❛❝❡ ❜② ✜rst ❝♦♠♣✉t✐♥❣ L(p H 0 (x)) = G -1 2 γ(x) ❛♥❞ ❝♦♠♣✉t❡ ❞✐r❡❝t•② ✐♥♥❡r ♣r♦❞✉❝ts ✇✐t❤ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ✈❡❝t♦rs ✐♥ R N ✳ ✹✳✸✳✸ ▼♦❞❡s ❛♥❞ ❝•❛ss✐✜❝❛t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥❛• ❛t•❛s❡s ❲❡ s♣❡❝✐❢② ❛♥❞ ❞❡t❛✐• t❤❡ ♣r❡✈✐♦✉s P❈❆ ❛♥❞ ▲❉❆ ❛♥❛•②s✐s t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❞❛t❛ ✐s ♥♦✇ ❛ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥ ♠♦♠❡♥t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ♦♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ❛s ♣r♦✈✐❞❡❞ ❜② t❤❡ ♣r❡✈✐♦✉s ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❲❡ s❤❛•• r❡str✐❝t t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡ E = R 3 ❛s ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✱ ❛•t❤♦✉❣❤ ✇❤❛t ✇❡ ♣r❡s❡♥t r❡♠❛✐♥s ✈❛•✐❞ ✐♥ ❛♥② ❞✐♠❡♥s✐♦♥✳ ❋r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✇❡ ❛r❡ ♠❛♥✐♣✉•❛t✐♥❣ ♦❜•❡❝ts t❤❛t •✐✈❡ ✐♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ❍✐•❜❡rt s♣❛❝❡s ✭t❤❡ ❞✉❛• V ′ ♦❢ t❤❡ |❑❍❙ ♦❢ ✈❡❝t♦r ✜❡•❞s ❢♦r ♠♦♠❡♥t❛✱ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ r❡s✐❞✉❛•s✮✳ ■♥ ♣r❛❝t✐❝❡✱ t❤❡s❡ ❛r❡ s❛♠♣•❡❞ ♦♥ t❤❡ ♣♦✐♥ts ♦❢ t❤❡ t❡♠♣•❛t❡ ❜✉t st✐•• ❜❡•♦♥❣ t♦ ✈❡r② ❤✐❣❤ ❞✐♠❡♥s✐♦♥❛• ✈❡❝t♦r s♣❛❝❡s✳ ❚❤❡ s❡❝♦♥❞ ♣♦✐♥t ✐s t❤❡ ♠❡tr✐❝ t❤❛t ✐s ❝❤♦s❡♥ t♦ ❝♦♠♣✉t❡ ❝♦rr❡•❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❞❛t❛✳ ❲❡ ❡①❛♠✐♥❡ t❤♦s❡ ✐ss✉❡s ❢♦r ❜♦t❤ ❢❡❛t✉r❡s ✐♥ t❤❡ ❢♦••♦✇✐♥❣✳ ✶✺✾ ▼♦♠❡♥t❛ ❛♥❛•②s✐s ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❞❛t❛s❡t t♦ ❛♥❛•②③❡ ✐s t❤❡ s❡t ♦❢ ✐♥✐t✐❛• ♠♦♠❡♥t❛ p i 0 ❢♦r i = 1, .., N ✱ N ❜❡✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜•❡❝ts ❛♥❞ ❡❛❝❤ ♦❢ t❤❡ p i 0 ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ✈❡❝t♦r p i 0,k ∈ R 3 ❛tt❛❝❤❡❞ t♦ ❡✈❡r② ♣♦✐♥t (q k ) k=1,..,n ♦❢ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ s❤❛♣❡ ❛♥❞ t❤✉s t♦ ❛ •❛r❣❡ ✈❡❝t♦r ♦❢ R 3n ✳ ❍♦✇❡✈❡r✱ t❤❡s❡ ♠♦♠❡♥t❛ ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❡•❡♠❡♥ts ♦❢ V ′ ❛♥❞ ❝♦♥s❡q✉❡♥t•② ♦♥❡ ❤❛s t♦ ❡✈❛•✉❛t❡ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♥♦t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ L 2 ✲♠❡tr✐❝ ✐♥ R 3n ✭✇❤✐❝❤ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② ✬♥❛t✉r❛•✬ ♠❡❛♥✐♥❣ ❤❡r❡✮ ❜✉t ✇✐t❤ t❤❡ |❑❍❙ ♠❡tr✐❝ ♦♥ V ′ ✱ t❤❛t ✐s t♦ s❛② ✿ p i 0 , p j 0 V ′ = Kp i 0 , Kp j 0 V = v i 0 , v j 0 V ✇❤❡r❡ ✇❡ r❡♠✐♥❞ t❤❛t v i 0 (x) = n k=1 K V (q k , x)p i 0,k ✐s t❤❡ ✐♥✐t✐❛• ✈❡•♦❝✐t② ✜❡•❞ ❝♦rr❡✲ s♣♦♥❞✐♥❣ t♦ t❤❡ ♠♦♠❡♥t✉♠ p i 0 ✳ ❚❤✉s✱ ❢r♦♠ t❤❡ ♥✉♠❡r✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✉s✐♥❣ ✉s✉❛• ❦❡r♥❡• ❝♦♠♣✉t❛t✐♦♥s✱ ❢♦r ❛•• ♣❛✐rs ♦❢ s✉❜•❡❝ts i, j ∈ {1, .., N } ✇❡ ❤❛✈❡ ✿ p i 0 , p j 0 V ′ = n k,l=1 (p i 0,k ) T K V (q k , q l )p j 0,l . ✭✹✳✷✹✮ ❚❤❡s❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ♦❢ ♥✉♠❡r✐❝❛• ❝♦♠♣•❡①✐t② O(n 2 ) ❢♦r ❡❛❝❤ ✐♥♥❡r ♣r♦❞✉❝t ❜✉t ❝❛♥ ❜❡ ❛•s♦ ❞♦♥❡ ✉s✐♥❣ ❛•• t❤❡ ❛rs❡♥❛• ♦❢ ❢❛st ❦❡r♥❡• ❝♦♠♣✉t❛t✐♦♥s ❞❡✈❡•♦♣❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✸✳✸✳ ❇❛s❡❞ ♦♥ ❡q✳✭✹✳✷✹✮✱ ✇❡ ❝❛♥ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ •r❛♠ ♠❛tr✐① ♦❢ t❤❡ ❞❛t❛ G = ( p i 0 , p j 0 V ′ ) i,j=1,..,N . ✭✹✳✷✺✮ ■♥ ♦✉r s✐t✉❛t✐♦♥s✱ ✇❡ ❤❛✈❡ ❣❡♥❡r❛••② N ≪ n✱ ❛♥❞ t❤✉s ❝♦♠♣✉t✐♥❣ ❡✐❣❡♥✈❛•✉❡s ♦❢ G ❛s ♥❡❡❞❡❞ ❢♦r t❤❡ P❈❆ ✐s ♥✉♠❡r✐❝❛••② ✈❡r② ❢❛st ❝♦♠♣❛r❡❞ t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ G✳ ❆♣♣•②✐♥❣ P❈❆ ✐♥ t❤✐s s❡tt✐♥❣ •✉st ❝♦♥s✐sts ✐♥ ❝♦♠♣✉t✐♥❣ t❤❡ •❛r❣❡st ❡✐❣❡♥✈❛•✉❡s ♦❢ t❤❡ ♠❛tr✐① G✳ ■t ♣r♦✈✐❞❡s ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs w s ∈ R N ✇❤✐❝❤ ❛r❡ t❤❡ ♣r✐♥❝✐♣❛• ♠♦❞❡s ♦❢ ❣❡♦♠❡tr✐❝❛• ✈❛r✐❛❜✐•✐t② ✐♥ t❤❡ ❞❛t❛s❡t✳ ❈♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ s♣❛❝❡ ♦❢ ♠♦♠❡♥t❛✱ t❤❡s❡ ♠♦❞❡s ❛r❡ ❣✐✈❡♥ ❜② p P CA s = N i=1 w s,i p i 0 ❙❤♦♦t✐♥❣ t❤❡ t❡♠♣•❛t❡ ✉s✐♥❣ s✉❝❤ ♠♦♠❡♥t❛ t❤✉s ❛••♦✇s t♦ ✈✐s✉❛•✐③❡ ✇❤❛t t❤❡s❡ ♠♦❞❡s ❛r❡ ✐♥ t❡r♠ ♦❢ ❞❡❢♦r♠❛t✐♦♥✱ ✇❤✐❝❤ ✇❡ ✇✐•• s❤♦✇ ♦♥ t❤❡ r❡t✐♥❛ ❞❛t❛s❡t ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ ❝❛s❡ ♦❢ ▲❉❆ ❝❛♥ ❜❡ tr❡❛t❡❞ ❛s ❡①♣•❛✐♥❡❞ ✐♥ s✉❜s❡❝t✐♦♥ ✹✳✸✳✷ ✭❢♦r H = V ′ ✮✳ ❆❞♦♣t✐♥❣ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s✱ ✇❡ ✜rst ❝♦♠♣✉t❡ t❤❡ •r❛♠ ♠❛tr✐① ❛s ✐♥ ❡q✳✭✹✳✷✺✮ ❛♥❞ ✐ts sq✉❛r❡ r♦♦t G 1 2 ✱ t❤❡♥ ❛♣♣•② ✉s✉❛• ▲❉❆ ✐♥ R N ✳ ■t ♣r♦✈✐❞❡s ❛ ❞✐s❝r✐♠✐♥❛♥t ✈❡❝t♦r a ∈ R N ✇❤✐❝❤✱ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ♠♦♠❡♥t❛✱ tr❛♥s•❛t❡s t♦ ✿ p LDA = L † (a) = N i=1 α i p i 0 , α = G -1 2 a ❆❣❛✐♥✱ t❤✐s ❞✐s❝r✐♠✐♥❛♥t ❞✐r❡❝t✐♦♥ ❝❛♥ ❜❡ ✈✐s✉❛•✐③❡❞ ❜② s❤♦♦t✐♥❣ t❤❡ t❡♠♣•❛t❡ ✇✐t❤ ✐♥✐t✐❛• ♠♦♠❡♥t✉♠ p LDA ✳ ❖♥❡ ❝❛♥ t❤❡♥ ♣r♦•❡❝t t❤❡ ❞❛t❛ ♦♥ t❤❡ •✐♥❡ ❙♣❛♥(p LDA ) ❜② ❝♦♠♣✉t✐♥❣ p i 0 , p LDA V ′ ✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ t❤❡ s♣❛❝❡ R N s✐♥❝❡ ✿ p i 0 , p LDA V ′ = N j=1 α j G ij . ✶✻✵ ❆♥② ♥❡✇ ❞❛t❛ p 0 ♦✉ts✐❞❡ t❤❡ tr❛✐♥✐♥❣ s❡t ❝❛♥ ❜❡ ❛•s♦ ♣r♦•❡❝t❡❞ t♦ t❤✐s s✉❜s♣❛❝❡✱ ❜② ✜rst ❝♦♠♣✉t✐♥❣ L(p 0 ) ❛♥❞ ✇r✐t❡ ✿ p 0 , p LDA V ′ = N j=1 u j L(p 0 ) j ❈•❛ss✐✜❝❛t✐♦♥ ❢♦r ❜♦t❤ P❈❆ ❛♥❞ ▲❉❆ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♣r♦•❡❝t✐♦♥s ♦❢ t❤❡ ❞❛t❛ ♦♥t♦ t❤❡ •♦✇ ❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ♣r♦✈✐❞❡❞ ❜② t❤❡ ♠❡t❤♦❞s✳ ■❢ t❤❡s❡ s✉❜s♣❛❝❡s ❛r❡ s✉✣❝✐❡♥t•② ❞✐s❝r✐♠✐♥❛t✐✈❡✱ ♦♥❡ ❝❛♥ ❡①♣❡❝t t❤❛t t❤❡ ❞❛t❛s❡t s♣•✐ts ✐♥t♦ s❡✈❡r❛• ❝•✉st❡rs✱ ❡❛❝❤ ♦♥❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♦♥❡ ❝•❛ss✳ ❚❤❡ ❝•❛ss✐✜❝❛t✐♦♥ r✉•❡ ❝❛♥ t❤❡♥ ❝♦♥s✐st ✐♥ s✐♠♣•② ✜♥❞✐♥❣ t❤❡ ❝•♦s❡st ❝•✉st❡r ❝❡♥t❡r t♦ t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ ❛ ❣✐✈❡♥ ❞❛t❛✱ ♣♦ss✐❜•② ❜② ❝♦♠♣✉t✐♥❣ ▼❛❤❛•❛♥♦❜✐s ❞✐st❛♥❝❡s ✐♥ ♦r❞❡r t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❝•✉st❡rs✬ ✈❛r✐❛♥❝❡s✳ |❡s✐❞✉❛•s ❛♥❛•②s✐s ❚❤❡ tr❡❛t♠❡♥t ♦❢ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ❢♦••♦✇s t❤❡ ❡①❛❝t s❛♠❡ ♣❛tt❡r♥ ❛s ♠♦♠❡♥t❛✳ ❖♥❡ ♦♥•② ♥❡❡❞s t♦ ✐♥tr♦❞✉❝❡ t❤❡ r✐❣❤t ♠❡tr✐❝ t♦ ❝♦♠♣✉t❡ ❝♦✈❛r✐❛♥❝❡ ❜❡t✇❡❡♥ s✉❜•❡❝ts✳ ❚❤❡s❡ r❡s✐❞✉❛•s (ζ i ) i=1,..,N ❛r❡ ❛•• ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ❝♦♠♠♦♥ t❡♠♣•❛t❡ s❤❛♣❡ X✳ ❋♦r t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡✱ ✇❡ ✉s❡❞ t❤❡ L 2 ♠❡tr✐❝ ♦♥ X ❛s ❛ ♣❡♥❛•t② t❡r♠ ✐♥ t❤❡ ❢✉♥❝t✐♦♥❛•✳ ❚❤✐s ♠❡tr✐❝ ❝♦✉•❞ ❜❡ ❛•s♦ ✉s❡❞ ❛s t❤❡ ♠❡❛s✉r❡ ♦❢ ❝♦rr❡•❛t✐♦♥✳ ❲✐t❤ t❤❡ ♥♦t❛t✐♦♥s ♦❢ s❡❝t✐♦♥ ✹✳✷✳✷✱ ✐t ✇r✐t❡s ✿ ζ i , ζ j L 2 (X) = X ζ i (x)ζ j (x)dH d (x) = m l=1 ζi l ζj l |ξ l | . ✭✹✳✷✻✮ ❚❤✐s ♠❡tr✐❝✱ ❤♦✇❡✈❡r✱ ❝♦✉•❞ ❜❡ ❛r❣✉❡❞ t♦ ❜❡ t♦♦ str♦♥❣ ✐♥ t❤✐s ♣✉r♣♦s❡ ❜❡❝❛✉s❡ ✐t ❝♦♠♣❛r❡s ✈❛•✉❡s ♦❢ r❡s✐❞✉❛•s ♦♥•② ❛t ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ♦♥ t❤❡ t❡♠♣•❛t❡✳ ❖♥❡ ❡❛s② ♣♦ss✐❜✐•✐t② t♦ ♦✈❡r❝♦♠❡ t❤❛t ✐s t♦ r❡❣✉•❛r✐③❡ t❤❡ L 2 ♠❡tr✐❝ ❜② ❛ •❛✉ss✐❛♥ e -|x 1 -x 2 | 2 σ 2 ✱ ✇❤✐❝❤ ✇♦✉•❞ ❣✐✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❝♦rr❡•❛t✐♦♥ ♠❡❛s✉r❡ ❜❡t✇❡❡♥ r❡s✐❞✉❛•s ✿ ζ i , ζ j = X×X e -|x 1 -x 2 | 2 σ 2 ζ i (x 1 )ζ j (x 2 )dH d (x 1 )dH d (x 2 ) = m k,l=1 e -|x k -x l | 2 σ 2 ζ i k ζ j l |ξ k ||ξ l | . ✭✹✳✷✼✮ ❆ t❤✐r❞ ♣♦ss✐❜✐•✐t② ❝♦✉•❞ ❜❡ t♦ •❡❛♥ ❛❣❛✐♥ ♦♥ t❤❡ ❦❡r♥❡• ♥♦r♠s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✭♦r ❡✈❡♥ ♦♥ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s✮ t❤❛t ✇❡r❡ ✉s❡❞ ❛s ❞❛t❛ ❛tt❛❝❤♠❡♥t ❞✐st❛♥❝❡s ✐♥ r❡❣✐str❛t✐♦♥ ❛♥❞ ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❍✐•❜❡rt s♣❛❝❡ H ✐♥ ✇❤✐❝❤ t❤❡ P❈❆ ❛♥❞ ▲❉❆ ❛r❡ ❞♦♥❡ ✐s t❤❡ s♣❛❝❡ W ′ ❜② ✐❞❡♥t✐❢②✐♥❣ ❛ r❡s✐❞✉❛• ζ ✇✐t❤ t❤❡ ❢❝✉rr❡♥t C (X,f +ζ) ∈ W ′ ✳ ◆❛t✉r❛• ❝❛♥❞✐❞❛t❡s ❢♦r ❦❡r♥❡•s ❛r❡ ❛•s♦ ♣r♦❞✉❝ts ♦❢ ❛ •❛✉ss✐❛♥ K g (x 1 , x 2 ) = e - |x 1 -x 2 | 2 σ 2 g ■❞ ✇✐t❤ k f (f 1 , f 2 ) = e - |f 1 -f 2 | 2 σ 2 f ✳ ❚❤❡ ❝♦rr❡•❛t✐♦♥ ♠❡❛s✉r❡ ❜❡t✇❡❡♥ r❡s✐❞✉❛•s ✐s t❤❡♥ ✿ C (X,ζ i ) , C (X,ζ j ) W ′ = X×X e - |x 1 -x 2 | 2 σ 2 g e - |ζ i (x 1 )-ζ j (x 2 )| 2 σ 2 f ξ(x 1 ), ξ(x 2 ) dH d (x 1 )dH d (x 2 ) = m k,l=1 e - |x k -x l | 2 σ 2 g e - |ζ i k -ζ i l | 2 σ 2 f ξ k , ξ l ✭✹✳✷✽✮ ✶✻✶ ✇❤❡r❡ ✇❡ ❛❞♦♣t❡❞ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s ❛s ✐♥ ❝❤❛♣t❡r ✷✱ ξ(x) = ξ(x)/|ξ(x)| ✐s t❤❡ ✉♥✐t d✲✈❡❝t♦r ❛t ❡❛❝❤ ♣♦✐♥t x ∈ X✳ ◆♦t❡ t❤❛t ♦♥❡ ❝❛♥ t❛❦❡ ♦t❤❡r ❦❡r♥❡• ✇✐❞t❤s σ g ❛♥❞ σ f t❤❛♥ t❤❡ ♦♥❡ ✉s❡❞ ❢♦r t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣❛rt✱ t❤✉s ❛••♦✇✐♥❣ ❛ ❝❡rt❛✐♥ ✢❡①✐❜✐•✐t② ♦❢ t❤❡ ❛♥❛•②s✐s ✇✐t❤ r❡s♣❡❝t t♦ ✈❛r✐♦✉s s❝❛•❡s ✭❛t t❤❡ ♣r✐❝❡ ♦❢ ❛❞❞✐t✐♦♥❛• ♣❛r❛♠❡t❡rs✮✳ ❙✉❝❤ ♠❡tr✐❝s ❤❛✈❡ t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❛❦✐♥❣ ♠♦r❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ s✉♣♣♦rt X ♦♥ ✇❤✐❝❤ ❛•• t❤❡ ζ i ❛r❡ ❞❡✜♥❡❞✳ ❇✉t t❤❡r❡ ❛r❡ t✇♦ ❝♦✉♥t❡r♣❛rts ✿ ✜rst✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ♠♦r❡ ❡①♣❡♥s✐✈❡ ❝♦♠♣❛r❡❞ t♦ L 2 ♥♦r♠ ❛♥❞ s❡❝♦♥❞✱ t❤❡ ✈❛r✐❛t✐♦♥ ♠♦❞❡s ❛♥❞ ❞✐s❝r✐♠✐♥❛♥t ❞✐r❡❝t✐♦♥s ❛r❡ ❜❛s✐❝❛••② •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❢❝✉rr❡♥ts C (X,f +ζ i ) ✇❤✐❝❤ ❝❛♥♥♦t ❜❡ ✐♥t❡r♣r❡t❡❞ ❛♥❞ ✈✐s✉❛•✐③❡❞ ❞✐r❡❝t•② ❛s ❛ s✐❣♥❛• ❞❡✜♥❡❞ ♦♥ X ❞✉❡ t♦ t❤❡ ♥♦♥✲•✐♥❡❛r✐t② ♦❢ k f ✳ ❖♥❝❡ t❤❡ ♠❡tr✐❝ ✐s ❝❤♦s❡♥ ❛♥❞ t❤❡ •r❛♠ ♠❛tr✐① ♦❢ t❤❡ ❞❛t❛s❡t ❝♦♠♣✉t❡❞✱ P❈❆ ❛♥❞ ▲❉❆ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ❡①❛❝t•② ❛s ♣r❡✈✐♦✉s•② ❡①♣•❛✐♥❡❞ ❢♦r ♠♦♠❡♥t❛✳ ■♥ ❜♦t❤ ❝❛s❡s✱ ✇❡ ♦❜t❛✐♥ ♠♦❞❡s ♦❢ ✈❛r✐❛t✐♦♥s t❤❛t ❛r❡ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ζ i ✬s ❛♥❞ t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s r❡s✐❞✉❛• ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t❡♠♣•❛t❡✳ ❈•❛ss✐✜❝❛t✐♦♥ ✐s t❤❡♥ ❞♦♥❡ ❛❣❛✐♥ ❜② ♣r♦•❡❝t✐♥❣ t❤❡ tr❛✐♥✐♥❣ ❛♥❞ t❡st✐♥❣ ❞❛t❛s❡ts ♦♥ t❤❡ •♦✇❡r✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡s ♣r♦✈✐❞❡❞ ❜② P❈❆ ❛♥❞ ▲❉❆✳ ✹✳✸✳✹ ❈•❛ss✐✜❝❛t✐♦♥ r❡s✉•ts ♦♥ t❤❡ ❖❈❚ ❞❛t❛s❡t ❲❡ ♥♦✇ s❤♦✇ ❛ ❢❡✇ r❡s✉•ts ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢r❛♠❡✇♦r❦ ♦♥ t❤❡ r❡t✐♥❛ t❤✐❝❦♥❡ss ♠❛♣s ♦❢ s❡❝t✐♦♥ ✹✳✷✳✸✳ ❲❡ ❤❛❞ ♦❜t❛✐♥❡❞✱ ✇✐t❤ ❛•❣♦r✐t❤♠ ✸✱ ❛♥ ❛t•❛s ♦❢ t❤❡ ◆❋▲ s✉r❢❛❝❡s ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❢✉♥❝t✐♦♥❛• s✉r❢❛❝❡ ❜❛s❡•✐♥❡ ✭t❤❡ t❡♠♣•❛t❡✮ ♣•✉s ♠♦♠❡♥t❛ p i ❛♥❞ r❡s✐❞✲ ✉❛•s t❤✐❝❦♥❡ss❡s ζ i ❢♦r ❡❛❝❤ ♦❢ t❤❡ N = 30 s✉❜•❡❝ts✳ ❇❛s❡❞ ♦♥ t❤❡s❡✱ ✇❡ ✐♥t❡♥❞ t♦ ✐••✉str❛t❡ t❤❡ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ ❣❡♦♠❡tr✐❝❛• ❛♥❞ ❢✉♥❝t✐♦♥❛• ✈❛r✐❛❜✐•✐t② ❞❡✈❡•♦♣❡❞ ✐♥ t❤✐s s❡❝t✐♦♥✳ ❙t❛rt✐♥❣ ✇✐t❤ t❤❡ ❛♥❛•②s✐s ♦♥ ♠♦♠❡♥t❛✱ ✐♥ ✜❣✉r❡ ✹✳✶✸✱ ✇❡ s❤♦✇ t❤❡ t❡♠♣•❛t❡ ❞❡✲ ❢♦r♠❡❞ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ t✇♦ ✜rst ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts✳ ❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ✜rst ❝♦♠♣♦♥❡♥t ✐s ❡ss❡♥t✐❛••② ❛ ❞✐•❛t❛t✐♦♥✴s❤r✐♥❦❛❣❡ ♦❢ t❤❡ ✇❤♦•❡ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✇❤✐•❡ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ♠♦r❡ ♦r •❡ss ❛ ❞✐s♣•❛❝❡♠❡♥t ❛♥❞ ❞✐st♦rt✐♦♥ ♦❢ t❤❡ ♦♣❡♥✐♥❣✳ ❆s ✇❡ ❝♦✉•❞ s✉♣♣♦s❡✱ t❤❡s❡ ❛r❡ ♠❛✐♥•② ❡①♣•❛✐♥❡❞ ❜② t❤❡ ✈❛r✐❛❜✐•✐t② ❝❛✉s❡❞ ❜♦t❤ ❜② ❛❝q✉✐s✐t✐♦♥ ✭♣♦s✐t✐♦♥ ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡②❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞❡✈✐❝❡ ❞✉r✐♥❣ t❤❡ ♠❡❛s✉r❡♠❡♥t✮ ❛♥❞ ❜② ❣•♦❜❛• ✈❛r✐❛t✐♦♥s ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ r❡t✐♥❛ ❛♥❞ ♦♣❡♥✐♥❣s ❛❝r♦ss ✐♥❞✐✈✐❞✉❛•s✳ ❋✐❣✉r❡ ✹✳✸✳✹ s❤♦✇s t❤❡ ♣r♦•❡❝t✐♦♥ ♦❢ t❤❡ ❞❛t❛s❡t ♦♥ t❤❡ ♣•❛♥❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡s❡ ♣r✐♥❝✐♣❛• ❝♦♠♣♦♥❡♥ts ❛♥❞ ❝♦♥✜r♠s t❤❛t t❤❡s❡ ❞♦ ♥♦t ❞✐s❝r✐♠✐✲ ♥❛t❡ s❛t✐s❢②✐♥❣•② ❜❡t✇❡❡♥ ❝♦♥tr♦•s ❛♥❞ ❣•❛✉❝♦♠❛t♦✉s✳ ❖♥❡ ✇♦✉•❞ ♦❜✈✐♦✉s•② ♥❡❡❞ t♦ ❝♦♠♣✉t❡ ♠♦r❡ P❈❆ ✈❡❝t♦rs t♦ ❣❡t ❞✐s❝r✐♠✐♥❛t✐✈❡ ❞✐r❡❝t✐♦♥s ❜✉t t❤❡ ✜rst ❝♦♠♣♦♥❡♥ts ❛r❡ ❝•❡❛r•② ♥♦t ❡♥♦✉❣❤✳ ❚❤✐s ✐s ❝♦♥tr❛st✐♥❣ ✇✐t❤ t❤❡ ▲❉❆ ✇❤✐❝❤ ♣r♦✈✐❞❡s ❛ ❝♦♠♣♦♥❡♥t ✈❡❝t♦r ❞❡t❡❝t✐♥❣ t❤❡ ❜❡t✇❡❡♥✲❝•❛ss ✈❛r✐❛❜✐•✐t②✳ ❆s ❢♦r P❈❆✱ ✇❡ s❤♦✇ t❤✐s ❞✐s❝r✐♠✐♥❛t✐✈❡ ❞✐r❡❝t✐♦♥ ♦♥ ✜❣✉r❡ ✹✳✶✺✳ ■t ✇❛s ♦❜t❛✐♥❡❞ ❜② r❡❣✉•❛r✐③❡❞ ▲❉❆ ✇✐t❤ ǫ = 0.1✳ ❖❜s❡r✈❡ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✲ ✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ✐s ✈❡r② ❞✐✛❡r❡♥t t♦ t❤❡ ♦♥❡s ♣r♦✈✐❞❡❞ ❜② t❤❡ ✜rst P❈❆ ❝♦♠♣♦♥❡♥ts✳ ❚❤❡ ❞✐s♣•❛❝❡♠❡♥ts ❛r❡ ❡ss❡♥t✐❛••② ❝♦♥❝❡♥tr❛t❡❞ ❛r♦✉♥❞ t❤❡ ♦❝✉✲ •❛r ♥❡r✈❡ ♦♣❡♥✐♥❣ ❛♥❞ s❡❡♠ t♦ ❝♦rr❡s♣♦♥❞ t♦ ❛♥ ✐♥❝r❡❛s❡ ❛♥❞ ❞❡❝r❡❛s❡ ♦❢ t❤❡ •❛②❡r ❞✐❣❣✐♥❣✳ ❆s ✇❡ s❡❡ t❤❛♥❦s t♦ ✜❣✉r❡ ✹✳✸✳✹✱ t❤✐s ♠♦❞❡ ♦❢ ✈❛r✐❛t✐♦♥ ✐s ❛♣♣❛r❡♥t•② ♠♦r❡ ❞✐s❝r✐♠✐♥❛♥t ❜❡t✇❡❡♥ ❣•❛✉❝♦♠❛t♦✉s ❛♥❞ ❝♦♥tr♦•s ❜❡❝❛✉s❡ ✐t s❡♣❛r❛t❡s t❤❡ ❝•❛ss❡s ✐♥ t✇♦ ♥❡❛r•② ❞✐st✐♥❝t ❝•✉st❡rs✳ ❚❤❡ s❛♠❡ ❛♥❛•②s✐s ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ❢♦r t❤❡ r❡s✐❞✉❛• t❤✐❝❦♥❡ss❡s ♦♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡✳ ❚❤✐s t✐♠❡✱ t❤❡ ❝♦♠♣♦♥❡♥t ✈❡❝t♦rs ♣r♦✈✐❞❡❞ ❜② P❈❆ ♦r ▲❉❆✱ ❛s ❞❡t❛✐•❡❞ ♣r❡✈✐✲ ♦✉s•②✱ ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ✇❤✐❝❤ ✐s ❛❞❞❡❞✴s✉❜tr❛❝t❡❞ t♦ t❤❡ ❜❛s❡•✐♥❡ t❤✐❝❦♥❡ss✳ ❲❡ ❤❛✈❡ ✉s❡❞ t❤❡ s✐♠♣•❡st ❝♦rr❡•❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② t❤❡ L 2 ✶✻✷ t❂✲✶ t❂✲✵✳✺ t❂✵ t❂✵✳✺ t❂✶ ❋✐❣✉r❡ ✹✳✶✸✿ ❚❤❡ ✜rst ✭♠✐❞❞•❡ r♦✇✮ ❛♥❞ s❡❝♦♥❞ ✭♠✐❞❞•❡ ❝♦•✉♠♥✮ ♣r✐♥❝✐♣❛• ❝♦♠♣♦✲ ♥❡♥ts ♦❢ t❤❡ ♠♦♠❡♥t❛✳ ❲❤❛t ✇❡ s❤♦✇ ✐s t❤❡ t❡♠♣•❛t❡ s❤♦♦t❡❞ ❢♦r ❜♦t❤ ♥❡❣❛t✐✈❡ ❛♥❞ ♣♦s✐t✐✈❡ t✐♠❡s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ P❈❆ ✈❡❝t♦rs✱ ❛s ❡①♣•❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ✶✻✸ ❋✐❣✉r❡ ✹✳✶✹✿ Pr♦•❡❝t✐♦♥s ❛♥❞ ❝•❛ss ♠❡❛♥s ♦❢ t❤❡ ❞❛t❛s❡t ♦♥ t❤❡ ♣•❛♥❡ ❣✐✈❡♥ ❜② t❤❡ t✇♦ ✜rst P❈❆ ✈❡❝t♦rs ❢♦r t❤❡ ♠♦♠❡♥t❛✳ ❈•❛ss ✵ ❝♦rr❡s♣♦♥❞s t♦ ❝♦♥tr♦•s ❛♥❞ ✶ t♦ ❣•❛✉❝♦♠❛t♦✉s s✉❜•❡❝ts✳ ❱✐❡✇ ✶ ❱✐❡✇ ✷ t❂✲✶ t❂✲✵✳✺ t❂✵ t❂✵✳✺ t❂✶ ❋✐❣✉r❡ ✹✳✶✺✿ |❡s✉•t✐♥❣ ▲❉❆ ❞✐r❡❝t✐♦♥ ♦♥ t❤❡ ♠♦♠❡♥t❛ ✭♦❜t❛✐♥❡❞ ✇✐t❤ ❛ r❡❣✉•❛r✐③❛✲ t✐♦♥ ♣❛r❛♠❡t❡r ǫ = 0.1✮✳ ❋✐❣✉r❡ ✹✳✶✻✿ Pr♦•❡❝t✐♦♥s ❛♥❞ ❝•❛ss ♠❡❛♥s ♦❢ t❤❡ ❞❛t❛s❡t ♦♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ❣✐✈❡♥ ❜② t❤❡ ▲❉❆ ♦♥ ♠♦♠❡♥t❛✳ ✶✻✹ ❇❛s❡•✐♥❡ ✲ P❈❆ r❡s✐❞✉❛• ❇❛s❡•✐♥❡ ❇❛s❡•✐♥❡ ✰ P❈❆ r❡s✐❞✉❛• ❋✐❣✉r❡ ✹✳✶✼✿ |❡s✉•ts ♦❢ t❤❡ P❈❆ ♦♥ t❤❡ t❤✐❝❦♥❡ss r❡s✐❞✉❛•s✳ ❲❡ ❝♦♠♣✉t❡❞ t✇♦ ❝♦♠♣♦♥❡♥t ✈❡❝t♦rs ❢♦r t❤❡ P❈❆✳ ❖♥ t❤❡ t♦♣ ✜❣✉r❡✱ ✇❡ r❡♣r❡s❡♥t t❤❡ ✜rst ♦♥❡ ❜② s❤♦✇✐♥❣ t❤❡ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ♦❢ t❤❡ r❡s✐❞✉❛• t♦ t❤❡ ❜❛s❡•✐♥❡ t❤✐❝❦♥❡ss✳ ❖♥ t❤❡ ❜♦tt♦♠ ✜❣✉r❡ ❛r❡ t❤❡ ♣r♦•❡❝t❡❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ r❡s✐❞✉❛•s ❢r♦♠ t❤❡ ❞❛t❛s❡t ♦♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛• s✉❜s♣❛❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ P❈❆ ✈❡❝t♦rs✳ ✶✻✺ ❇❛s❡•✐♥❡ ✲ ▲❉❆ r❡s✐❞✉❛• ❇❛s❡•✐♥❡ ❇❛s❡•✐♥❡ ✰ ▲❉❆ r❡s✐❞✉❛• ❋✐❣✉r❡ ✹✳✶✽✿ |❡s✉•ts ♦❢ t❤❡ r❡❣✉•❛r✐③❡❞ ▲❉❆ ♦♥ t❤❡ t❤✐❝❦♥❡ss r❡s✐❞✉❛•s ✭ǫ = 0.1✮✳ ■t ♣r♦✈✐❞❡s ♦♥❡ ❝♦♠♣♦♥❡♥t ✈❡❝t♦r ✇❤✐❝❤ ✇❡ r❡♣r❡s❡♥t ♦♥ t❤❡ t♦♣ ✜❣✉r❡✳ ❚❤❡ ♣r♦•❡❝t✐♦♥s ♦♥ t❤✐s ❞✐r❡❝t✐♦♥ ❛r❡ s❤♦✇♥ ♦♥ t❤❡ •♦
♥ ♠♦♠❡♥t❛ ❛♥❞ t❤✐❝❦♥❡ss ♠❛♣s s❤♦✉•❞ ♣r♦✈✐❞❡ ❛ s✐♠♣•❡ ✇❛② t♦ ❝•❛ss✐❢②✱ ❛t •❡❛st ✇✐t❤ ❛ r❡❛s♦♥❛❜•❡ ❡✣❝✐❡♥❝②✱ ❜❡t✇❡❡♥ ❝♦♥tr♦•s ❛♥❞ ❣•❛✉❝♦♠❛✳ ❚❤✐s ❝♦✉•❞ ❜❡ ❛ss❡ss❡❞ ♠♦r❡ r✐❣♦r♦✉s•② ❜② ❛♣♣•②✐♥❣ t❤❡ ♣r❡❞✐❝t✐♦♥ r✉•❡ •❡❛r♥❡❞ ❜② ▲❉❆ ♦♥ ❛ t❡st✐♥❣ s❡t ♦❢ ♥❡✇•② ❛❝q✉✐r❡❞ s✉❜•❡❝ts ❛s ✇❡ ❡①♣•❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤✐s ♣❛rt✐❝✉•❛r ❛♣♣•✐❝❛t✐♦♥✱ ✇❡ ❞✐❞ ♥♦t ❤❛✈❡ ②❡t ♥❡✇ s✉❜•❡❝ts ❛✈❛✐•❛❜•❡ t♦ ✇♦r❦ ♦♥ ✭t❤✐s ✐s ❛ ❝✉rr❡♥t ✇♦r❦ ❢r♦♠ ♦✉r ❝♦••❛❜♦r❛t♦rs✬ s✐❞❡ ✐♥ ❙❋❯✮✳ ■♥st❡❛❞✱ ✇❡ t❡st❡❞ t❤❡ q✉❛•✐t② ♦❢ ♣r❡❞✐❝t✐♦♥ ❜② ❝r♦ss✲✈❛•✐❞❛t✐♦♥ t❡sts ♦♥ t❤❡ •❡❛r♥✐♥❣ ❞❛t❛s❡t ✐ts❡•❢✳ ◆❛♠❡•②✱ ✇❡ ❛♣♣•✐❡❞ •❡❛✈❡✲♦♥❡ ♦✉t ♣r♦❝❡❞✉r❡s ✇❤✐❝❤ ❝♦♥s✐st ✐♥ ❡①❝•✉❞✐♥❣ ♦♥❡ ✭♦r ♠♦r❡ ❣❡♥❡r❛••② p✮ s✉❜•❡❝t ❢r♦♠ t❤❡ ❞❛t❛s❡t ❛♥❞ ♣r❡❞✐❝t t❤❡ ❝•❛ss ♦❢ t❤❡s❡ s✉❜•❡❝ts ❜② ✉s✐♥❣ t❤❡ r❡♠❛✐♥❞❡r✳ ❚❤✐s ✐s ❡①❛❝t•② ❛s ✐❢ ♦♥❡ ❛rt✐✜❝✐❛••② s❡♣❛r❛t❡s t❤❡ ❞❛t❛s❡t ✐♥t♦ tr❛✐♥✐♥❣ ❛♥❞ t❡st✐♥❣ s❡ts✳ ■♥ ♣r❛❝t✐❝❛• t❡r♠s✱ ✇❡ r❛♥❞♦♠•② ❝❤♦♦s❡ p s✉❜•❡❝ts ❛♠♦♥❣ t❤❡ N = 30 ❛✈❛✐•❛❜•❡✱ ❡st✐♠❛t❡ ❛ t❡♠♣•❛t❡ ❛♥❞ ▲❉❆ ❛①✐s ✇✐t❤ t❤❡ N -p ♦t❤❡rs✱ t❤❡♥ ♣r♦•❡❝t t❤❡ p ✬t❡st✬ s✉❜•❡❝ts ♦♥ t❤❡ ▲❉❆ s✉❜s♣❛❝❡ ❛♥❞ ❛✛❡❝t ❡❛❝❤ ♦❢ t❤❡♠ t♦ t❤❡ ❝•❛ss ♦❢ t❤❡ ❝•♦s❡st ❝•❛ss ♠❡❛♥✳ ❇② r❡♣❡❛t✐♥❣ t❤✐s ♣r♦❝❡❞✉r❡ ❢♦r ♥✉♠❡r♦✉s ❞r❛✇s ♦❢ p ❛♠♦♥❣ N ✱ ♦♥❡ ❝❛♥ ✶✻✻ ❉❛t❛ ♣❂✶ ♣❂✷ ♣❂✸ ♣❂✹ ▼♦♠❡♥t❛ ✵✳✶✵✵ ✵✳✵✽✷ ✵✳✵✾✵ ✵✳✵✽✾ ❚❤✐❝❦♥❡ss r❡s✐❞✉❛•s ✵✳✵✸✸ ✵✳✵✹✵ ✵✳✵✹✶ ✵✳✵✺✻ ▼♦♠❡♥t❛ ❛♥❞ t❤✐❝❦♥❡ss ✵✳✵✵✵ ✵✳✵✵✽ ✵✳✵✷✶ ✵✳✵✸✷ ❚❛❜•❡ ✹✳✶✿ ▲❉❆ ❝•❛ss✐✜❝❛t✐♦♥ ❡rr♦r r❛t❡s ✐♥ ❝r♦ss ✈❛•✐❞❛t✐♦♥ ✭N = 30✮ ❢♦r •❡❛✈❡✲p✲♦✉t t❡sts✳ ❉❛t❛ ♣❂✶ ♣❂✷ ♣❂ ✸ ♣❂✹ ▼♦♠❡♥t❛ ✵✳✶✺✵ ✵✳✶✼✹ ✵✳✶✽✸ ✵✳✷✵✼ ❚❤✐❝❦♥❡ss r❡s✐❞✉❛•s ✵✳✵✵✵ ✵✳✵✵✽ ✵✳✵✶✽ ✵✳✵✸✺ ▼♦♠❡♥t❛ ❛♥❞ t❤✐❝❦♥❡ss ✵✳✵✺✵ ✵✳✵✺✸ ✵✳✵✺✼ ✵✳✵✻✼ ❚❛❜•❡ ✹✳✷✿ ▲❉❆ ❝•❛ss✐✜❝❛t✐♦♥ ❡rr♦r r❛t❡s ♦♥ t❤❡ ♦•❞ s✉❜•❡❝t s✉❜✲❞❛t❛s❡t ✭N = 20✮✳ t❤❡♥ ♠❡❛s✉r❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛•s❡ ❝•❛ss ♣r❡❞✐❝t✐♦♥s✳ ◆♦t❡ t❤❛t ✇❡ ❛•s♦ tr② s❡✈❡r❛• r❡❣✉•❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡rs t♦ ♦♣t✐♠✐③❡ t❤❡ ❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s✳ ❚❤✐❝❦♥❡ss ❛♥❞ ♠♦✲ ♠❡♥t❛ ❝♦♦r❞✐♥❛t❡s ❝❛♥ ❜❡ ❛•s♦ ❝♦♠❜✐♥❡❞ ✐♥ t❤❡ ❝•❛ss✐✜❝❛t✐♦♥ ✿ ❤❡r❡ ✇❡ ❤❛✈❡ tr✐❡❞ ❛ s✐♠♣•❡ ℓ 1 ❞✐st❛♥❝❡ t♦ ❝•❛ss ❝❡♥t❡rs ✐♥ t❤❡ ✷❉ s♣❛❝❡ ♣r♦✈✐❞❡❞ ❜② ▲❉❆ ❝♦♦r❞✐♥❛t❡s ❢♦r ❜♦t❤ ♠♦♠❡♥t❛ ❛♥❞ r❡s✐❞✉❛•s✳ ❚❤❡ r❡s✉•ts ❛r❡ s✉♠♠❡❞ ✉♣ ✐♥ t❛❜•❡ ✹✳✶✳ ❈•❛ss✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ♠♦♠❡♥t❛ ♦r t❤✐❝❦♥❡ss❡s •❡❛❞s t♦ ♠✐s❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s ♦❢ t❤❡ ♦r❞❡r ♦❢ 9%✱ ✇❤✐❝❤ ♦❢ ❝♦✉rs❡ ✐♥❝r❡❛s❡s s•♦✇•② ❛s ✇❡ r❡♠♦✈❡ ♠♦r❡ s✉❜•❡❝ts ❢r♦♠ t❤❡ tr❛✐♥✐♥❣ s❡t ✭✐✳❡ ✇❤❡♥ p ✐♥❝r❡❛s❡s✮✳ ❲❡ r❡♠❛r❦ t❤❛t ❡rr♦r r❛t❡s ❛r❡ ❛•s♦ s•✐❣❤t•② ❜❡tt❡r ✉s✐♥❣ t❤✐❝❦✲ ♥❡ss❡s✳ ❍♦✇❡✈❡r✱ t❤❡ ♠✐s❝•❛ss✐✜❝❛t✐♦♥s ❞♦ ♥♦t ♦❝❝✉r ❣❡♥❡r❛••② ❢♦r t❤❡ s❛♠❡ s✉❜•❡❝ts ✐♥ t❤❡ t✇♦ ♠❡t❤♦❞s✳ ■t r❡s✉•ts t❤❛t t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ♣r♦✈✐❞❡s ✈❡r② s♦•✐❞ ❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s ✐♥ t❤❛t ❝❛s❡ s✐♥❝❡ ✇❡ ❣❡t ❛♥ ❡rr♦r ✉♥❞❡r 1% ❢♦r p = 1, 2✳ ◆❛t✉r❛••②✱ t❤❡s❡ ❛r❡ st✐•• ❝r✉❞❡ r❡s✉•ts ❛t t❤❡ ♣r❡s❡♥t t✐♠❡✳ ❆♥ ✐♠♣♦rt❛♥t ❢♦••♦✇✲✉♣ st✉❞② ✐s t♦ ✐♥t❡r♣r❡t ❢r♦♠ ❛ ♠♦r❡ ❝•✐♥✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇ t❤❡s❡ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♠♦❞❡s ♦❢ ✈❛r✐❛t✐♦♥s ❢♦✉♥❞ ❜② ▲❉❆✱ ❡s♣❡❝✐❛••② t❤❡ ❣❡♦♠❡tr✐❝❛• ❞❡❢♦r♠❛t✐♦♥s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐t✬s ❛•s♦ ✐♥t❡r❡st✐♥❣ t♦ ❛♥❛•②③❡ ✐♥❝✐❞❡♥t❛• ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❞❛t❛s❡t ✐♥ ♦r❞❡r t♦ ❡•✐♠✐♥❛t❡ ❤✐❞❞❡♥ ❡✛❡❝ts t❤❛t ❛r❡ ♥♦t s♣❡❝✐✜❝❛••② ❝♦rr❡•❛t❡❞ t♦ t❤❡ ❞❡✈❡•♦♣♠❡♥t ♦❢ ❣•❛✉❝♦♠❛✳ ❚❤❡ s✉❜•❡❝ts✬ ❛❣❡ ✐s t②♣✐❝❛••② ❛♥ ✐♠♣♦rt❛♥t ♦♥❡ ❜❡❝❛✉s❡ ✐t ❣❡♥❡r❛••② ❛✛❡❝ts t❤❡ s❤❛♣❡ ♦❢ t❤❡ r❡t✐♥❛✳ ❙✐♥❝❡ ✐t ✐s ❛ ❦♥♦✇♥ ❞❛t❛✱ ✇❡ ❛•s♦ ❝♦♥❞✉❝t❡❞ t❤❡ s❛♠❡ ❝•❛ss✐✜❝❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ❜✉t r❡str❛✐♥❡❞ t♦ ♦♥•② ♦•❞ s✉❜•❡❝ts ❢r♦♠ t❤❡ ❞❛t❛s❡t ✭✇❤✐❝❤ ❣✐✈❡s ✷✵ s✉❜•❡❝ts✱ ❝♦♥tr♦•s ❛♥❞ ❣•❛✉❝♦♠❛t♦✉s✮✳ ❚❤❡ ❝r♦ss✲✈❛•✐❞❛t✐♦♥ ❝•❛ss✐✜✲ ❝❛t✐♦♥ r❡s✉•ts ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❛❜•❡ ✹✳✷✳ ❲❡ s❡❡ t❤❛t t❤❡ ❝•❛ss✐✜❝❛t✐♦♥ ♣❡r❢♦r♠❛♥❝❡ ❜❛s❡❞ ♦♥ ❣❡♦♠❡tr② ✇✐t❤ t❤❡ ♠♦♠❡♥t❛ ✐s •❡ss ❛❝❝✉r❛t❡ ✐♥ t❤❛t s✐t✉❛t✐♦♥ ✭❛r♦✉♥❞ 17% ❡rr♦r✮✱ ✇❤❡r❡❛s t❤❡ ♦♥❡ ❜❛s❡❞ ♦♥ t❤❡ t❤✐❝❦♥❡ss ❛♥❛•②s✐s ✐♠♣r♦✈❡s q✉✐t❡ s♣❡❝t❛❝✉•❛r•②✱ s✉❣❣❡st✐♥❣ t❤❛t t❤✐❝❦♥❡ss ❞❡❝r❡❛s❡ ✐s ♣r♦❜❛❜•② ❛ ❜❡tt❡r ❜✐♦♠❛r❦❡r ♦❢ t❤❡ ❞✐s❡❛s❡ t❤❛♥ s❤❛♣❡ ✈❛r✐❛t✐♦♥s ✇❤✐❝❤ ❝♦✉•❞ ❜❡ ♣❛rt•② ❝❛✉s❡❞ ❜② ❛❣❡✳ ❚❤✐s ✐s ❛ st✉❞② t♦ ❜❡ ❝♦♥✲ t✐♥✉❡❞ ❛s ❛ ❢✉t✉r❡ ✇♦r❦✱ ♦♥❡ ♣♦ss✐❜•❡ tr❛✐• ❜❡✐♥❣ t♦ s❡♣❛r❛t❡ t❤❡ ❞❛t❛s❡t ✐♥t♦ ♠♦r❡ ❝•❛ss❡s ❢♦r ✐♥st❛♥❝❡ ❜② ❛❞❞✐♥❣ ❛❣❡ ❛♥❞ ♣❡r❢♦r♠ ♠✉•t✐✲❝•❛ss ▲❉❆ ✿ t❤✐s ✇♦✉•❞ ❜❡❝♦♠❡ ♣❛rt✐❝✉•❛r•② r❡•❡✈❛♥t st❛t✐st✐❝❛••② ✐❢ t❤❡ ❞❛t❛s❡t ✐s ❡♥r✐❝❤❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛• s✉❜•❡❝ts✬ ❛❝q✉✐s✐t✐♦♥s✳ ✶✻✼ ✹✳✹ ❆t•❛s ❡st✐♠❛t✐♦♥ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✈❛r✐❢♦•❞s ❛♥❞ ❛♣✲ ♣•✐❝❛t✐♦♥s ❚❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡s ♦❢ s❡❝t✐♦♥ ✹✳✷ ❝❛♥ ❜❡ r❡❛❞✐•② tr❛♥s♣♦s❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ ✈❛r✐❢♦•❞s ❛♥❞ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s t❤❛t ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ❆•• ✐t ♥❡❡❞s ❜❛s✐❝❛••② ✐s t♦ r❡♣•❛❝❡ ❢❝✉rr❡♥ts✬ ♥♦r♠s ❜② ✈❛r✐❢♦•❞s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❣r❛❞✐❡♥ts ❢♦r t❤❡ ✉♣❞❛t❡ ♦❢ t❤❡ t❡♠♣•❛t❡ ♣♦s✐t✐♦♥✱ ♠♦♠❡♥t❛ ❛♥❞ r❡s✐❞✉❛•s✳ ❈♦♠♣❛r❡❞ t♦ t❤❡ ♣r❡✈✐♦✉s ❛•❣♦r✐t❤♠s✱ t❤❡ ❛❞✈❛♥t❛❣❡s ❛r❡ t❤❡ ♦♥❡ ❞❡t❛✐•❡❞ ✐♥ ❝❤❛♣t❡r ✸ ✿ t❤❡ ♣♦ss✐❜✐•✐t② t♦ ✇♦r❦ ✇✐t❤ ♥♦♥✲♦r✐❡♥t❡❞ ♠❛♥✐❢♦•❞s ❛♥❞ t❤❡ ❣r❡❛t❡r st❛❜✐•✐t② ♦❢ t❤❡ ❡st✐♠❛t✐♦♥ ❛t ❜✐❣❣❡r s❝❛•❡s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❡ss❡♥t✐❛••② ♣r❡s❡♥t t✇♦ ♦♥❣♦✐♥❣ ❛♣♣•✐❝❛t✐♦♥s ♦❢ t❤❛t ♦♥ r❡❛• ❞❛t❛s❡ts ♦❢ ♣✉r❡•② ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s ✿ ❛ ✜rst ♦♥❡ ❝♦♥❝❡r♥s t❤❡ st✉❞② ♦❢ ❝❡••s✬ ❞✐✈✐s✐♦♥s ✐♥ ♠✐❝❡ ❤❡❛rt ❡♥✈❡•♦♣❡s ✇❤✐•❡ t❤❡ s❡❝♦♥❞ ✇✐•• ♣r❡s❡♥t ❛t•❛s❡s ♦❢ ❈♦❝❤•❡❛✬s s✉r❢❛❝❡s✳ ✹✳✹✳✶ ❆•❣♦r✐t❤♠ ❚❤❡ ❛•❣♦r✐t❤♠s ❛r❡ ✈❡r② s✐♠✐•❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡s s♦ ✇❡ ✇✐•• •✉st ❜r✐❡✢② ❤✐❣❤•✐❣❤t t❤❡ ❢❡✇ ❞✐✛❡r❡♥❝❡s✳ ■❢ ✇❡ ❝♦♥s✐❞❡r ♣✉r❡ ❣❡♦♠❡tr✐❡s ✇✐t❤ ♥♦ s✐❣♥❛•s✱ ❛t•❛s ❡st✐♠❛t✐♦♥ ✐♥ t❤❛t ❝♦♥t❡①t ✇r✐t❡s ✿ min (X,p i 0 ) J(X, (p i 0 )) . = N i=1 v i 0 2 V + γ µ X i -µ φ i (X) 2 W ′ ✭✹✳✷✾✮ µ X ❞❡♥♦t❡s t❤❡ ✈❛r✐❢♦•❞ ❛ss♦❝✐❛t❡❞ t♦ X ❛♥❞ . W ′ ✐s ❛ |❑❍❙ ♥♦r♠ ♦♥ ✈❛r✐❢♦•❞s ❛s ❞❡✜♥❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ◆♦✇ t❤❡ ❣r❛❞✐❡♥ts ♦❢ J ✇✐t❤ r❡s♣❡❝t t♦ X ❛♥❞ t❤❡ ♠♦♠❡♥t❛ p i 0 ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛•s♦ ❜② t❤❡ s❛♠❡ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❞②♥❛♠✐❝s✳ ❖♥•② t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ ❛❞•♦✐♥t ✈❛r✐❛❜•❡s Q i ✐♥ t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥ ❡q✳✭✹✳✶✵✮ ♠✉st ❜❡ ❝❤❛♥❣❡❞ t♦ Q i 1 = -∇ q g i (q i 1 ) ✇❤❡r❡ g i ✐s t❤❡ sq✉❛r❡❞ ❞✐st❛♥❝❡ t♦ t❤❡ i th s✉❜•❡❝t ❜✉t ♥♦✇ ❢♦r t❤❡ ✈❛r✐❢♦•❞ ♠❡tr✐❝✳ ❚❤❡s❡ ❣r❛❞✐❡♥ts ❛r❡ ❞✐r❡❝t•② ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❝♦♠♣✉✲ t❛t✐♦♥s ♣r❡s❡♥t❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ❚❤❡ ♥♦r♠❛•✐③❛t✐♦♥s ♦❢ ❣r❛❞✐❡♥ts ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✷✳✶ ❛♥❞ ✹✳✷✳✶ ❛r❡ st✐•• ✈❛•✐❞ ❢♦r ✈❛r✐❢♦•❞s✳ ❆♥❞ ✜♥❛••②✱ t❤❡ ✈❛r✐❛t✐♦♥ ❢♦r♠✉•❛ s❤♦✇♥ ✐♥ t❤❡♦r❡♠ ✸✳✹✳✷ ❜r✐♥❣s ♦✉t s✐♥❣✉•❛r✐t✐❡s ♦♥ t❤❡ ❜♦✉♥❞❛r② ❢♦r t❤❡ ❣r❛❞✐❡♥t ✇❤✐❝❤ ❝❛♥ ❜❡ t❛❝❦•❡❞✱ ❛s ✐♥ ✹✳✷✳✶✱ ❜② r❡❣✉•❛r✐③❛t✐♦♥✳ ◆♦✇ t❤❡ ❢✉♥❝t✐♦♥❛• ❝♦♠♣♦♥❡♥t ❝❛♥ ❜❡ ❛•s♦ ❛❞❞❡❞ t♦ t❤❡ ✈❛r✐❢♦•❞ ❢r❛♠❡✇♦r❦✱ ❢♦••♦✇✲ ✐♥❣ ✇❤❛t ✇❛s ❡①♣♦s❡❞ ✐♥ ❝❤❛♣t❡r ✸ s❡❝t✐♦♥ ✸✳✻✳✷✳ ■t r❡s✉•ts t❤❛t ❛•❣♦r✐t❤♠ ✸ ✐s ❛•s♦ tr❛♥s♣♦s❛❜•❡ ✇✐t❤ ♠✐♥✐♠❛• ❝❤❛♥❣❡s t♦ ✈❛r✐❢♦•❞s✱ ♣r♦✈✐❞✐♥❣ ❛♥ ❛•t❡r♥❛t✐✈❡ t♦ ❢❝✉rr❡♥ts ❢♦r ❡st✐♠❛t✐♥❣ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❛t•❛s❡s✳ ✹✳✹✳✷ ❲♦r❦s ♦♥ ❡♠❜r②♦♥✐❝ ♠♦✉s❡ ❤❡❛rt ❆s ❛ ✜rst ✐••✉str❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡r❡st ♦❢ ✈❛r✐❢♦•❞s✬ ❛t•❛s ❡st✐♠❛t✐♦♥ ❢r♦♠ ❛ ♠♦r❡ ❛♣♣•✐❡❞ ♣♦✐♥t ♦❢ ✈✐❡✇ ✐♥ t❤❡ ✜❡•❞ ♦❢ ❜✐♦•♦❣✐❝❛• ✐♠❛❣✐♥❣✱ ✇❡ ♣r❡s❡♥t ❛ ✜rst s❡r✐❡s ♦❢ r❡s✉•ts ❢r♦♠ ❛♥ ♦♥❣♦✐♥❣ ♣r♦•❡❝t ✐♥ ❝♦••❛❜♦r❛t✐♦♥ ✇✐t❤ ❙✳ ▼❡✐•❤❛❝ ❛♥❞ ❏✳❋ ▲❡ •❛rr❡❝ ❢r♦♠ t❤❡ t❡❛♠ ▼♦•❡❝✉•❛r •❡♥❡t✐❝s ♦❢ ❉❡✈❡•♦♣♠❡♥t ✐♥ ■♥st✐t✉t P❛st❡✉r✳ ■t ❝♦♥s✐sts ✐♥ t❤❡ ❛♥❛•②s✐s ♦❢ ❝❡••s✬ ❞✐r❡❝t✐♦♥s ♦❢ ❞✐✈✐s✐♦♥s ❞✉r✐♥❣ t❤❡ ❣r♦✇t❤ ♦❢ ❤❡❛rts ✐♥ ♠✐❝❡ ❡♠❜r②♦s✱ ❛s ♣✐♦♥❡❡r❡❞ ✐♥ ❬✺✸(✳ ❚❤❡ ❞❛t❛s❡t ✐ts❡•❢ ✐s ♠❛❞❡ ♦❢ s❡✈❡r❛• s✉❜•❡❝ts✬ ❡♥✈❡•♦♣❡s✱ ✇❤✐❝❤ ❛r❡ s✉r❢❛❝❡s ✐♥ t❤❡ ✸❉ s♣❛❝❡✳ ❚❤❡s❡ ✇❡r❡ ❛❝q✉✐r❡❞ ❜② ❝♦♥❢♦❝❛• ♠✐❝r♦s❝♦♣② ❛♥❞ ❞❡•✐♠✐t ♦♥•② ❛ s❡❝t✐♦♥ ♦♥ t♦♣ ♦❢ t❤❡ ❤❡❛rt ♦❢ t❤❡ ❡♠❜r②♦ ✭✉♥t✐• ❛ ❞❡♣t❤ •✐♠✐t❡❞ ❜② t❤❡ ❛❝q✉✐s✐t✐♦♥ ❞❡✈✐❝❡✮✳ ❚❤❡ ❞❡♣t❤ ♦❢ ❛❝q✉✐s✐t✐♦♥ ✐s r❛t❤❡r s♠❛•• ❛♥❞ t❤✉s t❤❡ s✉r❢❛❝❡s ❛r❡ t❤✐♥ ♠❡♠❜r❛♥❡s✳ ■♥s✐❞❡ t❤❡s❡ ❡♥✈❡•♦♣❡s✱ ✇❡r❡ ♠❡❛s✉r❡❞✱ ❛t t❤❡ t✐♠❡ ♦❢ ❡♠❜r②♦✬s ✶✻✽ s❛♠♣•✐♥❣✱ t❤❡ ❞✐✈✐s✐♦♥s ❛①❡s✬ ✭✉♥♦r✐❡♥t❡❞✮ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❝❡••s ✇❤✐❝❤ ✇❡r❡ ✐♥ ❛ ♠✐t♦s✐s ♣r♦❝❡ss ❛t t❤✐s ♣r❡❝✐s❡ ♠♦♠❡♥t✳ ❲❡ s❤♦✇ ✐♥ ✜❣✉r❡ ✹✳✶✾ t❤❡ ❢✉•• ❞❛t❛s❡t ✇✐t❤ t❤❡ ♥✐♥❡ s✉❜•❡❝ts ❡♥✈❡•♦♣❡s ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐✈✐s✐♦♥ ❛①❡s ✐♥s✐❞❡ ✭❡①❝❡♣t ❢♦r ❛ ❢❡✇ ❛rt✐❢❛❝ts •♦❝❛t❡❞ ♦✉ts✐❞❡ t❤❡ ❡♥✈❡•♦♣❡s ♦♥ s♦♠❡ s✉❜•❡❝ts✮✳ ❚❤❡ ❜✐♦•♦❣✐❝❛• ✐ss✉❡s ❛t st❛❦❡ ❛r❡ s❡✈❡r❛•✳ ❆s ✇❡ ❝❛♥ s❡❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ♠❡❛s✉r❡❞ ❞✐✈✐s✐♦♥s✬ ❛①❡s ✇✐t❤✐♥ ❛ s✉❜•❡❝t ✐s r❛t❤❡r s♠❛•• ✐♥ ❛ ❣✐✈❡♥ ❡♥✈❡•♦♣❡✳ ❖♥❡ ♦❢ t❤❡ q✉❡s✲ t✐♦♥ r❛✐s❡❞ ❜② ❜✐♦•♦❣✐sts ✐s t♦ ✜♥❞ ✐❢ s♦♠❡ r❡❣✐♦♥s s❤♦✇ s♦♠❡ ♠❡❛♥✐♥❣❢✉• ❝♦rr❡•❛t✐♦♥s ♦❢ t❤❡ ❞✐✈✐s✐♦♥s✬ ❞✐r❡❝t✐♦♥s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❢❡❛t✉r❡s ♦❢ t❤❡ ❤❡❛rt✬s ❣r♦✇t❤✳ ❍♦✇✲ ❡✈❡r✱ ❡st✐♠❛t✐♥❣ ❝♦rr❡•❛t✐♦♥s ♦♥ s✉❜•❡❝ts s❡♣❛r❛t❡•② ✇♦✉•❞ •❡❛❞ t♦ ✈❡r② ✉♥st❛❜•❡ ❛♥❞ ♥♦♥✲r❡♣r♦❞✉❝✐❜•❡ r❡s✉•ts ❞✉❡ t♦ t❤❡ •♦✇ ♥✉♠❜❡r ♦❢ ❛①❡s ❢♦r ❡❛❝❤ ♦♥❡✳ ❚❤✉s ♦♥❡ ♥❡❡❞s ❛ ❢r❛♠❡✇♦r❦ t♦ ❣❛t❤❡r t❤❡ ✐♥❢♦r♠❛t✐♦♥s ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t s✉❜•❡❝ts ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ ❞❡♥s❡r ❞✐str✐❜✉t✐♦♥ ♦❢ ❛①❡s✳ ❚❤❡ s❡❝♦♥❞ ✐♠♣♦rt❛♥t ✐ss✉❡✱ t❤❛t ✇❡ ❤❛✈❡ •✉st t♦✉❝❤❡❞ ✉♣♦♥ ❢♦r t❤❡ t✐♠❡ ❜❡✐♥❣✱ ✐s t♦ st✉❞② t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ❛ ♠✉t❛t✐♦♥ ♦♥ ❛ ♣❛rt✐❝✉•❛r ❣❡♥❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s✬ r❡♣❛rt✐t✐♦♥✳ ❚❤❡ ❞❛t❛s❡t ❝♦♥t❛✐♥s ❛❝t✉❛••② t❤r❡❡ s✉❜•❡❝ts ♦❢ ❡❛❝❤ ❝•❛ss ✿ ✇✐•❞ s♣❡❝✐❡s ✭✇✐t❤ t✇♦ ♥♦♥✲♠✉t❛t❡❞ ❣❡♥❡s✮✱ ❤❡t❡r♦③②❣♦✉s ❛♥❞ ♠✉t❛♥ts✳ ■♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ ❛ ✇❛② t♦ ❣❛t❤❡r t♦❣❡t❤❡r ❛①❡s ❞❛t❛ ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t s✉❜•❡❝ts ✐♥ ♦♥❡ ❝♦♠♠♦♥ s❡t ♦❢ ❞✐✈✐s✐♦♥s✳ ❆ r❡❛s♦♥❛❜•❡ ✇❛② ✐s t♦ ♥♦r♠❛•✐③❡ ❣❡♦♠❡tr✐❡s ♦❢ t❤❡ s✉❜•❡❝ts ❜② ❝♦♠♣✉t✐♥❣ ❞❡❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❡♥✈❡•♦♣❡s ❛♥❞ tr❛♥s♣♦rt t❤❡ ❞✐r❡❝t✐♦♥s t❤r♦✉❣❤ t❤❡s❡ ❞❡❢♦r♠❛t✐♦♥s✱ ✐✳❡ t❤r♦✉❣❤ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠s ❛s ❛•r❡❛❞② ❞❡s❝r✐❜❡❞ ❜② ❡q✳✭✷✳✸✮✳ ■♥st❡❛❞ ♦❢ ❝❤♦♦s✐♥❣ ♦♥❡ ♦❢ t❤❡ s✉❜✲ •❡❝ts ❛s ❛ r❡❢❡r❡♥❝❡ ✇✐t❤ t❤❡ ❜✐❛s t❤❛t t❤✐s ✇♦✉•❞ ✐♥tr♦❞✉❝❡ ✐♥ t❤❡ ♠❡t❤♦❞✱ ✇❡ ♣r♦♣♦s❡ ✐♥st❡❛❞ t♦ ❝♦♠♣✉t❡ ✜rst ❛ t❡♠♣•❛t❡ ♦❢ t❤❡ ❡♥✈❡•♦♣❡ s❤❛♣❡s✳ ❚❤❡ ❡ss❡♥t✐❛• ❞✐✣❝✉•t✐❡s ❤❡r❡ ❝♦♠❡s ❢r♦♠ ❛ ❢❡✇ ❛rt✐❢❛❝ts ♦♥ t❤❡ ❡♥✈❡•♦♣❡s✬ tr✐❛♥❣✉•❛t✐♦♥s ❛♥❞ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ ♠❡♠❜r❛♥❡s ❛r❡ ✉s✉❛••② ✈❡r② t❤✐♥✱ ✇❤✐❝❤ ❝❛✉s❡s ✐♠♣♦rt❛♥t ❝❛♥❝❡••❛t✐♦♥ ♣r♦❜•❡♠s ✇❤❡♥ ✉s✐♥❣ ✉s✉❛• ❝✉rr❡♥ts ✭❝❢ ❞✐s❝✉ss✐♦♥ ♦❢ ❝❤❛♣t❡r ✸ s❡❝t✐♦♥ ✸✳✶✳✷✮✳ ❋r♦♠ t❤✐s ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ✈❛r✐❢♦•❞s ❛♥❞ t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❢r❛♠❡✲ ✇♦r❦ ♣r❡s❡♥t❡❞ •✉st ♣r❡✈✐♦✉s•② s❡❡♠s ♠♦r❡ ❛❞❛♣t❡❞✳ ❲❡ ✐♥✐t✐❛•✐③❡ t❤❡ t❡♠♣•❛t❡ t♦ ❛♥ ❡••✐♣s♦✐❞ t♦ ❛✈♦✐❞ ❜✐❛s ❛s ♠✉❝❤ ❛s ♣♦ss✐❜•❡ ❛♥❞ r✉♥ t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ ✈❛r✐❢♦•❞s ✿ t❤✐s •❡❛❞s t♦ t❤❡ t❡♠♣•❛t❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✹✳✷✵✳ ❆ s❡❝♦♥❞ st❡♣ ✐s t♦ ❝♦♠♣✉t❡ t❤❡ ❞❡❢♦r♠❛t✐♦♥s ❢r♦♠ ❡❛❝❤ s✉❜•❡❝t t♦ t❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ✭♥♦t❡ t❤❛t t❤❡s❡ ❛r❡ ♠♦r❡ ♦r •❡ss t❤❡ ✐♥✈❡rs❡s ♦❢ t❤❡ ♠❛♣s φ i ❣✐✈❡♥ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥✮✳ ❋✐♥❛••②✱ ✇❡ ♣✉•• t❤❡ ❛①❡s✬ ❞✐r❡❝t✐♦♥s ♦❢ ❡❛❝❤ s✉❜•❡❝t ❜❛❝❦ ♦♥ t❤❡ t❡♠♣•❛t❡ ❜② ❛♣♣•②✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛•s ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞✐✛❡♦♠♦r♣❤✐s♠s✳ ■♥ t❤❡ ❡♥❞✱ ✇❡ ❣❡t ❛ t❡♠♣•❛t❡ ❡♥✈❡•♦♣❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ ❛①❡s ❢r♦♠ t❤❡ ♥✐♥❡ s✉❜•❡❝ts t♦❣❡t❤❡r✱ ✇❤✐❝❤ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ ✜❣✉r❡ ✹✳✷✶✳ ❙✐♥❝❡ t❤❡ t♦t❛• ♥✉♠❜❡r ♦❢ ❛①❡s ❞❛t❛ tr❛♥s♣♦rt❡❞ ✐♥ t❤❡ t❡♠♣•❛t❡ ❡♥✈❡•♦♣❡ ✐s ♥♦✇ ♠♦r❡ t❤❛♥ ✶✵✵✵✱ st❛t✐st✐❝s ♦❢ s♣❛t✐❛• ❝♦rr❡•❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞✐r❡❝t✐♦♥s ♠❛❦❡s ❢❛r ♠♦r❡ s❡♥s❡ t❤❛♥ s❡♣❛r❛t❡ st❛t✐st✐❝s ♦♥ ❡❛❝❤ s✉❜•❡❝t✳ ❚❤❡ r❡s✉•t ♦❢ ✜❣✉r❡ ✹✳✷✶ ✇❡r❡ ❛♥❛•②③❡❞ ❜② ❜✐♦•♦❣✐sts ✉s✐♥❣ ❝•✉st❡r✐♥❣ ♠❡t❤♦❞s ❝♦♠❜✐♥❡❞ ✇✐t❤ |❛②•❡✐❣❤ st❛t✐st✐❝❛• t❡sts ♦♥ t❤❡ s❡t ♦❢ ❞✐r❡❝t✐♦♥s ✿ t❤✐s st✉❞② ❤❛s ❡✈✐❞❡♥❝❡❞ t❤r❡❡ r❡❣✐♦♥s ♦❢ t❤❡ t❡♠♣•❛t❡ t❤❛t ♣r❡s❡♥t s✐❣♥✐✜❝❛♥t ❝♦rr❡•❛t✐♦♥ ♦❢ t❤❡ ❛①❡s ❛•♦♥❣ ❛ ♣❛rt✐❝✉•❛r ❞✐r❡❝t✐♦♥✳ ▼❛t❝❤✐♥❣ ♣✉rs✉✐t ❛•❣♦r✐t❤♠ •✐❦❡ t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ❢♦r ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✐♥ ❝❤❛♣t❡r ✷ ✇❛s ❛•s♦ ❛❞❛♣t❡❞ t♦ ✈❛r✐❢♦•❞s ❛♥❞ ✉s❡❞ t♦ ✈✐s✉❛•✐③❡ t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❝♦rr❡•❛t✐♦♥s✳ ❆•t❤♦✉❣❤ q✉✐t❡ ♣r❡•✐♠✐♥❛r② ❛t t❤✐s st❛❣❡✱ t❤❡s❡ ✜rst s❡ts ♦❢ r❡s✉•ts s❡❡♠❡❞ ❡♥❝♦✉r❛❣✐♥❣ ❡♥♦✉❣❤ t♦ ❜❡❣✐♥ ♠♦r❡ ♣r❡❝✐s❡ ✐♥✈❡st✐❣❛t✐♦♥s ♦♥ t❤❡ ❞❡♣❡♥❞❡♥❝② ♦❢ t❤❡s❡ r❡❣✐♦♥s ❛♥❞ ❞✐r❡❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❣❡♥❡ ♠✉t❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❝✉rr❡♥t•② ❛t ✇♦r❦ ❜✉t ✇♦✉•❞ ♠♦st •✐❦❡•② r❡q✉✐r❡ t❤❡ ❛❝q✉✐s✐t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛• s✉❜•❡❝ts✳ ❋r♦♠ t❤❡ ♠❡t❤♦❞♦•♦❣✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✇❡ ❜❡•✐❡✈❡ t❤❛t t❤✐s ❞❛t❛s❡t ♦✛❡rs s♦♠❡ ♠♦r❡ ♣❡rs♣❡❝t✐✈❡s ❛♥❞ ♦♣❡♥ q✉❡st✐♦♥s t❤❛t ❞❡s❡r✈❡ t♦ ❜❡ ♠❡♥t✐♦♥❡❞✳ ❚❤❡ ✈❛r✐❢♦•❞ ✶✻✾ ❋✐❣✉r❡ ✹✳✶✾✿ ❚❤❡ ♥✐♥❡ ❞✐✛❡r❡♥t s✉❜•❡❝ts ♦❢ P❛st❡✉r ❞❛t❛s❡t s❤♦✇✐♥❣ t❤❡ tr✐❛♥❣✉•❛t❡❞ ❡♥✈❡•♦♣❡s✬ ✸❉ s✉r❢❛❝❡s s❡❡♥ ❢r♦♠ ❛❜♦✈❡ ❛♥❞ t❤❡ ❞✐✈✐s✐♦♥ ❛①❡s ✐♥ r❡❞✳ ❖♥ t❤❡ ✜rst r♦✇ ❛r❡ t❤❡ t❤r❡❡ ✇✐•❞ s✉❜•❡❝ts✱ ♦♥ t❤❡ s❡❝♦♥❞ t❤❡ t❤r❡❡ ❤❡t❡r♦③②❣♦✉s ❛♥❞ ♦♥ t❤❡ •❛st ♦♥❡ t❤❡ ♠✉t❛♥t s✉❜•❡❝ts✳ ■♥ ♠♦st s✉❜•❡❝ts✱ ♦♥❡ s❡❡s t❤❡ ❞❡✈❡•♦♣♠❡♥t ♦❢ t✇♦ ✈❡♥tr✐❝•❡s ✭✉♣ ❛♥❞ ❞♦✇♥✮ s❡♣❛r❛t❡❞ ❜② t❤❡ ✐♥t❡r✲✈❡♥tr✐❝✉•❛r r❡❣✐♦♥✳ ✶✼✵ ■♥✐t✐❛• t❡♠♣•❛t❡ ■t❡r❛t✐♦♥ ✺ ■t❡r❛t✐♦♥ ✶✵ ❋✐♥❛• t❡♠♣•❛t❡ ❋✐❣✉r❡ ✹✳✷✵✿ ❊✈♦•✉t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ❡♥✈❡•♦♣❡ t❤r♦✉❣❤ t❤❡ ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣r♦❝❡ss✳ ❋✐❣✉r❡ ✹✳✷✶✿ ❚❤❡ t❡♠♣•❛t❡ ❡♥✈❡•♦♣❡ ✇✐t❤ t❤❡ tr❛♥s♣♦rt❡❞ ❞✐✈✐s✐♦♥ ❛①❡s ❢r♦♠ t❤❡ ♥✐♥❡ s✉❜•❡❝ts✳ ✶✼✶ r❡♣r❡s❡♥t❛t✐♦♥ ✇❛s ✉s❡❞ ❤❡r❡ ♦♥•② ♦♥ t❤❡ ❡♥✈❡•♦♣❡s✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s ♣❡r❝❡♣t✐✈❡•② ❜❡tt❡r r❡s✉•ts ♦❢ t❡♠♣•❛t❡ ❛♥❞ r❡❣✐str❛t✐♦♥ t❤❛♥ ❝✉rr❡♥ts✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ s❡❡ ♦♥ ✜❣✉r❡ ✹✳✶✾✱ ♦♥ ❛ ❢❡✇ s✉❜•❡❝ts✱ s♦♠❡ ❜♦✉♥❞❛r✐❡s ❛r❡ ❛rt✐✜❝✐❛• ❜❡❝❛✉s❡ t❤❡② ❝♦rr❡s♣♦♥❞ ♦♥•② t♦ t❤❡ •✐♠✐t ♦❢ t❤❡ ❛❝q✉✐s✐t✐♦♥ ❜♦① r❛t❤❡r t❤❛♥ t♦ t❤❡ r❡❛• •✐♠✐t ♦❢ t❤❡ ❡♥✈❡•♦♣❡✳ ❚❤✐s ✐ss✉❡ ♦❢ ♣❛rt✐❛• ♦❜s❡r✈❛t✐♦♥ ❞✉❡ t♦ ❛❝q✉✐s✐t✐♦♥ ❞❡✈✐❝❡s✱ ❡✈❡♥ ✐❢ ♥♦t t♦♦ ❞❡t❡r♠✐♥❛♥t ✐♥ t❤❛t ❝❛s❡✱ ✐s ❛ r❡❝✉rr❡♥t ♣r♦❜•❡♠ ✐♥ ♠❛♥② ♠❡❞✐❝❛• ✐♠❛❣✐♥❣ ❞❛t❛s❡ts ❛♥❞ ✐s ♥♦t ②❡t ❞✐•✐❣❡♥t•② ❛❞❞r❡ss❡❞ ❜② t❤❡ ♣r❡s❡♥t r❡❣✐str❛t✐♦♥ ❛♥❞ t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s✳ ❆ s❡❝♦♥❞ ✐♥t❡r❡st✐♥❣ ♣♦✐♥t ✐s t❤❡ ❛①❡s ❞✐r❡❝t✐♦♥s✬ ❛♥❛•②s✐s✳ ❙✐♥❝❡ t❤❡s❡ ❛r❡ ❜❛s✐❝❛••② ❞✐str✐❜✉t✐♦♥s ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ ❞✐r❡❝t✐♦♥s ✐♥ t❤❡ s♣❛❝❡✱ ✈❛r✐❢♦•❞s ❛♥❞ ❦❡r♥❡• ♠❡tr✐❝s ♦✛❡rs ❛❣❛✐♥ ❛ ♥❛t✉r❛• r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❢r❛♠❡✇♦r❦ t❤❛t ❝♦✉•❞ ❜❡ ✉s❡❞ t♦ ❡st✐♠❛t❡ ♠❡❛♥✐♥❣❢✉• st❛t✐st✐❝❛• ❝♦rr❡•❛t✐♦♥s✳ ❖♥❡ ♣♦ss✐❜✐•✐t②✱ ②❡t t♦t❛••② ♣r♦s♣❡❝t✐✈❡ ❛t t❤❡ ♠♦♠❡♥t✱ ✇♦✉•❞ ❜❡ t♦ ❡st✐♠❛t❡ ❛ t❡♠♣•❛t❡ ❛s ❛♥ ❡♥✈❡•♦♣❡ s❤❛♣❡ ❝♦♥t❛✐♥✐♥❣ ❛ ❞❡♥s❡ ♠❛♣ ♦❢ ❞✐r❡❝t✐♦♥s ❡st✐♠❛t❡❞ ❢r♦♠ t❤❡ ❞❛t❛❜❛s❡✳ ✹✳✹✳✸ ❚❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ♦♥ ❈♦❝❤•❡❛❡ ❞❛t❛s❡t ❲❡ s❤♦✇ ♥♦✇ ❛ s❡❝♦♥❞ ✐••✉str❛t✐♦♥ ♦❢ ✈❛r✐❢♦•❞ ❛t•❛s ❡st✐♠❛t✐♦♥ ♦♥ ❛ ❝♦♠♣•❡t❡•② ❞✐✛❡r✲ ❡♥t ❞❛t❛s❡t✳ ❲❡ ❣r❛t❡❢✉••② t❤❛♥❦ ♣r♦❢❡ss♦r ❏♦s❡ ❇r❛❣❛ ❢r♦♠ ❯♥✐✈❡rs✐t② P❛✉• ❙❛❜❛t✐❡r ✐♥ ❚♦✉•♦✉s❡✱ ♣r♦❢❡ss♦r ❏❡❛♥✲▲✉❝ ❑❛❤♥✱ ❝✉r❛t♦r ❛t t❤❡ ♠✉s❡✉♠ ♦❢ ✬■♥st✐t✉t ❞✬❛♥❛t♦♠✐❡ ♥♦r♠❛•❡ ❡t ♣❛t❤♦•♦❣✐q✉❡✬ ✐♥ ❙tr❛s❜♦✉r❣ ❛♥❞ ❞♦❝t♦r ❙t❛♥•❡② ❉✉rr•❡♠❛♥ ❢r♦♠ ✬■♥st✐t✉t ❞✉ ❈❡r✈❡❛✉ ❡t ❞❡ •❛ ▼♦❡••❡ é♣✐♥✐èr❡✬ ✐♥ P❛r✐s ❢♦r ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛t❡r✐❛• ❞❛t❛ ✉s❡❞ ✐♥ t❤✐s s❡❝t✐♦♥✳ ❚♦ ❜r✐❡✢② ❡①♣•❛✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤✐s st✉❞② ❛♥❞ t❤❡ ♦✈❡r❛•• ♣r♦•❡❝t ❜❡❤✐♥❞ ✐t✱ ❛♥t❤r♦♣♦•♦❣✐sts ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❡✈♦•✉t✐♦♥ ✐♥ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ✐♥♥❡r ❡❛r ✭♦r ❈♦❝❤•❡❛✮ ✐♥ ❛ ♣❤②•♦❣❡♥❡t✐❝ tr❡❡ ❛♥❞ r❡•❛t❡ s♦♠❡ s❤❛♣❡ ❢❡❛t✉r❡s ♦❢ t❤❡ ❈♦❝❤•❡❛ t♦ ♦t❤❡r ♠♦r♣❤♦•♦❣✐❝❛• ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ s♣❡❝✐❡s s✉❝❤ ❛s t❤❡ ❜♦❞② ♠❛ss ✐♥❞❡① ♦r t❤❡ ❛✉❞✐♦❣r❛♠✳ ❋✐❣✉r❡ ✹✳✷✷✿ ❚❤❡ ❡✐❣❤t ❞✐✛❡r❡♥t s✉❜•❡❝ts ♦❢ t❤❡ ❍♦♠♦ ❙❛♣✐❡♥s ❈♦❝❤•❡❛ ❞❛t❛s❡t✳ ❚❤❡ s✉r❢❛❝❡s ✇❡ s❤♦✇ ❤❡r❡ ✇❡r❡ s✉❜s❛♠♣•❡❞ ❢r♦♠ t❤❡ ♦r✐❣✐♥❛• ❞❛t❛ t♦ ❛♣♣r♦①✐♠❛t❡•② ✼✺✵✵ ♣♦✐♥ts ❡❛❝❤✱ ✉s✐♥❣ ✈❡rt❡① ❞❡❝✐♠❛t✐♦♥ ✜•t❡r ✐♥ P❛r❛✈✐❡✇✳ ✶✼✷ ❋✐❣✉r❡ ✹✳✷✸✿ ❚❤r❡❡ ❞✐✛❡r❡♥t ✈✐❡✇s ♦❢ t❤❡ ✜rst s✉❜•❡❝t✳ ◆♦t❡ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ✬s♥❛✐•✬ s❤❛♣❡ ♦❢ t❤❡ ❈♦❝❤•❡❛ ❛♥❞ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤r❡❡ ❝❛♥❛•s ✭♣♦st❡r✐♦r✱ s✉♣❡r✐♦r ❛♥❞ ❤♦r✐③♦♥t❛•✮✳ ❋✐❣✉r❡ ✹✳✷✹✿ ❚❡♠♣•❛t❡ ❡st✐♠❛t❡❞ ✇✐t❤ ❝✉rr❡♥ts ❢r♦♠ ❞✐✛❡r❡♥t ✈✐❡✇s✳ ❋✐❣✉r❡ ✹✳✷✺✿ ❚❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ✭♦♥ t❤❡ •❡❢t✮ ❛♥❞ t❤❡ t❡♠♣•❛t❡ s❤❛♣❡ ❡st✐♠❛t❡❞ ❜② t❤❡ ✈❛r✐❢♦•❞ ❛•❣♦r✐t❤♠ ✭♦♥ t❤❡ r✐❣❤t✮✱ ✇✐t❤ s❡✈❡r❛• ✐♥t❡r♠❡❞✐❛t❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t st❡♣s✳ ❚❤❡ ✇♦r❦ ✇❡ ♣r❡s❡♥t ❤❡r❡ ✐s •❡ss ❛♠❜✐t✐♦✉s ❢♦r t❤❡ t✐♠❡ ❜❡✐♥❣✳ ❲❡ ❢♦❝✉s❡❞ ♦♥•② ♦♥ t❤❡ ❈♦❝❤•❡❛ ♦❢ ❍♦♠♦ ❙❛♣✐❡♥s ❢♦r ✇❤✐❝❤ t❤❡ ❞❛t❛❜❛s❡ ❝♦♥t❛✐♥❡❞ t❤❡ tr✐❛♥❣✉•❛t❡❞ s✉r❢❛❝❡s ♦❢ ❡✐❣❤t s✉❜•❡❝ts✱ ✇❤✐❝❤ ✇❡ s❤♦✇ ✐♥ ✜❣✉r❡s ✹✳✷✷ ❛♥❞ ✹✳✷✸✳ ❚❤❡ ✐♥t❡r✲s✉❜•❡❝t s❤❛♣❡ ✈❛r✐❛❜✐•✐t② ❡✈❡♥ ❛♠♦♥❣ ♦♥❡ ♣❛rt✐❝✉•❛r s♣❡❝✐❡ ✐s ❛•s♦ ❛♥ ✐♥t❡r❡st✐♥❣ ❛♥❞ st✐•• ♥♦t ✈❡r② ✇❡••✲❦♥♦✇♥ ❢❡❛t✉r❡✳ ❚❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s ✇❡ ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t✐♥❣ s♦ ❢❛r ❝♦♥st✐t✉t❡ ❛ ♣r♦❝❡❞✉r❡ t♦ ❡st✐♠❛t❡ t❤✐s ✈❛r✐❛❜✐•✐t②✳ ❉✉❡ t♦ t❤❡ ❝♦♠♣•❡① ❣❡♦♠❡tr② ♦❢ t❤❡ ❈♦❝❤•❡❛ s❤❛♣❡✱ ♥♦t❛❜•② t❤❡ ♣r❡s❡♥❝❡ ♦❢
♣•❛t❡ ❛♥❞ ❞❡❢♦r♠❛t✐♦♥s ♦❜t❛✐♥❡❞ ✇✐t❤ ✈❛r✐❢♦•❞s ❛r❡ ❛❜•❡ t♦ ❝❛♣t✉r❡ s♦♠❡ ♦❢ t❤❡ s❤❛♣❡ ❣•♦❜❛• ❢❡❛t✉r❡s ❛♥❞ ✈❛r✐❛❜✐•✐t② ❜❡t✇❡❡♥ ✐♥❞✐✈✐❞✉❛•s✱ ✇❤✐❝❤ ❝♦✉•❞ ❜❡ st✉❞✐❡❞ ✐♥ ❛ ❞❡❡♣❡r ✇❛② ✉s✐♥❣ ❢♦r ✐♥st❛♥❝❡ t❤❡ ❢r❛♠❡✇♦r❦ ♣r♦♣♦s❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✸✳ ❚❤❡ •❛②♦✉t ♦❢ t❤❡ t❤r❡❡ ❝❛♥❛•s ✐s ♦❜✈✐♦✉s•② q✉✐t❡ ✇❡•• r❡❝♦✈❡r❡❞ ❜② t❤❡ ❛•❣♦r✐t❤♠✳ ❍♦✇❡✈❡r✱ t❤❡ ❝♦✐•✐♥❣ ♥✉♠❜❡r ✐♥ t❤❡ s♥❛✐• ♣❛rt✱ ✇❤✐❝❤ ✐s ❛•s♦ ❛♥ ✐♠♣♦rt❛♥t ❜✐♦•♦❣✐❝❛• ♠❛r❦❡r✱ ✐s st✐•• ❞✐✣❝✉•t t♦ ❝❛t❝❤ ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✱ ❡ss❡♥t✐❛••② ❜❡❝❛✉s❡ ✐t ✐♥✈♦•✈❡s ✐♠♣♦rt❛♥t ❞❡❢♦r♠❛t✐♦♥s ❛t ✈❡r② s♠❛•• s❝❛•❡s✳ ■t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ q✉✐t❡ ✇❡•• t❤❡ t❡♠♣•❛t❡ ✐s ✐♥✐t✐❛•✐③❡❞ t♦ ♦♥❡ s✉❜•❡❝t ♦❢ t❤❡ ❞❛t❛❜❛s❡✳ ❚♦ ❡✈❛•✉❛t❡ t❤✐s ❞❡♣❡♥❞❡♥❝② ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥✐t✐❛• t❡♠♣•❛t❡✱ ✐♥ ✜❣✉r❡ ✹✳✷✼✱ ✇❡ s❤♦✇ t❤❡ ♦❜t❛✐♥❡❞ t❡♠♣•❛t❡ ❢♦r ❞✐✛❡r❡♥t ✐♥✐t✐❛•✐③❛t✐♦♥s✳ ❚❤❡ ✐♥t❡r❡st✐♥❣ ♣♦✐♥t ✐s t❤❛t✱ ❢♦r t❤✐s ❞❛t❛s❡t✱ ✇❡ s❡❡ ❛ r❛t❤❡r s❛t✐s❢②✐♥❣ ❝♦♥s✐st❡♥❝② ✐♥ t❤❡ ❡st✐♠❛t❡❞ s❤❛♣❡✱ ❡✈❡♥ ❢♦r ❞✐✛❡r❡♥t ✐♥✐t✐❛•✐③❛t✐♦♥s s♦ t❤❛t t❤❡ ❜✐❛s t♦✇❛r❞ ♣❛rt✐❝✉•❛r ✐♥❞✐✈✐❞✉❛•s ❞♦❡s ♥♦t s❡❡♠ ❛ ✈❡r② •✐♠✐t❛t✐♥❣ ❢❛❝t♦r ✐♥ t❤✐s ❝❛s❡✳ ❋✐♥❛••②✱ t♦ ❣✐✈❡ ❛ r♦✉❣❤ ✐❞❡❛ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡s ❢♦r s✉❝❤ ❡①❛♠♣•❡s✱ •❡t✬s ♠❡♥t✐♦♥ t❤❛t t❤❡ ♣r❡✈✐♦✉s ❛t•❛s❡s ✇❡r❡ ♦❜t❛✐♥❡❞ ✐♥ ❛❜♦✉t ♦♥❡ ❤♦✉r ♦♥ t❤❡ ❈▼▲❆ ❝♦♠♣✉t❛✲ t✐♦♥❛• s❡r✈❡r ❢♦r ❈♦❝❤•❡❛ s❤❛♣❡s s✉❜s❛♠♣•❡❞ t♦ ❛♣♣r♦①✐♠❛t❡•② ✽✵✵✵ ♣♦✐♥ts ❡❛❝❤✳ ✶✼✹ ■♥✐t✐❛•✐③❛t✐♦♥ ❚❡♠♣•❛t❡ ✭✈✐❡✇ ✶✮ ❚❡♠♣•❛t❡ ✭✈✐❡✇ ✷✮ ❚❡♠♣•❛t❡ ✭✈✐❡✇ ✸✮ ❋✐❣✉r❡ ✹✳✷✼✿ ❚❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ ❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❛•❣♦r✐t❤♠ ❢♦r ❞✐✛❡r❡♥t ✐♥✐t✐❛•✐③❛t✐♦♥s✳ ❖♥ t❤❡ ✜rst r♦✇✱ t❤❡ t❡♠♣•❛t❡ ♦❢ ✜❣✉r❡ ✹✳✷✺✳ ❖♥ t❤❡ s❡❝♦♥❞ ♦♥❡✱ t❤❡ t❡♠♣•❛t❡ ♦❜t❛✐♥❡❞ ❜② ✐♥✐t✐❛•✐③✐♥❣ ✇✐t❤ s✉❜•❡❝t ✶✳ ❚❤❡ •❛st r♦✇ ✇✐t❤ ✐♥✐t✐❛•✐③❛t✐♦♥ t♦ s✉❜•❡❝t ✽✳ ✹✳✺ ❈♦♥❝•✉s✐♦♥ ❛♥❞ ❡①t❡♥s✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❤❛✈❡ ❢♦❝✉s❡❞ ♦♥ t❤❡ ♣r♦❜•❡♠ ♦❢ ❡st✐♠❛t✐♥❣ ❛♥❛t♦♠✐❝ ❛♥❞ ❛♥❛t♦♠♦✲ ❢✉♥❝t✐♦♥❛• ❛t•❛s❡s ♦♥ ❛ ♣♦♣✉•❛t✐♦♥ ♦❢ s✉❜•❡❝ts✳ ❚❤✐s ✐s ❛ ✇❛② t♦ ❡①t❡♥❞ t❤❡ s❝♦♣❡ ♦❢ r❡❣✐str❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ✐♥❞✐✈✐❞✉❛•s✱ ✐♥ t❤❡ ♣❡rs♣❡❝t✐✈❡ ♦❢ st✉❞②✐♥❣ ❣r♦✉♣ ✈❛r✐❛❜✐•✐t② ♦❢ ❛ ✇❤♦•❡ ❞❛t❛s❡t✳ ■♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ ♣r♦♣♦s❡❞ ❛•❣♦r✐t❤♠s t♦ ❝♦♠♣✉t❡ ❛t•❛s❡s ♦♥ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❝❛rr②✐♥❣ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s✱ ❜❛s❡❞ ♦♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♦❢ ❝❤❛♣t❡r ✷✳ ❲❡ s❤♦✇❡❞ t❤❛t t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❡st✐♠❛t✐♥❣ ✐t❡r❛t✐✈❡•② ❛♥❞ s✐♠✉•t❛♥❡✲ ♦✉s•② ❛ t❡♠♣•❛t❡ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ (X, f ) ❢r♦♠ t❤❡ ❞❛t❛s❡t ❛♥❞ ❛ s❡t ♦❢ ♠♦♠❡♥t❛ p i t❤❛t ❞❡❢♦r♠s t❤❡ t❡♠♣•❛t❡ t♦ ❡❛❝❤ s✉❜•❡❝t✱ ✇❤✐❝❤ ✐s t❤❡ ♣✉r♣♦s❡ ♦❢ ❛•❣♦r✐t❤♠ ✷✳ ❚❤✐s ❝❛♥ ❜❡ t❤❡♥ ❡①t❡♥❞❡❞ ❜② ❡st✐♠❛t✐♥❣ ✐♥ ❛❞❞✐t✐♦♥ s♦♠❡ r❡s✐❞✉❛• s✐❣♥❛•s ζ i ♦♥ t❤❡ ❜❛s❡✲ •✐♥❡ s❤❛♣❡✱ ❡♥❝♦❞✐♥❣ t❤❡ ❢✉♥❝t✐♦♥❛• ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ ❞✐✛❡r❡♥t s✉❜•❡❝ts ✿ t❤✐s r❡s✉•t❡❞ ✐♥ ❛•❣♦r✐t❤♠ ✸✳ ❇♦t❤ ❛•❣♦r✐t❤♠s ✐♥✈♦•✈❡❞ s♦♠❡ ✐♠♣♦rt❛♥t ❣r❛✲ ❞✐❡♥t ♥♦r♠❛•✐③❛t✐♦♥s ❛♥❞ r❡❣✉•❛r✐③❛t✐♦♥s t♦ ❜❡ r♦❜✉st t♦ r❡s❝❛•✐♥❣✱ s❛♠♣•✐♥❣ ❝❤❛♥❣❡s ♦r ❜♦✉♥❞❛r② ❡✛❡❝ts✳ ❆s ❛•r❡❛❞② ♠❡♥t✐♦♥❡❞ ❢♦r r❡❣✐str❛t✐♦♥✱ s♦♠❡ ♥❛t✉r❛• ❛♥❞ ♣♦✲ ✶✼✺ t❡♥t✐❛••② q✉✐t❡ r❡❛❝❤❛❜•❡ ❡①t❡♥s✐♦♥s ♦❢ s✉❝❤ ❛•❣♦r✐t❤♠s ✇♦✉•❞ ❝♦♥s✐st ✐♥ ❣❡♥❡r❛•✐③✐♥❣ t❤❡♠ t♦ ♦t❤❡r s✐❣♥❛• s♣❛❝❡s t❤❛♥ R✱ ✈❡❝t♦rs ♦r t❡♥s♦rs ❜❡✐♥❣ ♣r♦❜❛❜•② t❤❡ ♠♦st ✐♥✲ t❡r❡st✐♥❣ ❝❛s❡s ❢♦r ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✳ ❆♥♦t❤❡r ❛•❣♦r✐t❤♠✐❝ ✐♠♣r♦✈❡♠❡♥t ❝♦✉•❞ ❜❡ ❛•s♦ ❜r♦✉❣❤t t♦ t❤❡ ♣r❡s❡♥t ❝♦❞❡ ❜② ✐♥❝♦r♣♦r❛t✐♥❣ t❤❡ ❝♦♥tr♦• ♣♦✐♥t ❢♦r♠✉•❛t✐♦♥ ♦❢ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥s✱ ❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✷✾✱ ✸✸✱ ✸✹(✱ ✇❤✐❝❤ ✇♦✉•❞ ❜❡ ❜❡♥❡✜❝✐❛• ✐♥ t❤❛t ✐t ✇♦✉•❞ s♣❡❡❞ ✉♣ ❝♦♠♣✉t❛t✐♦♥s r❡•❛t❡❞ t♦ ❞✐✛❡♦♠♦r♣❤✐s♠s ❛♥❞ ❡♥❛❜•❡ s♣❛rs❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❞❡❢♦r♠❛t✐♦♥s✳ ❋r♦♠ t❤❡ t❤❡♦r❡t✐❝❛• ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✇❡ ❛❞❞r❡ss❡❞ ✐♥ ✹✳✷✳✹ t❤❡ ❝❡♥tr❛• ✐ss✉❡ ♦❢ s❡tt✐♥❣ ❛ ❣❡♥❡r❛• ❝♦♥t✐♥✉♦✉s ❢r❛♠❡✇♦r❦ ❢♦r ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❛t•❛s ❡st✐♠❛t✐♦♥✱ t❤❛t ❡♥✲ s✉r❡s ❡①✐st❡♥❝❡ ♦❢ s♦•✉t✐♦♥s✳ ❲❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ❤♦•❞s ✇✐t❤ t❤❡ r❛t❤❡r ✇❡❛❦ L 2 ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ s✐❣♥❛• ♣❛rts✱ ❜✉t ♣r♦✈✐❞❡❞ ♣❡♥❛•t② ❝♦♥st❛♥ts ❛r❡ ❝❤♦s❡♥ ❤✐❣❤ ❡♥♦✉❣❤✳ ❆ ♣❡♥❞✐♥❣ q✉❡st✐♦♥ ✇♦✉•❞ ❜❡ t♦ ❡①❛♠✐♥❡ t❤❡ t❤❡♦r❡t✐❝❛• ❜❡♥❡✜ts ♦❢ ❛ss✉♠✐♥❣ ♠♦r❡ r❡❣✉•❛r✐t② ♦♥ t❤❡ s✐❣♥❛•s ✭❜② ✇♦r❦✐♥❣ ❢♦r ✐♥st❛♥❝❡ ♦♥ ❙♦❜♦•❡✈ s♣❛❝❡s ♦r ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥s✬ ❢✉♥❝t✐♦♥s✮✳ (❡t✱ t❤❡ ✉s❡ ♦❢ s✉❝❤ s♣❛❝❡s ✇♦✉•❞ ♦❜✈✐♦✉s•② ✐♥tr♦❞✉❝❡ ♥❡✇ ♥✉♠❡r✐❝❛• ❝❤❛••❡♥❣❡s✱ ❛•• t❤❡ ♠♦r❡ t❤❛t t❤❡ s✉♣♣♦rts ❛r❡✱ ✐♥ ♦✉r ❝❛s❡✱ ✈❛r②✐♥❣ ❣❡♦♠❡tr✐❝❛• s❤❛♣❡s✳ ❚❤❡ s❡❝♦♥❞ ✐♠♣♦rt❛♥t ♣♦✐♥t✱ t❤❛t s❡❡♠s ❛❝❤✐❡✈❛❜•❡ ✐♥ ❛ ♥❡❛r ❢✉t✉r❡✱ ✐s t♦ ❢♦r♠✉•❛t❡ ❛♥❞ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❛t•❛s❡s ✐♥ t❤❡ s•✐❣❤t•② ♠♦r❡ ❝♦♠♣•❡① |✐❡♠❛♥♥✐❛♥ ❢r❛♠❡✇♦r❦ ♦❢ ♠❡t❛♠♦r♣❤♦s✐s t❤❛t ✇❛s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✜rst ❝❤❛♣t❡r✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❛•s♦ ❡①❛♠✐♥❡❞ t❤❡ q✉❡st✐♦♥ ♦❢ st❛t✐st✐❝❛• ✐♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❝•❛s✲ s✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤♦s❡ ❛t•❛s❡s✳ ❚❤✐s ♣❛rt ✐s t❤❡ ♠♦st •✐❦❡•② t♦ •❡❛❞ t♦ ❢✉t✉r❡ ✐♠♣r♦✈❡♠❡♥ts ♦r ❡①t❡♥s✐♦♥s✳ ■♥❞❡❡❞✱ ✇❡ ❝❤♦s❡ t♦ ❢♦❝✉s ♦♥ s✐♠♣•❡ st❛t✐st✐❝❛• ❛♥❛•②s✐s t❡❝❤♥✐q✉❡s s✉❝❤ ❛s t❤❡ P❈❆ ❛♥❞ ✐ts s✉♣❡r✈✐s❡❞ ✈❡rs✐♦♥ ❢♦r ♠✉•t✐✲❝•❛ss ❞❛t❛s❡ts✱ t❤❡ ▲❉❆✳ ❲❡ ♣r♦♣♦s❡❞ ❛❞❛♣t❛t✐♦♥s ♦❢ t❤❡s❡ ♠❡t❤♦❞s t♦ st✉❞② t❤❡ st❛t✐st✐❝❛• ✈❛r✐❛❜✐•✲ ✐t② ♦❢ s❤❛♣❡ ❛♥❞ s✐❣♥❛• ❜❛s❡❞ ♦♥ ♠♦♠❡♥t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s✳ ❊✈❡♥ t❤♦✉❣❤ ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ st❛t✐st✐❝❛• •❡❛r♥✐♥❣ t♦♦•s ❝♦✉•❞ ❜❡ ✉s❡❞ ✐♥st❡❛❞✱ ✇❡ s❤♦✇❡❞ t❤❛t t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❛t•❛s ❡st✐♠❛t✐♦♥ ❛♥❞ ▲❉❆ ❛♥❛•②s✐s ♦❢ ♠♦✲ ♠❡♥t❛ ❛♥❞ t❤✐❝❦♥❡ss❡s •❡❛❞s t♦ ✐♥t❡r❡st✐♥❣ ❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s ♦♥ t❤❡ ❞❡t❡❝t✐♦♥ ♦❢ ❣•❛✉❝♦♠❛ ✐♥ t❤❡ r❡t✐♥❛ ❞❛t❛❜❛s❡✳ ❯♥❞♦✉❜t❡❞•②✱ ❛♣♣•✐❝❛t✐♦♥s t♦ ♦t❤❡r ❞❛t❛s❡ts ♦❢ t❤✐s ❦✐♥❞ ❡♥t❡rs ✐♥ t❤❡ r❛♥❣❡ ♦❢ ♦✉r ❢r❛♠❡✇♦r❦ ❛♥❞ s❤❛•• ❜❡ t❡st❡❞ ✐♥ ❛ ♥❡❛r ❢✉t✉r❡✳ ❋✐♥❛••②✱ ❢♦••♦✇✐♥❣ t❤❡ ✐ss✉❡s r❛✐s❡❞ ✐♥ ❝❤❛♣t❡r ✸✱ t❤❡ ❝❛s❡ ♦❢ ✉♥♦r✐❡♥t❡❞ s❤❛♣❡s ✇❛s ❛•s♦ tr❡❛t❡❞ ✐♥ t❤❡ ❛t•❛s ❡st✐♠❛t✐♦♥ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❚❤✐s ❝❛♥ ❜❡ ❡❛s✐•② ❞♦♥❡ ✐♥ t❤❡ ✈❛r✐❢♦•❞ s❡tt✐♥❣✱ ❛♥❞ ♣r♦✈✐❞❡s t❡♠♣•❛t❡ ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠s ✈❡r② ❝•♦s❡ t♦ ❝✉rr❡♥ts ❛♥❞ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✳ ❲❡ s❤♦✇❡❞ ❛ ❢❡✇ r❡s✉•ts ❛♥❞ ❜❡♥❡✜ts ♦❢ s✉❝❤ ♠❡t❤♦❞♦•♦❣✐❡s ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ ♦r✐❡♥t❡❞ ♦♥❡s ♦♥ ❛♣♣•✐❝❛t✐♦♥s t♦ ❡♠❜r②♦♥✐❝ ♠✐❝❡ ❤❡❛rt ♠❡♠❜r❛♥❡s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ♦♥ t❤❡ st✉❞② ♦❢ ❈♦❝❤•❡❛ s❤❛♣❡ ♦♥ t❤❡ ♦t❤❡r✳ ✶✼✻ ❈♦♥❝•✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ ❖♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❈❤❛♣t❡r ✸ ✇❛s ❢♦❝✉s❡❞ ♦♥ t❤❡ ✐ss✉❡ ♦❢ ♦r✐❡♥t❛t✐♦♥ ✇❤✐❝❤ ❤❛s ❜❡❡♥ ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉rr❡♥ts ✐♥ ❛♣♣•✐❝❛t✐♦♥s t♦ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠②✳ ■♠♣♦s✐♥❣ ❛ ❝♦♥s✐st❡♥t ♦r✐❡♥t❛t✐♦♥ ♦❢ s❤❛♣❡s✱ ❛s ✇❡ ♣♦✐♥t❡❞ ♦✉t✱ ✐s ❛ ❝•❡❛r •✐♠✐t❛t✐♦♥ ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐♥ tr❡❛t✐♥❣ s❡✈❡r❛• s♦rts ♦❢ ❞❛t❛s❡ts✱ ✐♥❝•✉❞✐♥❣ ✜❜❡r ❜✉♥❞•❡s ❛♥❞ ♥♦♥✲ ❝♦♥♥❡❝t❡❞ s❤❛♣❡s✳ ❲❡ ❤❛✈❡ ❛•s♦ ❡✈✐❞❡♥❝❡❞ t❤❡ ❝✉rr❡♥ts✬ ❝❛♥❝❡••❛t✐♦♥ ♣❤❡♥♦♠❡♥♦♥ ✇❤✐❝❤ ❤❛s s❡✈❡r❛• ♥❡❣❛t✐✈❡ ❝♦♥s❡q✉❡♥❝❡s ✐♥ ♣r❛❝t✐❝❛• ❝❛s❡s ♦❢ r❡❣✐str❛t✐♦♥✳ ❚❤❡ ❛♣♣r♦❛❝❤ t❤❛t ✇❡ ❤❛✈❡ ❞❡✈❡•♦♣❡❞ ✐♥ t❤✐s t❤❡s✐s✱ ❛s ❛♥ ❛•t❡r♥❛t✐✈❡ ♦r ❝♦♠♣•❡♠❡♥t t♦ ❝✉rr❡♥ts✱ r❡•✐❡s ♦♥ ❛♥♦t❤❡r ❝♦♥❝❡♣t ❢r♦♠ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r② ❝❛••❡❞ ✈❛r✐❢♦•❞s✳ ❚❤❡ ❝❤❛••❡♥❣❡ ♦❢ ❝❤❛♣t❡r ✸ ✇❛s t♦ s❤♦✇ t❤❛t ✈❛r✐❢♦•❞s ❝❛♥ ❜❡ ❛•s♦ tr❛♥s♣♦s❡❞ t♦ t❤❡ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥s✬ s❡tt✐♥❣ ❛♥❞ t♦ ❞❡t❛✐• t❤❡ str❡♥❣t❤s ❛♥❞ ✇❡❛❦♥❡ss❡s ♦❢ t❤✐s ♥❡✇ ❢r❛♠❡✇♦r❦✳ ❲❡ ❤❛✈❡ ♣r♦✈❡❞ ✐♥ ♣❛rt✐❝✉•❛r t❤❛t t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❢♦•❞s ❡♠❜❡❞s ❛•• ♥♦♥✲ ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ ♣r♦✈✐❞❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ♠❡tr✐❝s t❤❛t ❞♦ ♥♦t
(E × R, Λ d E * )✱ t❤❛t X ❛♥❞ (X i ) 1≤i≤N ❛r❡ ✜♥✐t❡ ✈♦•✉♠❡ ❜♦✉♥❞❡❞ ♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ t❤❛t f i ∈ L 2 (X i ) ❢♦r 1 ≤ i ≤ N ✳ ❆ss✉♠❡ t❤❛t γ f /γ W ✐s •❛r❣❡ ❡♥♦✉❣❤ ✇✐t❤ γ f , γ W ≥ 0✳ ❚❤❡♥ J X (f ) . = γ f 2 X |f (x)| 2 dH d (x) + 1 2 N i=1 γ W C X i ,f i -C X,f 2
= γ f 2 ν(|f | 2 ) + γ W 2 N i=1 C X i ,f i -C ν 2 W ′ ✭❆✳✶✶✮ ✇❤❡r❡ ν(|f | 2 ) ✐s t❤❡ ♥♦t❛t✐♦♥ ✇❡ s❤❛•• ✉s❡ ❢♦r |f | 2 dν✳ ◆♦✇ •❡t ν ∈ M X (F )✳ ❈♦♥s✐❞❡r ❢♦r t ∈ R✱ J t . = J(ν t ) ✇❤❡r❡ ν t ✐s t❤❡ ♣r❡✈✐♦✉s•② ❞❡✜♥❡❞ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ν ❛♥❞ ❛ss✉♠❡ t❤❛t J 0 < ∞ ✭✇❤✐❝❤ ✐s ❡q✉✐✈❛•❡♥t t♦ s❛② t❤❛t ν(|f | 2 ) < ∞✮✳ ❲❡ r❡❝❛•• t❤❛t C X i ,f i -C νt 2 W ′ = (C X i ,f i -C νt )(ω i ) ✇❤❡r❡ ω i = K W (C X i ,f i -C νt ) ∈ W ✳ ❚❤❡♥ ♦♥❡ ❡❛s✐•② ❝❤❡❝❦s t❤❛t J t < ∞ ❛♥❞✱ s✐♥❝❡ ✇❡ ❛ss✉♠❡ t❤❛t W ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥t♦ C 2 0 (E × R, Λ d E * )✱ ✇✐t❤ ❡①✐st✐♥❣ ❞❡r✐✈❛t✐✈❡ J ′ t ❛t ❛♥② •♦❝❛t✐♦♥ t ❣✐✈❡♥ ❜② J ′ t = ν d dt (ρ t (f )) γ f ρ t (f ) + γ W N i=1 ∂ω i (x,ρt(f )) ∂f (ξ) ✭❆✳✶✷✮ ❯s✐♥❣ ❛❣❛✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❡♠❜❡❞❞✐♥❣ ♦❢ W ✐♥t♦ C 2 0 (E × R, Λ d E * ) ✇❡ ❣❡t ❢♦r ❛ ❝♦♥st❛♥t C > 0 t❤❛t ∂ω i (x,ρt(f )) ∂f (ξ) ≤ C ω i W ≤ K C X i ,f i W ′ + C νt W ′ . ▼♦r❡♦✈❡r✱ ✐t ✐s ❛ tr✐✈✐❛• ✈❡r✐✜❝❛t✐♦♥ t❤❛t ✇❡ ❤❛✈❡✱ ❜② ❛♥❛•♦❣② t♦ ❡q✳✭✶✳✶✼✮ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝✉rr❡♥ts✱ C X i ,f i W ′ ≤ ❝t❡.H d (X i )✳ ❲✐t❤ s✐♠✐•❛r ❛r❣✉♠❡♥ts✱ C νt W ′ ≤ ❝t❡.ν t (E × Λ d (E) × R) ❛♥❞✱ s✐♥❝❡ ν t ∈ M X (F )✱ ✇❡ ❤❛✈❡ ν t (E × Λ d (E) × R) = H d (X) ❛♥❞ ❝♦♥s❡q✉❡♥t•② C νt W ′ ≤ ❝t❡.H d (X)✳ ❚❤✉s✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t K > 0 s✉❝❤ t❤❛t ✿ N i=1 ∂ω i (x,ρt(f )) ∂f (ξ) ≤ K N i=1 H d (X i ) + H d (X) ✭❆✳✶✸✮ ◆♦t✐❝✐♥❣ ♥♦✇ t❤❛t d dt (ρ t (f )) ρ t (f ) ≤ 0✱ t❤❛t | d dt (ρ t (f )) | = 0 ❢♦r |f | ≤ a ❛♥❞ t❤❛t |ρ t (f )| ≥ a ❢♦r |f | ≥ a ❛♥❞ t ∈ [0, 1]✱ ✇❡ ❣❡t ❢♦r t ∈ [0, 1] J ′ t ≤ ν - d dt (ρ t (f )) 1 |f |>a γ f a -γ W K N i=1 H d (X i ) + H d (X) ✭❆✳✶✹✮ ✶✽✸ s♦ t❤❛t J(ν 1 ) ≤ J(ν 0 ) ✐❢ a ≥ K γ W γ f N i=1 H d (X i ) + H d (X) .
✇ ♥♦✇ t❤❛t ❢♦r γ f /γ W •❛r❣❡ ❡♥♦✉❣❤ t❤❡r❡ ❡①✐sts f * ∈ L 2 (X) s✉❝❤ t❤❛t ν * = ν X,f * s♦ t❤❛t ✇❡ ✇✐•• ❞❡❞✉❝❡ J X (f * ) = J(ν * ) ≤ inf f ∈L 2 (X) J X (f ) ✭❆✳✷✵✮ ❛♥❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r ♦♥ L 2 (X)✳ ▲❡t δf ∈ C b (F, R) ❛♥❞ ❢♦r ❛♥② t ∈ R ❝♦♥s✐❞❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ν t ∈ M X (F ) ♦❢ ν ∈ M X,C (F ) s✉❝❤ t❤❛t ❢♦r ❛•• g ∈ C b (F, R) ✇❡ ❤❛✈❡ ✿ ν t (g)
. = g(x, ξ, f + tδf (x, ξ, f ))dν(x, ξ, f ) .

✭❆✳✷✶✮

❍❡r❡ ❛❣❛✐♥✱ t❤❡ ❢✉♥❝t✐♦♥ t → J(ν t ) ✐s ❞✐✛❡r❡♥t✐❛❜•❡ ❡✈❡r②✇❤❡r❡ ❛♥❞ ✇❡ ❤❛✈❡ ❢♦r

ω i . = K W (C X i ,f i -C ν ) d dt J(ν t )| t=0 = ν γ f f + γ W N i=1 ∂ω i (x,f ) ∂f (
❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ✉♥✐❢♦r♠•② ❜② K N i=1 H d (X i ) + H d (X) ✳ ■t r❡s✉•ts t❤❛t ❢♦r γ f /γ W •❛r❣❡ ❡♥♦✉❣❤ ❛♥❞ ❢♦r ❛•• (x, V ) ∈ E × Λ d E✱ f → γ f f + γ W A(x, ξ, f ) ✐s ❛ str✐❝t•② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ❣♦✐♥❣ t♦ -∞ ✭r❡s♣ +∞✮ ❛t -∞ ✭r❡s♣ +∞✮ ❛♥❞ t❤✉s t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s♦•✉t✐♦♥ f (x, ξ) t♦ ✭❆✳✷✷✮✳ ◆♦✇✱ s✐♥❝❡ t❤❡ ❛♣♣•✐❝❛t✐♦♥ f → γ f f +γ W A(x, ξ, f ) ✐s ❛•s♦ C 1 ♦♥ E × Λ d E × R✱ ✇❡ ❞❡❞✉❝❡ ❢r♦♠ t❤❡ ✐♠♣•✐❝✐t ❢✉♥❝t✐♦♥ t❤❡♦r❡♠ t❤❛t f ✐s ❛ C 1 ❢✉♥❝t✐♦♥ ♦♥ E × Λ d E✳ •♦✐♥❣ ❜❛❝❦ t♦ t❤❡ s♦•✉t✐♦♥ ν * ✱ ✇❡ ❦♥♦✇ t❤❛t ❢♦r ν * ❛•♠♦st ❡✈❡r② (x, ξ, f ) ∈ E × Λ d E × R✱ ✇❡ ❤❛✈❡ (x, ξ, f ) ∈ C ❛♥❞ f = f (x, V )✱ s♦ t❤❛t | f | ≤ a ❛✳❡✳ ❋♦r ❛♥② ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ g ✿ ν * (g) = g(x,
V 0 ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥ C 2 0 (E, E)✱ ❛♥ ✐♥✐t✐❛• ❤②♣❡rt❡♠♣•❛t❡ X 0 ❛♥❞ ❝♦♥s✐❞❡r ❛•s♦ ❛♥ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ X ✐♥ t❤❡ ♦r❜✐t ♦❢ X 0 ✉♥❞❡r t❤❡ ❛❝✲ t✐♦♥ ♦❢ φ v 0 ∈ G V 0 ✱ t❤❡ ❣r♦✉♣ ♦❢ ❞✐✛❡♦♠♦r♣❤✐s♠s ❛ss♦❝✐❛t❡❞ ✇✐t❤ V 0 ✐✳❡✳ X = φ v 0 (X 0 )✳ ❚♦ ♣r♦✈❡ ❡①✐st❡♥❝❡ r❡s✉•t✱ ✇❡ ✇✐•• ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ ❛ ♣❡♥❛•✐③❛t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ X 0 ❛♥❞ X ✐✳❡✳ ♦♥ d G V 0 (■❞, φ v 0 )✳ ❆ t②♣✐❝❛• ❢✉♥❝t✐♦♥❛• ✇♦✉•❞ ❜❡✱ ✐❢ X 0 ✐s t❤❡ ♦r❜✐t ♦❢ X 0 ✉♥❞❡r G V 0 ❛♥❞ d X 0 (X, X ′ ) .
= inf φ∈G V 0 ,φ(X)=X ′ d G V 0 (■❞, φ) ✐s t❤❡ ✐♥❞✉❝❡❞ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ t❡♠♣•❛t❡s ✐♥ t❤❡ ♦r❜✐t✱ ❲❡ ♥♦✇ ✐♥tr♦❞✉❝❡ t❤❡ ❡①t❡♥❞❡❞ ❢✉♥❝t✐♦♥❛• ✿

J 1 (X, f ) = γ 0 2 d X 0 (X 0 , X) 2 + γ f 2 X |f (x)| 2 dH d (x) + γ W 2 N i=1 C X i ,f i -C X,f 2 W ′ ✭❆✳✷✹✮ ✶✽✺ ❙✐♥❝❡ ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ φ 0 ∈ G V 0 ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ✢♦✇ ♦❢ ❛ t✐♠❡✲✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞ v 0 ∈ L 2 ([0, 1], V 0 )✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❢✉♥❝t✐♦♥❛• ✿ J 2 (v 0 , f ) = γ 0 2 v 0 2 L 2 V 0 + γ f 2 X |f (x)| 2 dH d (x) + γ W 2 N i=1 C X i ,f i -C X,f 2 W ′ ✭❆✳✷✺✮ ❢♦r v 0 ∈ L 2 V 0 ✱ f ∈ L 2 (X) ✇✐t❤ X = φ v 0 (X)✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❢♦r X = φ v 0 (X 0 ) t♦ ❝♦♥s✐❞❡r t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ f → f 0 = f • φ v 0 ❢r♦♠ L 2 (X) → L 2 (X 0 ) s♦ t❤❛t ✇❡ ❦❡❡♣ ✇♦r❦✐♥❣ ✐♥ ❛ ✜①❡❞ s♣❛❝❡ L 2 (X 0 )✳ ❍❡♥❝❡ ✇❡ ❡♥❞ ✉♣ ✇✐t❤ ❢✉♥❝t✐♦♥❛• J 3 (v 0 , f 0 ) = γ 0 2 v 0 2 L 2 V 0 ♣❡♥❛•✐③❛t✐♦♥ ♦♥ X + γ f 2 X 0 |f 0 (x)| 2 |d x φ v 0 .ξ(x)|dH d (x) = X |f (x)| 2 dH d (x) ❢♦r f =f 0 •(φ v 0 ) -1 ∈L 2 (X) + γ W 2 N i=1 C X i ,f i -(φ v 0 1 ) * C X 0 ,f 0 2 W ′ ✭❆✳✷✻✮ ✇✐t❤ f 0 ∈ L 2 (X 0 )✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r (v 0 * , f 0, * ) ❢♦r J 3 ❣✐✈❡s ✐♠♠❡❞✐❛t❡•② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r (X * , f * ) ❢♦r J 1 ✇✐t❤ X * . = φ v 0 * 1 (X 0 )✱ f * . = f 0, * • (φ v 0 * 1 ) -1 ∈ L 2 (X * )✳ ❚❤❡ ❡①✐st❡♥❝❡ r❡s✉•t t❤❡♥ ❜❡❝♦♠❡s ✿ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✷✳ ❆ss✉♠❡ t❤❛t W ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥ C 2 0 (E × R, Λ d E * )✱ t❤❛t X ❛♥❞ (X i ) 1≤i≤N ❛r❡ ✜♥✐t❡ ✈♦•✉♠❡ ❜♦✉♥❞❡❞ ♦r✐❡♥t❡❞ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡ts ❛♥❞ t❤❛t f i ∈ L 2 (X i ) ❢♦r 1 ≤ i ≤ N ✳ ❆ss✉♠❡ t❤❛t γ f /γ W ✐s •❛r❣❡ ❡♥♦✉❣❤ ✇✐t❤ γ f , γ W ≥ 0 ❛♥❞ γ 0 > 0✳ ❚❤❡♥ J(X, f ) . = γ 0 2 d X 0 (X 0 , X) 2 + γ f 2 X |f (x)| 2 dH d (x) + γ W 2 N i=1 C X i ,f i -C X,f 2 W ′ ❛❝❤✐❡✈❡s ✐ts ♠✐♥✐♠✉♠ ♦♥ {(X, f | X ∈ X 0 , f ∈ L 2 (X)} ❛♥❞ ❛♥② ♠✐♥✐♠✐③❡r (X * , f * ) ✐s s✉❝❤ t❤❛t f * ∈ L ∞ (X * )✳ ▼♦r❡♦✈❡r✱ ✐❢ X 0 ✐s ❛ C p s✉❜♠❛♥✐❢♦•❞ ❛♥❞ W ֒→ C m 0 (E × R, Λ d E * ) ✇✐t❤ m ≥ max{p, 2}✱ f * ∈ C p-1 (X * )✳ ❚❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✶ ❝❛♥ ❜❡ ✐♥ ❢❛❝t ❡❛s✐•② ❛❞❛♣t❡❞ t♦ t❤✐s ♥❡✇ s✐t✉❛✲ t✐♦♥✳ ❲❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❢♦r♠✉•❛t✐♦♥ ♦❢ ❡q✉❛t✐♦♥ ✭❆✳✷✻✮ ✇✐t❤ J 3 ❛ ❢✉♥❝t✐♦♥❛• ♦♥ t❤❡ ✈❡❝t♦r ✜❡•❞ v 0 ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ f ∈ L 2 (X 0 )✳ ❲✐t❤ r❡s♣❡❝t t♦ v 0 ✱ t❤❛♥❦s t♦ t❤❡ ♣❡♥❛•✐③❛t✐♦♥ ✐♥ ✭❆✳✷✻✮✱ ✇❡ ❝❛♥ r❡str✐❝t t❤❡ s❡❛r❝❤ ♦❢ ❛ ♠✐♥✐♠✉♠ ♦♥ ❛ ❝•♦s❡❞ ❜❛•• B 0 ♦❢ ❣✐✈❡♥ r❛❞✐✉s b ✐♥ L 2 V 0 ✱ ✇❤✐❝❤ ❣✉❛r❛♥t❡❡s ❛t t❤❡ s❛♠❡ t✐♠❡ t❤❛t t❤❡ ❏❛✲ ❝♦❜✐❛♥s |d x φ v 0 .ξ(x)| ❛r❡ ✉♥✐❢♦r♠•② •♦✇❡r ❜♦✉♥❞❡❞✳ ❚❤✐s ❝•♦s❡❞ ❜❛•• ✐s ❛•s♦ ❝♦♠✲ ♣❛❝t ❢♦r t❤❡ ✇❡❛❦ t♦♣♦•♦❣② ✐♥ L 2 V 0 ✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐t ❢♦••♦✇s ❢r♦♠ ❝♦r♦••❛r② ✷✳✷✳✷ t❤❛t v 0 → N i=1 C X i ,f i -(φ v 0 1 ) * C X 0 ,f 0 2 W ′ = N i=1 C X i ,f i -C φ v 0 1 (X 0 ),f 0 •(φ v 0 1 ) -1 2 W ′ ✐s ✇❡❛❦•② ❝♦♥t✐♥✉♦✉s ♦♥ L 2 V 0 ❛♥❞ ✐t ✐s ❛•s♦ ❝•❛ss✐❝❛• t❤❛t v 0 → v 0
J X (f, (ζ i ), (v i )) . = γ f 2 X |f (x)| 2 dH d (x) + 1 2 N i=1 v i 2 L 2 V + γ ζ X |ζ i (x)| 2 dH d (x) + γ W C X i ,f i -C φ v i (X),f +ζ i 2 W ′ ◆♦✇✱ ❛s ❢♦r v 0 ✱ ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ♣❡♥❛•✐③❛t✐♦♥s v i 2 L 2 V ✱ ♦♥❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❛•• ✈❡❝t♦r ✜❡•❞s v i ❜❡•♦♥❣ t♦ ❛ ✜①❡❞ ❝•♦s❡❞ ❜❛•• B ♦❢ r❛❞✐✉s r > 0 ✐♥ L 2 V ✳ ❆s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ❆✳✶✳✶✱ ✇❡ ✜rst s❤♦✇ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r ✐♥ ❛ s♣❛❝❡ ♦❢ ♠❡❛s✉r❡s✳ ◆❛♠❡•②✱ ❡①t❡♥❞✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥s✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡ M X (F ) ♦❢ ♠❡❛s✉r❡s ν ♦♥ F . = E × Λ d E × R × R N t❤❛t ❛r❡ s✉♣♣♦rt❡❞ ♦♥ X✱ ✐✳❡ s✉❝❤ t❤❛t ❢♦r ❛•• ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ h ♦♥ E × Λ d E✱ ✇❡ ❤❛✈❡ ✿
J(ν, (v i )) . = γ f 2 ν(|f | 2 ) + 1 2 N i=1 v i 2 L 2 V + γ ζ ν(|ζ i | 2 ) + γ W C X i ,f i -(φ v i ) * C ν i 2 W ′ ❢♦r ν ∈ M X (F )✱ (v i ) ∈ (L 2 V ) N ❛♥❞ ✇❤❡r❡✱ ❢♦r ❛•• i ∈ {1, .., N }✱ ν i ✐s t❤❡ ♠❡❛s✉r❡ ♦♥ E × Λ d (E) × R ❞❡✜♥❡❞ ❢♦r ❛•• g ∈ C b (E × Λ d (E) × R, R) ❜② ✿ ν i (g) = g(x, ξ, f + ζ i )dν(x, ξ, f, (ζ j )) ✭❆✳✷✼✮ ❛♥❞ C ν i t❤❡ ❢❝✉rr❡♥t ❞❡✜♥❡❞ ❢♦r ❛•• ω ∈ W ❛s ✐♥ ✭❆✳✸✮ ✿ C ν i (ω) = ω (x,f +ζ i ) (ξ)dν i (x, ξ, f, (ζ i )) .
❆s ♣r❡✈✐♦✉s•②✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❢✉♥❝t✐♦♥ ρ t ❛❝t✐♥❣ ♦♥ s✐❣♥❛•s ❛♥❞ t❤❡ ♠❡❛s✉r❡s ν t (g) . = g(x, ξ, ρ t (f ), (ρ t (ζ i )))dν(x, ξ, f, (ζ i )) .

✶✽✼ ❉❡♥♦t✐♥❣ J t = J(ν t , (v i ))✱ ✇❡ ❤❛✈❡✱ ❢♦r t ∈ [0, 1]✱

J ′ t = ν   d dt (ρ t (f ))   γ f ρ t (f ) + γ W N i=1 ∂ω i (φ v i (x),ρt(f )+ρt(ζ i )) ∂f (d x φ v i .ξ)     + ν   N i=1 d dt (ρ t (ζ i ))   γ ζ ρ t (ζ i ) + γ W ∂ω i (φ v i (x),ρt(f )+ρt(ζ i )) ∂f (d x φ v i .ξ)     ✭❆✳✷✽✮ ✇❤❡r❡✱ ❢♦r ❛•• i ∈ {1, .., N }✱ ω i = K W (C X i ,f i -(φ v i ) * C ν i t )✳ ❖♥ t❤❡ ✜rst ❤❛♥❞✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t ❝t❡ s✉❝❤ t❤❛t ❢♦r ❛•• i✱ x ∈ E ❛♥❞ ξ ∈ Λ d E, |ξ| = 1✱ |d x φ v i .ξ| ≤ ❝t❡|dφ v i | ∞ ✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐t ✐s ❛ ❝•❛ss✐❝❛• r❡s✉•t ♦♥ ✢♦✇s ♦❢ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥s ✭❝❢ ❬✽✻(✮ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ C : R + → R + ✱ C(0) = 0✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ v ∈ L 2 V s✉❝❤ t❤❛t |dφ v | ∞ ≤ C( v L 2 V )✳ ❚❤❡r❡❢♦r❡✱ s✐♥❝❡ ❢♦r ❛•• i✱ v i L 2 V ≤ r ∂ω i (φ v i (x),ρt(f )+ρt(ζ i )) ∂f (d x φ v i .ξ) ≤ C( v i L 2 V ). ∂ω i ∂f ∞ ≤ C(r). ∂ω i ∂f ∞ .
◆♦✇✱ ✉s✐♥❣ t❤❡ s❛♠❡ ❝♦♥tr♦•s ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥s✱ ✇❡ ❤❛✈❡✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✿

∂ω i ∂f ∞ ≤ ❝t❡ ω i W ≤ ❝t❡( C X i ,f i W ′ + (φ v i ) * C ν i t W ′ ) ≤ ❝t❡(H d (X i ) + (φ v i ) * ν i t (E × Λ d E × R)) .
■t ✐s ❡❛s② t♦ ❝❤❡❝❦✱ ✉s✐♥❣ ❡q✳✭❆✳✺✮✱ t❤❛t

(φ v i ) * ν i t (E × Λ d E × R)) ≤ ❝t❡|dφ v i | ∞ ν i t (E × Λ d E × R) ❛♥❞✱ ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t ν ∈ M X (F ) ❛s ❛•r❡❛❞② ❛r❣✉❡❞ ✐♥ ❆✳✶✳✷✱ ν i t (E × Λ d E × R) = H d (X)✳ ❆•s♦✱ s✐♥❝❡ ❛•• v i ✬s ❛r❡ ❜♦✉♥❞❡❞ ❜② r ❢♦r t❤❡ L 2 V ✲♥♦r♠✱ t❤❡ t❡r♠s |dφ v i | ∞ ❝❛♥ ❜❡ ❛•s♦ ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♠♠♦♥ ❝♦♥st❛♥t ✭❝❢ •❡♠♠❛ ✷✳✷✳✸✮✳ ■t r❡s✉•ts✱ ❢r♦♠ ❛•• t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t✐❡s✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦♥st❛♥t K > 0 s✉❝❤ t❤❛t ❢♦r ❛•• i, x, V, f, ζ i ✿ ∂ω i (φ v i (x),ρt(f )+ρt(ζ i )) ∂f (d x φ v i .ξ) ≤ K.(H d (X i ) + H d (X)) ✭❆✳✷✾✮ ❋♦••♦✇✐♥❣ t❤❡ s❛♠❡ ♣❛t❤ t❤❛t ♣r❡✈✐♦✉s•② •❡❛❞ t♦ ✭❆✳✶✹✮ J ′ t ≤ ν - d dt (ρ t (f )) 1 |f |>a γ f a -γ W K N i=1 H d (X i ) + H d (X) + N i=1 ν - d dt ρ t (ζ i ) 1 |ζ i |>a γ ζ a -γ W K H d (X i ) + H d (X)
. ✭❆✳✸✵✮ ✶✽✽ ❏✉st ❛s ✐♥ ❆✳✶✳✷✱ t❤✐s ✐♠♣•✐❡s t❤❛t J(ν 1 , (v i )) ≤ J(ν 0 , (v i )) ❛s s♦♦♥ ❛s ✿

     a ≥ γ W γ f N i=1 K H d (X i ) + H d (X) ❛♥❞ a ≥ max i γ W γ ζ K H d (X i ) + H d (X)

  ✐t ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✲ ❚❤❡ ♣r❡✈✐♦✉s ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ k✲t✐♠❡s ❝♦♥t✐♥✉♦✉s•② ❞✐✛❡r❡♥t✐❛❜•❡ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✳ ❲❡ ✇✐•• ❝❛•• d t❤❡ ❝•❛ss✐❝❛• ❡①t❡r✐♦r ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦♣❡r❛t♦r ♦♥ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s ❛♥❞ ❞❡♥♦t❡ ❜② Ω p 0,k (E) t❤❡ s♣❛❝❡ ♦❢ k✲t✐♠❡s ❝♦♥t✐♥✉♦✉s•② ❞✐✛❡r❡♥✲ t✐❛❜•❡ ω s✉❝❤ t❤❛t ω ❛♥❞ ✐ts ❞✐✛❡r❡♥t✐❛•s ✉♣ t♦ ♦r❞❡r k ✈❛♥✐s❤ ❛t ✐♥✜♥✐t②✳ ✷✼ ❚❤✐s •❡❛❞s ♥♦✇ t♦ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ p✲❝✉rr❡♥ts ✿ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✳ ❚❤❡ s♣❛❝❡ ♦❢ p✲❝✉rr❡♥ts ♦♥ E ✐s t❤❡ s♣❛❝❡ Ω p 0 (E) ′ ♦❢ ❛•• ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✳ ❲❡ ❞r❛✇ ❛tt❡♥t✐♦♥ ♦♥ t❤❡ ❢❛❝t t❤❛t t❤✐s ❞❡✜♥✐t✐♦♥ ♦❢ ❝✉rr❡♥ts ✇❡ ❛❞♦♣t ✐♥ ♦✉r ❢r❛♠❡✲ ✇♦r❦ ✐s ❛❝t✉❛••② ❛ •✐tt•❡ ♠♦r❡ r❡str✐❝t✐✈❡ t❤❛♥ t❤❡ ❝•❛ss✐❝❛• ♦♥❡ ♦❢ ❬✸✼✱ ✻✼(✱ ✇❤❡r❡ ❛✉t❤♦rs ❞❡✜♥❡ ❝✉rr❡♥ts ❛s ❡•❡♠❡♥ts ♦❢ t❤❡ ❞✉❛• ♦❢ t❤❡ s♠❛••❡r s♣❛❝❡ ♦❢ C ∞ ❝♦♠♣❛❝t•②✲ s✉♣♣♦rt❡❞ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✳ ❖♥❡ r❡❛s♦♥ ❢♦r t❤✐s s•✐❣❤t ♥✉❛♥❝❡ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❡ss❡♥t✐❛••② t❤❛t ✐t ❞❡✜♥❡s ❝✉rr❡♥ts ❛s ❛ str✐❝t ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ ✜♥✐t❡ s✐❣♥❡❞ ❇♦r❡•

  1 ...du d . ❖❜s❡r✈❡ t❤❛t ❛ s✐♠♣•❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ❢♦r♠✉•❛ s❤♦✇s t❤❛t t❤✐s ❡①♣r❡ss✐♦♥ ✐s ✐♥❞❡❡❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛♥② ♣♦s✐t✐✈❡ ♣❛r❛♠❡tr✐③❛t✐♦♥✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ X ❜② t❤❡ ❝✉rr❡♥t T X ✐s t❤✉s ❣❡♦♠❡tr✐❝ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ♦♥•② ❞❡♣❡♥❞s ♦♥ t❤❡ s❤❛♣❡ ✐ts❡•❢ ❛♥❞ ♥♦t ✐ts ♣❛r❛♠❡tr✐③❛t✐♦♥✳ ❆s ♦♥❡ ❝❛♥ ✐♠❛❣✐♥❡✱ t❤❡ s♣❛❝❡ ♦❢ d✲❝✉rr❡♥ts ✐s ♠✉❝❤ •❛r❣❡r ❛

F

  ❚❤✐s ✐s ♣r♦✈❡❞ ❜② ❬✸✼( ✹✳✶✳✸✵✳ ❋✐♥❛••②✱ t❤✐s ❢♦r♠❛•✐s♠ ❛❞❛♣ts ✇✐t❤ ♥♦ ❝❤❛♥❣❡s t♦ t❤❡ ❝❛s❡ ♦❢ ❞✐s❝r❡t❡ ❣❡♦♠❡tr②✳ ❆♥② ❞✐s❝r❡t❡ s❤❛♣❡ ❣✐✈❡♥ ❛s ❛ s❡t ♦❢ ♣♦✐♥ts ✇✐t❤ ❛ ♠❡s❤ ✭t❤✉s ❛ ♣♦•②❤❡❞r♦♥✮ ❝❛♥ ❜❡ tr❛♥✲ s❝r✐❜❡❞ ✐♥t♦ ❛ ❝✉rr❡♥t ❜② ❛♣♣r♦①✐♠❛t✐♥❣ ❡❛❝❤ ❝❡•• ♦❢ t❤❡ ♠❡s❤ ✇✐t❤ ♦♥❡ ❉✐r❛❝ ❝✉rr❡♥t •♦❝❛t❡❞ ❛t t❤❡ ❝❡♥t❡r ✇✐t❤ ❛ s✐♠♣•❡ d✲✈❡❝t♦r ❡♥❝♦❞✐♥❣ t❤❡ •♦❝❛• ✈♦•✉♠❡ ❡•❡♠❡♥t✱ ❛s ❡①♣•❛✐♥❡❞ ✇✐t❤ ♠♦r❡ ❞❡t❛✐•s ✐♥ ❬✷✽( ❝❤❛♣t❡r ✶✳ ■t r❡s✉•ts t❤❛t ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛• s❤❛♣❡s ❝❛♥ ❜❡ ❛•✇❛②s ❛♣♣r♦①✐♠❛t❡❞ ❜② ✜♥✐t❡ s✉♠s ♦❢ ❉✐r❛❝ ❝✉rr❡♥ts ♦❢ t❤❡ ❢♦r♠ ♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❝❛♥ ❜❡ ❜❛s✐❝❛••② ❝♦♥tr♦••❡❞ ❜② t❤❡ ♠❛①✐♠✉♠ ❞✐❛♠❡t❡r ♦❢ t❤❡ ♠❡s❤ ❝❡••s ✭❝❢ ❬✷✽( ✶✳✸✳✷✮✳ ❚❤✉s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❝✉rr❡♥ts ❛••♦✇s ✉s t♦ ❡♠❜❡❞ ❛•• r❡❝t✐✜❛❜•❡ s❡ts ♦❢ ❣✐✈❡♥ ❞✐♠❡♥s✐♦♥ ✐♥t♦ ♦♥❡ ❝♦♠♠♦♥ ❢✉♥❝t✐♦♥❛• s♣❛❝❡ ♦♥ ✇❤✐❝❤ ♦♥❡ ❝❛♥ ❞❡✜♥❡ ♠♦r❡ ❡❛s✐•② ❛ ♠❡tr✐❝ str✉❝t✉r❡✳ ▼❡tr✐❝s ♦♥ ❝✉rr❡♥ts ❆s t❤❡ ❞✉❛• ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✱ t❤❡ s♣❛❝❡ ♦❢ ❝✉rr❡♥ts ♥❛t✉r❛••② ✐♥❤❡r✐ts ✐ts ♦♣❡r❛t♦r ♥♦r♠✱ ✇❤✐❝❤ ✐s ❝❛••❡❞ t❤❡ ♠❛ss ♥♦r♠ ✐♥ ❬✸✼✱ ✻✼(✳ ❋♦r ❛♥② ❝✉rr❡♥t T ∈ Ω d 0 (E) ′ ✱ ✐t ❡q✉❛•s ✿ ❡q✳✭✶✳✶✶✮ ❛♥❞ ✭✶✳✶✹✮✱ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t❤❛t✱ ❢♦r ❛♥② r❡❝t✐✜❛❜•❡ s❡t X✱ ✇❡ ❤❛✈❡ ✐♥ ♣❛rt✐❝✉•❛r ✿ M (T X ) = H d (X) (❡t t❤❡ ♠❛ss ♥♦r♠ ❞♦❡s ♥♦t ✐♥❞✉❝❡ ❛ s❛t✐s❢②✐♥❣ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ s❤❛♣❡s ❛s ♥♦t✐❝❡❞ ✐♥ ❬✷✽( ✶✳✺✳✶ ✿ ❢♦r ✐♥st❛♥❝❡✱ ✐❢ X ❛♥❞ Y ❛r❡ t✇♦ ❞✐st✐♥❝t s✉❜♠❛♥✐❢♦•❞s ✐♥ E t❤❡♥ ♦♥❡ ❡❛s✐•② s❡❡s t❤❛t M (T X -T Y ) = H d (X) + H d (Y ) ♥♦ ♠❛tt❡r ❤♦✇ ❝•♦s❡ ✐♥ ❞✐st❛♥❝❡ t❤❡ ♣♦✐♥ts ♦❢ X ♠✐❣❤t ❜❡ ❢r♦♠ t❤❡ ♦♥❡s ♦❢ Y ✳ ❚❤✐s ❞r❛✇❜❛❝❦ r❡s✉•ts ❡ss❡♥t✐❛••② ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ s❡t ♦❢ t❡st ❢✉♥❝t✐♦♥s ω ✐♥ ❡q✳✭✶✳✶✹✮ ✐s t♦♦ ❜✐❣✳ ❋♦r t❤❛t r❡❛s♦♥✱ t❤❡ ❛•t❡r♥❛t✐✈❡ ✢❛t ♥♦r♠ ✐s ♦❢t❡♥ ❝♦♥s✐❞❡r❡❞ ✐♥ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ t❤❡♦r②✳ ■t ✐s ❞❡✜♥❡❞ ❜② ✿ s t❤❡ t❡st ❞✐✛❡r❡♥t✐❛• ❢♦r♠s t♦ ❜❡ ✐♥ Ω d 0,1(E) ❛♥❞ ❛❞❞s ❛ ❝♦♥tr♦• ♦♥ dω✳ ■t ✐s ❛r❣✉❡❞ ✐♥ ❬✻✼( t❤❛t t❤❡ ✢❛t ♥♦r♠ ✐s ♠✉❝❤ ❜❡tt❡r ✜tt❡❞ t♦ s❤❛♣❡ ❝♦♠♣❛r✐s♦♥ ❜❡❝❛✉s❡ t❤❡ ❞✐st❛♥❝❡ ✐s ❜❛s✐❝❛••② ❝♦♥tr♦••❡❞ ❜② t❤❡

  t❡rs ✇❤❡r❡ t❤❡s❡ r❡♠❛r❦s ✇✐•• ❜❡ tr❡❛t❡❞ ✇✐t❤ ♠♦r❡ ❞❡t❛✐•✳ ■♥ s✉♠♠❛r②✱ ✇❡ ❜r✐❡✢② ✇❡♥t ♦✈❡r t❤❡ ❡ss❡♥t✐❛• ❛❞✈❛♥t❛❣❡s ♦❢ ❝✉rr❡♥ts ❛♥❞ |❑❍❙ ♥♦r♠s ♦♥ ❝✉rr❡♥ts ✐♥ ❜✉✐•❞✐♥❣ ❞❛t❛ ❛tt❛❝❤♠❡♥t ♠❡❛s✉r❡s ❢♦r ❞❛t❛s❡ts ❝♦♥s✐st✐♥❣ ♦❢ s✉❜♠❛♥✐❢♦•❞s✳ ❊①❝❡♣t ❢♦r t❤❡ ♣r♦❜•❡♠ ♦❢ ♥♦♥✲✐♥✈❛r✐❛♥❝❡ t♦ ♦r✐❡♥t❛t✐♦♥ ❝❤❛♥❣❡s t❤❛t s❤❛•• ❜❡ t❤❡ ♠❛✐♥ t♦♣✐❝ ♦❢ ❝❤❛♣t❡r ✸✱ t❤❡ r❡s✉•t✐♥❣ ❞✐st❛♥❝❡s ❝♦♠♣•② t♦ ♠♦st t❤❡ r❡✲ q✉✐r❡♠❡♥ts ✇❡ ♠❡♥t✐♦♥❡❞✳ ❙✉❝❤ ❛tt❛❝❤♠❡♥ts ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ✐♥ s❡✈❡r❛• r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠s •✐❦❡ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ♦❢ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ❛♥❞ ❤❛s ❜❡❡♥ ❛•r❡❛❞② ❛♣♣•✐❡❞ t♦ ❞❛t❛s❡ts ♦❢ ❝✉r✈❡s ❬✸✻✱ ✸✶(✱ ✜❜❡r ❜✉♥❞•❡s ❬✸✵( ❛♥❞ s✉r❢❛❝❡s ❬✹✸✱ ✸✺(✳ ✸✷ ✶✳✹ Pr♦❜•❡♠❛t✐❝s ❛♥❞ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤✐s ✇♦r❦ ❍❛✈✐♥❣ ♦✉t•✐♥❡❞ t❤❡ ❣❡♥❡r❛• ❢r❛♠❡✇♦r❦ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❛s ✇❡•• ❛s s♦♠❡ ♦❢ t❤❡ ❡①✐st✐♥❣ ♠❛t❤❡♠❛t✐❝❛• t♦♦•s r❡•❛t❡❞ t♦ t❤❡ t❤❡s✐s✱ ✇❡ ♥♦✇ ✐♥tr♦❞✉❝❡✱ ✐♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ❡ss❡♥t✐❛• ✐ss✉❡s t❤❛t ❛r❡ ❛❞❞r❡ss❡❞ ❜② t❤✐s ✇♦r❦✳ ✶✳✹✳✶ ❊①t❡♥s✐♦♥ t♦ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❛t❛ ❚❤❡ ❝❡♥tr❛• q✉❡st✐♦♥ st✉❞✐❡❞ ❤❡r❡ ✐s t❤❡ ❡①t❡♥s✐♦♥ ♦❢ ❛•• t❤❡ ♣r❡✈✐♦✉s ♠♦❞❡••✐♥❣ ❛♥❞ s❡tt✐♥❣ ❢♦r s❤❛♣❡ ❛♥❛•②s✐s t♦ t❤❡ ❝❛s❡ ♦❢ str✉❝t✉r❡s t❤❛t ♠✐① ❜♦t❤ ❣❡♦♠❡tr② ❛♥❞ s✐❣✲ ♥❛•✱ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ♥❛♠❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✭♦r ❢s❤❛♣❡s ✐♥ s❤♦rt✮✳ ❆s ✇❡ ✇✐•• s❡❡ ✐♥ ❝❤❛♣t❡r ✷✱ s✉❝❤ t②♣❡ ♦❢ ♦❜•❡❝ts ❛r❡ ♦❢ ❣r♦✇✐♥❣ ❢r❡q✉❡♥❝② ✐♥ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❛♥❞ ♠♦r❡ ♣❛rt✐❝✉•❛r•② ✐♥ ♠❡❞✐❝❛• ✐♠❛❣✐♥❣ s✐♥❝❡ t❤❡ ❛♣♣❛r✐t✐♦♥ ♦❢ ♥❡✇ ❞❛t❛ ❛❝q✉✐s✐✲ t✐♦♥ t❡❝❤♥♦•♦❣✐❡s t❤❛t ❡①tr❛❝t t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❛♥❛t♦♠② ✇❤✐•❡ ♠❡❛s✉r✐♥❣ ❛❞❞✐t✐♦♥❛• ❢✉♥❝t✐♦♥❛• s✐❣♥❛• ❞❛t❛ ♦♥ t❤❡s❡ s❤❛♣❡s✱ ❛s ❢♦r ✐♥st❛♥❝❡ ❢▼|■ ❞❛t❛s❡ts ❢♦r t❤❡ ❜r❛✐♥ ♦r t❤✐❝❦♥❡ss ♠❛♣s ❢♦r ✈❛r✐♦✉s ♦r❣❛♥s✳ ❈♦♥tr❛r✐•② t♦ ✉s✉❛• ✐♠❛❣❡s✱ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❛t❛s❡ts ❤❛✈❡ ✈❛r②✐♥❣ ❣❡♦♠❡tr✐❝❛• s✉♣♣♦rt ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✈❛r✐❛❜✐•✐t② ♦❢ t❤❡ s✐❣♥❛• ✐ts❡•❢✱ ✇❤✐❝❤ ✐s ❝❤❛••❡♥❣✐♥❣ ❛t s❡✈❡r❛• •❡✈❡•s ❢♦r ♦♥❡ t♦ ♣r♦♣♦s❡ ❛ ❝♦❤❡r❡♥t ❢r❛♠❡✇♦r❦ ♦❢ ❛♥❛•②s✐s ✐♥ t❤❡ s♣✐r✐t ♦❢ ▲❉❉▼▼✳ ❋✉♥❝t✐♦♥❛• s❤❛♣❡s ❛♥❞ ♠❡t❛♠♦r♣❤♦s❡s ■♥ t❤✐s ✐♥tr♦❞✉❝t✐♦♥✱ ✇❡ ✇✐•• ❢♦❝✉s ♦♥ t❤❡ s✐♠♣•❡st ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✱ ✇❤❡r❡ t❤❡ s✐❣♥❛•s ❝❛rr✐❡❞ ❜② s❤❛♣❡s ❛r❡ r❡❛•✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ♥♦t✐♦♥ s❤❛•• ❜❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ✐ts ❢✉•• ❣❡♥❡r❛•✐t② ✐♥ ❝❤❛♣t❡r ✷✱ ♥♦t❛❜•② t♦ ❡♥❝♦♠♣❛ss s✐t✉❛t✐♦♥s ✇❤❡r❡ s✐❣♥❛•s ❛r❡ ✈❡❝t♦rs ♦r t❡♥s♦rs ❢♦r ✐♥st❛♥❝❡✳ ■♥ t❤✐s ❝❛s❡✱ ❛ ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ❝❛♥ ❜❡ t❤♦✉❣❤t ♥❛✐✈❡•② ❛s ❛ r❡❛• ❢✉♥❝t✐♦♥ ♦♥ ❛ s❤❛♣❡ X✳ ■❢ ✇❡ ❛❞♦♣t t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✇❡ ❝♦✉•❞ s❛② ♠♦r❡ ♣r❡❝✐s❡•② t❤❛t ❛ d✲❞✐♠❡♥s✐♦♥❛• ❢✉♥❝t✐♦♥❛• s❤❛♣❡ ✐s ❛ ❝♦✉♣•❡ (X, f ) ✇❤❡r❡ X ✐s ❛ d✲❞✐♠❡♥s✐♦♥❛• r❡❝t✐✜❛❜•❡ s✉❜s❡t ♦❢ t❤❡ ✈❡❝t♦r s♣❛❝❡ E ❛♥❞ f ✐s ❛ r❡❛•✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ X✳ ❲❡ ❝❛♥ ❛ss✉♠❡ ❛•s♦ t❤❡ ❢✉♥❝t✐♦♥ f t♦ ❜❡•♦♥❣ t♦ t❤❡ s♣❛❝❡ L 2 (X) ✐✳❡ t❤❛t X f 2 (x)dH d (x) < +∞ .◆♦t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f ✐s ♥♦t ❛ss✉♠❡❞ t♦ ❜❡ ❞❡✜♥❡❞ ♦♥ E ❜✉t ♦♥•② ♦♥ t❤❡ ❣❡♦✲ ♠❡tr✐❝❛• s✉♣♣♦rt ❣✐✈❡♥ ❜② t❤❡ s❤❛♣❡ X✳ ❚❤✐s ✐s ❝♦♥tr❛st✐♥❣ ✇✐t❤ t❤❡ ❝❛s❡ ♦❢ ✐♠❛❣❡s ✇❤❡r❡ t❤❡ s✉♣♣♦rt ✐s ✜①❡❞ ♦♥❝❡ ❛♥❞ ❢♦r ❛••✳ ❍❡r❡✱ ✇❡ s❡❡ t❤❛t t❤❡ s♣❛❝❡s ♦♥ ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥s •✐✈❡ ❛r❡ ♥

  ❞❡r❛t✐♦♥ ♣❛r❛♠❡t❡rs ❜❡t✇❡❡♥ t❤❡ ❣❡♦♠❡tr✐❝❛• ❞❡❢♦r✲ ♠❛t✐♦♥ ♣❛rt ❛♥❞ t❤❡ r❡s✐❞✉❛• ♦♥❡✳ ◆♦t❡ t❤❛t t❤❡ ✐♥s✐❞❡ t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛• ❝❛♥ ❜❡ ❛•s♦ r❡✇r✐tt❡♥✱ t❤❛♥❦s t♦ t❤❡ ❛r❡❛ ❢♦r♠✉•❛ ❬✸✼( ❝♦r♦••❛r② ✸✳✷✳✷✵ ✭♦r s✐♠♣•② ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ X ✐s ❛ s✉❜♠❛♥✐❢♦•❞✮ ✿

✭✶✳✸✹✮

  ❚❤❡ r❡s✉•ts ♦❢ t❤❡ t✇♦ •❛st t❤❡♦r❡♠s s❤♦✇ t❤❛t ❝♦♠♣❛r✐s♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s t❤r♦✉❣❤ ♠❡t❛♠♦r♣❤♦s❡s ❝❛♥ ❜❡ ❞♦♥❡ ✇✐t❤✐♥ ❛ s✐♠✐•❛r ❢r❛♠❡✇♦r❦ ❛s t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❢♦r ✉s✉❛• s❤❛♣❡s ✿ ✐t ✐♥✈♦•✈❡s ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❞❡❢♦r♠❛t✐♦♥s✳ ❖♥❡ ❝♦✉•❞ ✐♥ ❛❞❞✐t✐♦♥✱ ❢♦••♦✇✐♥❣ t❤❡ s❛♠❡ ♣❛t❤ ❛s ✐♥ s❡❝t✐♦♥ ✶✳✷✳✺✱ ✇r✐t❡ t❤❡ ❍❛♠✐•t♦♥✐❛♥ ❡q✉❛t✐♦♥s r❡•❛t❡❞ t♦ t❤❡ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ✐♥ t❤✐s s❡tt✐♥❣✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❣❡♥❡r❛•✐③❡s✱ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞✱ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ s❤❛♣❡s ❛r❡ ♦♥•② ❞❡❢♦r♠❡❞ ❣❡♦♠❡tr✐❝❛••②✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥✜♥✐t❡ ♣❡♥❛•t② γ f → ∞ ♦♥ t❤❡ r❡s✐❞✉❛• ♣❛rt ♦❢ t❤❡ ❡♥❡r❣② ✐♥ ❡q✳✭✶✳✷✹✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐t✬s ❛•s♦ ❛ ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ t❤❡ ♠❡t❛♠♦r♣❤♦s✐s ✐❞❡❛ ❢♦r ✐♠❛❣❡s✱ ❢♦r ✇❤✐❝❤ ❜❛s✐❝❛••② ❛•• s❤❛♣❡ s✉♣♣♦rts ❡q✉❛• ❛ ✜①❡❞ ❞♦♠❛✐♥ ♦❢ t❤❡ s♣❛❝❡✳ ✸✼ ❆ t❛♥❣❡♥t✐❛• ♠♦❞❡• ■♥ t❤❡ ♣r❛❝t✐❝❛• ❝❛s❡s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ✇♦r❦✱ ✇❡ s❤❛•• ❜❡ ❞❡❛•✐♥❣ ✇✐t❤ ❛♥❡❛r•✐❡r ♠♦❞❡• t❤❛t s✐♠♣•✐✜❡s ❛ •✐tt•❡ ❜✐t t❤❡ ♣r❡✈✐♦✉s ♠❡t❛♠♦r♣❤♦s✐s ❢r❛♠❡✇♦r❦ ♦♥ ❢✉♥❝t✐♦

  ❞❛t❛ ❛tt❛❝❤♠❡♥t t❡r♠s ❜❡t✇❡❡♥ ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s ❛♥❞ st✉❞② r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ t❤✐s ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥✳ ❆s ❛ ❞✉❛• ♦❢ ❛ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s✱ ❍✐•❜❡rt ♠❡tr✐❝s ❜❛s❡❞ ♦♥ ❦❡r♥❡•s ❝❛♥ ❜❡ ✐♥❞❡❡❞ ❛❞❛♣t❡❞ ❢♦r ✈❛r✐❢♦•❞s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s s❤❛•• ❜❡ t❤♦r♦✉❣❤•② ❡①❛♠✐♥❡❞ ❛s ✇❡••✳ ✶✳✹✳✸ ❖r❣❛♥✐③❛t✐♦♥ ♦❢ t❤❡ ❝❤❛♣t❡rs ❚❤✐s t❤❡s✐s ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r ❝❤❛♣t❡rs✳ ❈❤❛♣t❡rs ✷ ❛♥❞ ✸ ❛r❡ ❛❞❞r❡ss✐♥❣ t❤❡ ♣r♦❜✲ •❡♠s ♦❢ ❞✐ss✐♠✐•❛r✐t② ♠❡❛s✉r❡s ❛♥❞ r❡❣✐str❛t✐♦♥ r❡s♣❡❝t✐✈❡•② ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❛♥❞ ❢♦r ♥♦♥✲♦r✐❡♥t❡❞ s❤❛♣❡s✳ ❚❤❡s❡ ❛r❡ ✇✐❞❡•② ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❝❛♥ ❜❡ r❡❛❞ s❡♣❛r❛t❡•② ❢♦r t❤❡ ♠♦st ♣❛rt✳ ❇❛s❡❞ ✉♣♦♥ t❤❡ s❡tt✐♥❣ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛♥❞ ✈❛r✐❢♦•❞s✱ t❤❡ •❛st ❝❤❛♣t❡r ❜r♦❛❞❡♥s t❤❡ ❛♣♣r♦❛❝❤ t♦ ❣r♦✉♣✲st✉❞② ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ ❢r❛♠❡✇♦r❦ ❢♦r st❛t✐st✐❝❛• ❛t•❛s ❡st✐♠❛t✐♦♥ ❛♥❞ ❛♥❛•②s✐s ♦❢ ♣♦♣✉•❛t✐♦♥s ♦❢ s✉❜•❡❝ts✳ ❚❤❡ ❝♦♥t❡♥ts ♦❢ ❡❛❝❤ ❝❤❛♣t❡r ❝❛♥ ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦••♦✇s ✿ • ❈❤❛♣t❡r ✷ ✿ t❤✐s ❝❤❛♣t❡r ✐♥tr♦❞✉❝❡s t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤✐♥ ❝♦♠✲ ♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ❛♥❞ ❣✐✈❡s ❛ ✇✐❞❡r ❞❡✜♥✐t✐♦♥ t❤❛♥ t❤❡ ♦♥❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ✐♥tr♦❞✉❝t✐♦♥✳ ■♥ ❞✐r❡❝t •✐♥❡ ♦❢ t❤❡ ❢s❤❛♣❡ ♠❡t❛♠♦r♣❤♦s✐s s❡tt✐♥❣ ♣r❡s❡♥t❡❞ ✐♥ ✶✳✹✳✶✱ t❤❡ ♠❛✐♥ ✐ss✉❡ t❤❛t r❡♠❛✐♥s ❢♦r ✐♥❡①❛❝t r❡❣✐str❛t✐♦♥ ✐s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❞❛t❛ ❛tt❛❝❤✲ ♠❡♥t t❡r♠s✳ ❚❤❡ ❢r❛♠❡✇♦r❦ ✇❡ ♣r♦♣♦s❡ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ❝✉rr❡♥ts✱ t❤❛t ✇❡ ❤❛✈❡ ❞❡s✐❣♥❛t❡❞ ❜② ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ✇❤✐❝❤ ❛✉❣♠❡♥ts t❤❡ s♣❛❝❡ ♦❢ ✉s✉❛• ❝✉rr❡♥ts ✇✐t❤ ❛ ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ s✐❣♥❛• s♣❛❝❡✳ ❚❤❡s❡ s♣❛❝❡s ❡♠❜❡❞ t❤❡ s❡t ♦❢ ♦r✐❡♥t❡❞ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❲❡ t❤❡♥ ❡①♣•❛✐♥ ❤♦✇✱ s✐♠✐•❛r•② t♦ ❝✉rr❡♥ts✱ ♦♥❡ ❝❛♥ ❜✉✐•❞ ❍✐•❜❡rt str✉❝t✉r❡s ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ❣✐✈❡♥ ❜② r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡•s✳ ❚❤❡ ❞✐st❛♥❝❡s t❤❛t ❛r❡ ✐♥❞✉❝❡❞ ♦♥ ❢s❤❛♣❡s ❛••♦✇ s✐♠✉•t❛♥❡♦✉s ❝♦♠♣❛r✐s♦♥ ♦❢ s❤❛♣❡ ❛♥❞ s✐❣♥❛• ✇✐t❤♦✉t ❡✈❡r r❡q✉✐r✐♥❣ ♣♦✐♥t✲t♦✲♣♦✐♥t ❝♦rr❡s♣♦♥❞❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ ♦❜•❡❝ts✳ ❲❡ s❤♦✇ ✐♥ ❛❞❞✐t✐♦♥ ❛ ❢❡✇ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥tr♦• ❢♦r t❤❡s❡ ♠❡tr✐❝s ❛♥❞ ❣❡♥❡r❛•✐③❡ ❈❛rt❛♥✬s ✈❛r✐✲ ❛t✐♦♥ ❢♦r♠✉•❛ t♦ ❢❝✉rr❡♥ts ✭t❤❡♦r❡♠ ✷✳✷✳✹✮✳ ❚❤❡ •❛st s❡❝t✐♦♥s ♦❢ ❝❤❛♣t❡r ✷ ❢♦❝✉s ♦♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ❛♥❞ ❞❡t❛✐• ✐♥❡①❛❝t r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠s ♦♥ ❢s❤❛♣❡s ✇✐t❤ ❛ ❢❡✇ r❡s✉•ts✳ ❚❤❡ ♥✉♠❡r✐❝❛• ✐ss✉❡s r❡•❛t❡❞ t♦ ❦❡r♥❡• ❝♦♠♣✉t❛t✐♦♥s ❢♦r ❞✐✛❡♦♠♦r♣❤✐s♠s ❛♥❞ ❢❝✉rr❡♥ts✬ ❞✐st❛♥❝❡s ❛r❡ ❛•s♦ ❞✐s❝✉ss❡❞✳

  47 Diracs σ g = 0.02, σ f = 0.4, 170 Diracs σ g = 0.01, σ f = 0.2, 565 Diracs σ g = 0.04, σ f = 0.2, 57 Diracs σ g = 0.02, σ f = 0.1, 161 Diracs σ g = 0.01, σ f = 0.1, 571 Diracs

  ✇✐t❤ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs✳ ❲❡ s❡❡ t❤❛t t❤❡ ❜r❛♥❝❤❡s ❛r❡ ✇❡•• r❡❝♦✈❡r❡❞ ✇✐t❤ ✈❛r✐❢♦•❞s ✇❤❡r❡❛s t❤❡ ❝✉rr❡♥t✬s ♠❡tr✐❝ ✐s ♥❡❛r•② ✐♥s❡♥s✐t✐✈❡ t♦ t❤❡♠✳ ❋✐❣✉r❡ ✸✳✼✿ ❚❤❡ s❛♠❡ ❡①❛♠♣•❡ ❢♦r ✈❛r✐❢♦•❞s ❜✉t ♣❡r❢♦r♠❡❞ ❛t ❞✐✛❡r❡♥t s❝❛•❡s σ e ✳ ✶✷✵ ❋✐❣✉r❡ ✸✳✽✿ |❡❣✐str❛t✐♦♥ r❡s✉•t ♦♥ ✜❜❡rs ✇✐t❤ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉rr❡♥ts ✭•❡❢t✮ ❛♥❞ ♦❢ ✈❛r✐❢♦•❞s ✭r✐❣❤t✮✳ ❆s ❡①♣❡❝t❡❞✱ ✈❛r✐❢♦•❞s ❝❛

  t❤❡ ♥❡r✈❡ ✜❜❡r •❛②❡r s✉r❢❛❝❡ ✭◆❋▲✮✱ t❤❡ ❇r♦♦❦✬s ♠❡♠❜r❛♥❡ ✭❇▼✮ ❛♥❞ t❤❡ ❈❤♦r♦✐❞ s✉r❢❛❝❡✳ ❚❤✐s ✐s s❤♦✇♥ ✐♥ t❤❡ ✐♠❛❣❡s ♦❢ ✜❣✉r❡ ✹✳✼✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❛t✱ ✇❛s ❝♦♠♣✉t❡❞ t✇♦ t❤✐❝❦♥❡ss ♠❛♣s ✿ ♦♥❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t❤✐❝❦♥❡ss ❜❡t✇❡❡♥ t❤❡ ■▲▼ ❛♥❞ ◆❋▲ s✉r❢❛❝❡s t❤❛t ✇❡ ✇✐•• ❝❛•• t❤❡ ◆❋▲ t❤✐❝❦♥❡ss ❛♥❞ t❤❡ ❈❤♦r♦✐❞❛• t❤✐❝❦♥❡ss ❜❡t✇❡❡♥ t❤❡ ❇▼ ❛♥❞ t❤❡ ❈❤♦r♦✐❞✳ ■♥ t❤✐s ♠❛♥✉s❝r✐♣t✱ ✇❡ ✇✐•• ❢♦❝✉s ♦♥•② ♦♥ t❤❡ ◆❋▲ ❛♥❞ ✐ts t❤✐❝❦♥❡ss ❡✈❡♥ t❤♦✉❣❤ s♦♠❡ ✇♦r❦ ✇❛s ❛•s♦ ✶✹✼ ❋✐❣✉r❡ ✹✳✼✿ ❚❤❡ ❞✐✛❡r❡♥t •❛②❡rs ❛♥❞ ♠❡♠❜r❛♥❡s ♦❜s❡r✈❡❞ ✐♥ t❤❡ r❡t✐♥❛ ✸❉ ✐♠❛❣❡s ❛❝q✉✐r❡❞ ❜② ❖❈❚✳ ❝❛rr✐❡❞ ♦♥ t❤❡ ❈❤♦r♦✐❞ ❛s ✇❡•• ❛s ♦♥ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ■▲▼✳ ❖♥❡ ♦❢ t❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ t❤✐❝❦♥❡ss ♦♥ t❤❡ ◆❋▲ ✐s ♠♦r❡ ❛❝❝✉r❛t❡ ❛♥❞ ♣r♦✈✐❞❡s ♣❛tt❡r♥s t❤❛t ❛r❡ ♠♦r❡ ❡❛s✐•② ✐♥t❡r♣r❡t❛❜•❡✳ ❚❤✉s✱ r❡str✐❝t❡❞ t♦ t❤❡ ◆❋▲✱ t❤❡ ❞❛t❛s❡t ♦❜t❛✐♥❡❞ ❛❢t❡r ♣r❡✲♣r♦❝❡ss✐♥❣ ✐s ♠❛❞❡ ♦❢ ❞✐✛❡r❡♥t s✉❜•❡❝ts✬ s✉r❢❛❝❡s ♦♥ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t✐♦♥❛• ❞❛t❛ r❡♣r❡s❡♥t✐♥❣ t❤❡ t❤✐❝❦✲ ♥❡ss ♦❢ t❤❡ •❛②❡r ❛t ❡❛❝❤ ♣♦✐♥t✳ ❚❤❡ ❞❛t❛❜❛s❡ ✇❡ ❤❛✈❡ ❜❡❡♥ ✇♦r❦✐♥❣ ♦♥ ❝♦♥t❛✐♥ s✉❝❤ ❢✉♥❝t✐♦♥❛• s✉r❢❛❝❡s ❢♦r ✸✵ ❞✐✛❡r❡♥t s✉❜•❡❝ts ♣✉t ✐♥ t❤❡ s❛♠❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✱ s♦♠❡ ♦❢ ✇❤✐❝❤ ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✹✳✽✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ s✉r❢❛❝❡s ✐s q✉✐t❡ ❝•♦s❡ t♦ ❜❡ ♣•❛♥❛r ✐♥ ♠♦st s✉❜•❡❝ts ❜✉t t❤❡ ♦r✐❡♥t❛t✐♦♥ ✈❛r✐❡s ❢r♦♠ s✉❜•❡❝t t♦ s✉❜•❡❝t✳ ❆♥♦t❤❡r ✐♠♣♦rt❛♥t ❡•❡♠❡♥t ✐s t❤❡ ♦♣❡♥✐♥❣ ✭t❤r♦✉❣❤ ✇❤✐❝❤ t❤❡ ♦♣t✐❝❛• ♥❡r✈❡ ♣❛ss❡s ✐♥ t❤❡ r❡t✐♥❛✮ t❤❛t ❛•s♦ ✈❛r✐❡s ✐♥ s✐③❡ ❛♥ ♣♦s✐t✐♦♥✇✐t❤✐♥ t❤❡ ❞❛t❛s❡t✳ ❚❤✐❝❦♥❡ss❡s ♦✛❡r ♠♦r❡ ♦r •❡ss ❝♦♠♠♦♥ ♣❛tt❡r♥s ❛s ✇❡•• ❜✉t ❛•s♦ s✐❣♥✐✜❝❛♥t •♦❝❛• ❞✐✛❡r❡♥❝❡s✳ ◆♦✇✱ t❤❡ ♠❛✐♥ ❝•✐♥✐❝❛• ✐ss✉❡ t❤❛t ♠❛❦❡s t❤❡ ✐♥t❡r❡st ♦❢ t❤✐s ❞❛t❛ ✐s t❤❡ st✉❞② ♦❢ t❤❡ ❣•❛✉❝♦♠❛✱ ❛♥ ♦❝✉•❛r ♣❛t❤♦•♦❣② t❤❛t ✐s r❡✢❡❝t❡❞ ❜② r❡t✐♥❛ ❛♥❞ ♦♣t✐❝❛• ♥❡r✈❡ ❞❛♠❛❣❡s •❡❛❞✐♥❣ ♣r♦❣r❡ss✐✈❡•② t♦ ♣❛rt✐❛• ♦r ❡✈❡♥ ❝♦♠♣•❡t❡ ✈✐s✉❛• ✜❡•❞ •♦ss✳ ❚❤❡ ❞❛t❛s❡t ✇❡ •✉st ♣r❡s❡♥t❡❞ ✇❛s ❛❝q✉✐r❡❞ ❢♦r ❜♦t❤ ❝♦♥tr♦•s ❛♥❞ ❣•❛✉❝♦♠❛t♦✉s s✉❜•❡❝ts✳ ❚❤❡ q✉❡st✐♦♥ r❛✐s❡❞ ✐s t❤❡ ♣♦ss✐❜✐•✐t② t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♣r❡s❡♥❝❡ ❛♥❞ st❛❣❡ ♦❢ t❤❡ ❞✐s❡❛s❡ ❜② t❤❡ s❤❛

  r♠❛t✐♦♥s ❛r♦✉♥❞ ❡❛❝❤ ❝•❛ss✱ ❛s ✐♥tr♦❞✉❝❡❞ ❢♦r ❣r❡②✲•❡✈❡• ✐♠❛❣❡s ✐♥ ✶✹✽ ❋✐❣✉r❡ ✹✳✽✿ ❚✇❡•✈❡ ◆❋▲ s✉r❢❛❝❡s ✇✐t❤ t❤❡✐r t❤✐❝❦♥❡ss ❢r♦♠ t❤❡ ❞❛t❛❜❛s❡ ♣r♦✈✐❞❡❞ ❜② ❇✐♦♠❡❞✐❝❛• ❖♣t✐❝s |❡s❡❛r❝❤ •r♦✉♣✳ ❚❤❡ t✇♦ ✜rst r♦✇s ❛r❡ ♥♦r♠❛• s✉❜•❡❝ts✱ t❤❡ t✇♦ •❛st ♦♥❡s ❛r❡ s✉❜•❡❝ts ❛✛❡❝t❡❞ ❜② ••❛✉❝♦♠❛✳ ❚❤❡ ❝❤♦s❡♥ ✈✐❡✇ ✐s t❤❡ s❛♠❡ ❢♦r ❛•• ✐♠❛❣❡s ❛s ✇❡•• ❛s t❤❡ ❝♦•♦r ♠❛♣ ✭❜•✉❡ ❢♦r s♠❛•• t❤✐❝❦♥❡ss✱ r❡❞ ❢♦r •❛r❣❡✮✳ ✶✹✾ ❚❡♠♣•❛t❡ ❢♦r ♥♦r♠❛• s✉❜•❡❝ts ❚❡♠♣•❛t❡ ❢♦r ❣•❛✉❝♦♠❛t♦✉s s✉❜•❡❝ts ❋✐❣✉r❡ ✹✳✾✿ ❚✇♦ ✈✐❡✇s ♦❢ t❤❡ t❡♠♣•❛t❡s ❢♦r ❜♦t❤ ❝•❛ss❡s ♦❢ s✉❜•❡❝ts ❡st✐♠❛t❡❞ ✇✐t❤

  ✇❡r ✜❣✉r❡✳ ♠❡tr✐❝ ♦♥ X ✭❝❢ ♣r❡✈✐♦✉s s❡❝t✐♦♥✮✳ ❲❡ ❞✐s♣•❛② t❤❡ r❡s✉•ts ❢♦r t❤❡ P❈❆ ✐♥ ✜❣✉r❡ ✹✳✶✼✳ ❚❤❡ ✜rst ❝♦♠♣♦♥❡♥t ✈❡❝t♦r ❜❛s✐❝❛••② s❤♦✇s ❛ t❤✐❝❦♥❡ss ❞❡❝r❡❛s❡ ❢♦r s✉❜•❡❝ts ✇✐t❤ ❣•❛✉❝♦♠❛ ♦♥ ❛ ❈✲s❤❛♣❡ ❝♦♥❝❡♥tr❛t❡❞ ♠❛✐♥•② ❛r♦✉♥❞ t❤❡ ♦♣❡♥✐♥❣✱ ✇❤✐❝❤ t❡♥❞s t♦ ❝♦♥✜r♠ t❤❡ ♦✈❡r❛•• ♦❜s❡r✈❛t✐♦♥s ♦❢ ❝•✐♥✐❝✐❛♥s✳ ❆s ✇❡ s❡❡ ♦♥ t❤❡ •♦✇❡r ♣•♦t✱ t❤❡ t✇♦ ✜rst P❈❆ ✈❡❝t♦rs ❢♦r t❤✐❝❦♥❡ss❡s s❡❡♠ t♦ s❡♣❛r❛t❡ t❤❡ ❝•❛ss❡s ✐♥ ❛ ♥✐❝❡r ✇❛② t❤❛♥ ♠♦♠❡♥t❛✳ ❚❤✐s ✐s ❡✈❡♥ ♠♦r❡ ♦❜✈✐♦✉s ✇✐t❤ t❤❡ ▲❉❆✳ ❚❤❡ ✈❛r✐❛t✐♦♥s ♦❢ t❤✐❝❦♥❡ss❡s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ▲❉❆ ✈❡❝t♦r ❢♦••♦✇ ❛ q✉✐t❡ s✐♠✐•❛r ♣❛tt❡r♥ ❛s t❤❡ ✜rst P❈❆ ✈❡❝t♦r✱ ❛s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✹✳✶✽✳ (❡t ♦♥❡ ❝❛♥ r❡♠❛r❦ ♦♥❝❡ ❛❣❛✐♥ t❤❡ ❣r❡❛t❡r ❝•❛ss s❡♣❛r❛t✐♦♥ ♣r♦♣❡rt② ♦❢ t❤❡ ▲❉❆✳ ❚❤❡s❡ ✜rst s❡ts ♦❢ r❡s✉•ts s✉❣❣❡st t❤❛t ❛♣♣•②✐♥❣ ▲❉❆ ✭r❛t❤❡r t❤❛♥ P❈❆✮ ♦

  t❤❡ t❤r❡❡ t❤✐♥ ❝❛♥❛•s✱ t❤❡ ❛•❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ❝✉rr❡♥ts ❝❛♥ ✈❡r② ❤❛r❞•② r❡❝♦✈❡r ❛ ♣r♦t♦t②♣❡ s❤❛♣❡ ❢r♦♠ t❤❡ ❞❛t❛s❡t ♠❛✐♥•② ❜❡❝❛✉s❡ t❤❡ ❝❛♥❝❡••❛t✐♦♥ ❡✛❡❝t ♦❢ ❝✉rr❡♥ts ♠❛❦❡s t❤❡ ❦❡r♥❡• ♠❡tr✐❝s ✐♥s❡♥s✐t✐✈❡ t♦ t❤♦s❡ ♣✐❡❝❡s✳ ■♥st❡❛❞✱ t❤❡ ✈❛r✐❢♦•❞ ❛t•❛s ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ♠♦r❡ r♦❜✉st ✐♥ t❤❛t r❡❣❛r❞✳ ❆ s♣❡❝✐❛• ❝❛r❡ ❢♦r t❤❡ ✐♥✐t✐❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡♠♣•❛t❡ ✐s ♥❡❡❞❡❞ ✶✼✸ ❋✐❣✉r❡ ✹✳✷✻✿ ❊st✐♠❛t❡❞ ❞❡❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ t❡♠♣•❛t❡ ❛♥❞ t❤❡ s✉❜•❡❝ts ❢r♦♠ t❤❡ ❞❛t❛s❡t✳ ❚❤❡ t❡♠♣•❛t❡ ✐s t❤❡ tr❛♥s♣❛r❡♥t s✉r❢❛❝❡ ❛♥❞ ✐ts ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ✐s t❤❡ ♦♣❛q✉❡ ♦♥❡✳ ❚r❛•❡❝t♦r✐❡s ♦❢ t❤❡ ♣♦✐♥ts ❛r❡ ❞✐s♣•❛②❡❞ ✐♥ r❡❞✳ ✐♥ t❤❛t s♣❡❝✐✜❝ ❞❛t❛s❡t ❜❡❝❛✉s❡ ♦❢ t❤❡ ♣❛rt✐❝✉•❛r t♦♣♦•♦❣② ♦❢ t❤❡ ✐♥♥❡r ❡❛r✳ ❆❢t❡r r❡❞✉❝✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥ t❤❡ tr✐❛♥❣✉•❛t✐♦♥s✱ ✇❡ ✐♥✐t✐❛•✐③❡❞ t❤❡ ❛•❣♦r✐t❤♠ t♦ t❤❡ s✐♠♣•❡ s❤❛♣❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✹✳✷✺✱ ✇❤✐❝❤ ❤❛s t❤❡ t♦♣♦•♦❣② ♦❢ ❛ tr✐♣•❡ t♦r✉s ❜✉t ✐s ♥♦t ❜✐❛s❡❞ t♦✇❛r❞ ♦♥❡ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛•✳ ❚❤❡ ❡st✐♠❛t❡❞ t❡♠♣•❛t❡ s❤❛♣❡ ✐s s❤♦✇♥ ♦♥ t❤❡ r✐❣❤t ♦❢ ✜❣✉r❡ ✹✳✷✺ ❛♥❞ t❤❡ ❞❡❢♦r♠❛t✐♦♥s ❢r♦♠ t❤❡ t❡♠♣•❛t❡ t♦ t❤❡ s✉❜•❡❝ts ♦❢ t❤❡ ❞❛t❛s❡t ✐♥ ✜❣✉r❡ ✹✳✷✻✳ ❆s ❛ ❝♦♠♣❛r✐s♦♥✱ ✇❡ s❤♦✇ t❤❡ t❡♠♣•❛t❡ ❡st✐♠❛t❡❞ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❝✉rr❡♥ts ✇✐t❤ ✐❞❡♥t✐❝❛• ♣❛r❛♠❡t❡rs ❢♦r t❤❡ ❦❡r♥❡•s ✭✜❣✉r❡ ✹✳✷✹✮✳ ❲❡ ♥♦t✐❝❡ ❛ ♣❛rt✐❝✉•❛r•② ✉♥s❛t✐s❢②✐♥❣ ❡st✐♠❛t✐♦♥ ✐♥ t❤❡ r❡❣✐♦♥ ♦❢ t❤❡ ❝❛♥❛•s✱ ✇❤✐❝❤ ✐s t❤❡ ♠♦st s✉❜•❡❝t t♦ ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s✳ ❆s ✇❡ s❡❡✱ t❤❡ t❡♠

  s✉✛❡r ❢r♦♠ s❤❛♣❡ ❛♥♥✐❤✐•❛t✐♦♥ ❛s ❝✉rr❡♥ts✳ |❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ✈❛r✐❢♦•❞s ✇❡r❡ t❤❡♥ ✐♠♣•❡♠❡♥t❡❞ ❛♥❞ t❡st❡❞ ♦♥ ❛ ❢❡✇ ❡①❛♠♣•❡s✳ ❆s ❜r✐❡✢② ♠❡♥t✐♦♥❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❝❤❛♣t❡r ✸✱ t❤❡r❡ ❛r❡ ❛ ❢❡✇ ✐♥t❡r❡st✐♥❣ ♦♣❡♥ q✉❡s✲ t✐♦♥s t❤❛t ❝♦✉•❞ ❢♦••♦✇ ❢r♦♠ t❤✐s ✇♦r❦✳ ❖♥❡ ♦❢ t❤❡♠ ✇♦✉•❞ ❜❡ t♦ ❡st❛❜•✐s❤ ♠❛t❤❡♠❛t✲ ✐❝❛••② ❛ r❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ✈❛r✐❢♦•❞s ❛♥❞ t❤❡ ♦t❤❡r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♥♦♥✲♦r✐❡♥t❡❞ s✉❜♠❛♥✐❢♦•❞s ❣✐✈❡♥ ❜② ♥♦r♠❛• ❝②❝•❡s ✭❝❢ s❡❝t✐♦♥ ✸✳✶✳✸✮✳ ■♥t✉✐t✐✈❡•②✱ ♥♦r♠❛• ❝②❝•❡s s❡❡♠ t♦ ❜❡ ❛ ❤✐❣❤❡r✲♦r❞❡r r❡♣r❡s❡♥t❛t✐♦♥ s✐♥❝❡ ✐t ♠♦❞❡•s ✐♥tr✐♥s✐❝❛••② ✈❛r✐❛t✐♦♥s ♦❢ t❛♥❣❡♥t s♣❛❝❡s✱ ❜✉t t❤✐s r❡♠❛✐♥s t♦ ❜❡ ❡✈❛•✉❛t❡❞ ✐♥ ❛ ♠♦r❡ r✐❣♦r♦✉s ✇❛②✳ ❆ s❡❝♦♥❞ ♣r♦♠✐s✐♥❣ tr❛❝❦✱ ✐♥ ♦✉r ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐s t♦ ❣♦ ❢✉rt❤❡r ♦♥ t❤❡ st✉❞② ♦❢ t❤❡ ❝♦♥str❛✐♥ts ♦♥ s❤❛♣❡s t❤❛t r❡s✉•t ❢r♦♠ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥ ❤✐st♦❣r❛♠s ✭❝❢ ✸✳✹✳✶✮✱ ✇❤✐❝❤✱ ❛s ✇❡ s❤♦✇❡❞✱ ❝♦rr❡s♣♦♥❞s t♦ |❑❍❙ ♦♥ ✈❛r✐❢♦•❞s ✇✐t❤ ✐♥✜♥✐t❡ s❝❛•❡ σ e ✳ ❖♥❡ ♦❢ t❤❡ t❤❡♦r❡t✐❝❛• q✉❡st✐♦♥ t❤❛t ❝♦✉•❞ ❜❡ ❡①❛♠✐♥❡❞ ❢♦r ❡①❛♠♣•❡ ✐s t♦ r❡•❛t❡ ❛ ❞❡❢♦r♠❛t✐♦♥ ♣❛t❤ ♦❢ ❛ ❣✐✈❡♥ s❤❛♣❡ ✐♥ E ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛t❤ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❤✐st♦❣r❛♠s ✿ t❤❡ r❡❛• ✐ss✉❡ t❤❡r❡ ❜❡✐♥❣ t♦ ✜♥❞ ❤♦✇ ♠✉❝❤ ✭♦r ✉♣ t♦ ✇❤❛t tr❛♥s❢♦r♠❛t✐♦♥s✮ ❛ ❤✐s✲ t♦❣r❛♠ ♣❛t❤ ❝❛♥ ❞❡t❡r♠✐♥❡ ❛ s❤❛♣❡ ❞❡❢♦r♠❛t✐♦♥ ♣❛t❤✳ ❚❤✐s ♣r♦❜•❡♠❛t✐❝ ✐s ♦❢ ❝♦✉rs❡ ♥♦t ♦♥•② r❡•❛t❡❞ t♦ ✈❛r✐❢♦•❞s ❜✉t✱ ✐♥ ♦✉r ❝♦♥t❡①t✱ ✐t ✇♦✉•❞ ❣✐✈❡ ❛ ✇❛② t♦ ❜❡tt❡r ✉♥❞❡r✲ st❛♥❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ✈❛r✐❢♦•❞✲❜❛s❡❞ r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠s ❛❝❝♦r❞✐♥❣•② t♦ t❤❡ s❝❛•❡ ♣❛r❛♠❡t❡rs ♦❢ ❦❡r♥❡•s✳ ❖♥ ❢s❤❛♣❡s✬ ♠♦❞❡••✐♥❣ ❛♥❞ ❞✐ss✐♠✐•❛r✐t② ♠❡❛s✉r❡s ❚❤❡ ♠❛✐♥ ♣r♦❜•❡♠❛t✐❝ ❛❞❞r❡ss❡❞ ❜② t❤✐s ❞✐ss❡rt❛t✐♦♥ ❝♦♥s✐st❡❞ ✐♥ ♣r♦♣♦s✐♥❣ ❛ ❢r❛♠❡✲ ✇♦r❦ t❤❛t ❝❛♥ ❡①t❡♥❞ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② t♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ❆s ✇❡ ❡①♣•❛✐♥❡❞ ✐♥ ❝❤❛♣t❡r ✶✱ ♦♥❡ ♦❢ t❤❡ ❡ss❡♥t✐❛• ❝❤❛••❡♥❣❡ ✇❛s t♦ ❤❛✈❡ ❛ r❡♣r❡✲ s❡♥t❛t✐♦♥ t❤❛t ♣r♦✈✐❞❡s ❞✐ss✐♠✐•❛r✐t② ♠❡tr✐❝s ❢♦r ❞❡✜♥✐♥❣ r❡•❡✈❛♥t ❞❛t❛ ❛tt❛❝❤♠❡♥t ✶✼✼ t❡r♠s ♦♥ t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s✳ ■♥ ❝❤❛♣t❡r ✷✱ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♣♦ss✐❜•❡ ❢r❛♠❡✇♦r❦ t♦ ❞♦ s♦✳ ❚❤❡ ❦❡② ✐❞❡❛ ✇❛s t♦ ❞❡✜♥❡ ❛ ♣r♦♣❡r ❣❡♥❡r❛•✐③❛t✐♦♥ ♦❢ ❝✉rr❡♥ts✱ ✇❡ ♥❛♠❡❞ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ t❤❛t ❡♠❜❡❞s ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ♦❢ ❣✐✈❡♥ ❞✐♠❡♥s✐♦♥ ✐♥t♦ ❛ ❝♦♠♠♦♥ ❢✉♥❝t✐♦♥❛• s♣❛❝❡ ✇❤✐❝❤ ✐s t❤❡ ❞✉❛• ♦❢ ❛ s♣❛❝❡ ♦❢ ❡①t❡♥❞❡❞ ❞✐✛❡r❡♥t✐❛• ❢♦r♠s✳ ▼❡tr✐❝s ✇❡r❡ t❤❡♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❜② ❝♦♥s✐❞❡r✐♥❣ ❛♣♣r♦♣r✐❛t❡ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡• ❍✐•❜❡rt s♣❛❝❡s✳ ❖♥ t❤❡ ♠♦❞❡••✐♥❣ s✐❞❡✱ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♦✛❡r s✐♠✐•❛r ❛❞✈❛♥t❛❣❡s t♦ ❝✉rr❡♥ts ✿ t❤❡ r❡s✉•t✐♥❣ ♠❡tr✐❝s ❞♦ ♥♦t r❡q✉✐r❡ ♣♦✐♥t t♦ ♣♦✐♥t ❝♦rr❡s♣♦♥❞❡♥❝❡s ❜❡t✇❡❡♥ ♦❜•❡❝ts✱ ❛r❡ r♦❜✉st t♦ ♥♦✐s❡ ❛♥❞ t♦♣♦•♦❣✐❝❛• ♣❡rt✉r❜❛t✐♦♥s ❛♥❞ ❛r❡ ❝♦♥tr♦••❡❞ ❜② ❣❡♦♠❡tr✐❝♦✲ ❢✉♥❝t✐♦♥❛• ❞❡❢♦r♠❛t✐♦♥s✱ ❛s s❤♦✇♥ ❜② t❤❡ r❡s✉•ts ♦❢ ✷✳✷✳✸✳ ❚❤✐s t❤❡♦r❡t✐❝❛• s❡tt✐♥❣ ✐s ❛•s♦ ❣❡♥❡r❛• ❡♥♦✉❣❤ t♦ tr❡❛t ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ♣♦ss✐❜•❡ s✐❣♥❛• t②♣❡s ✿ r❡❛•✱ ✈❡❝t♦r ♦r t❡♥s♦r✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥s ❛r❡ ❛♠♦♥❣ ✉s✉❛• ❡①❛♠♣•❡s✳ ◆✉♠❡r✐❝❛••②✱ ✇❡ ❤❛✈❡ ❢♦❝✉s❡❞ ♦♥ t❤❡ ❝❛s❡ ♦❢ r❡❛•✲✈❛•✉❡❞ s✐❣♥❛•s ❡①❝•✉s✐✈❡•② ❛♥❞ ♣r♦♣♦s❡❞ •❛r❣❡ ❞❡❢♦r♠❛t✐♦♥ r❡❣✐str❛t✐♦♥ ❛•❣♦r✐t❤♠s ❢♦r ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❜❛s❡❞ ♦♥ t❤❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t ♠❡tr✐❝s ♣r♦✈✐❞❡❞ ❜② ❢❝✉rr❡♥ts✳ ❆s ✇❡ ♠❡♥t✐♦♥❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❝❤❛♣t❡r ✸✱ t❤❡ ✈❛r✐❢♦•❞ s❡tt✐♥❣ ❝❛♥ ❜❡ ❛•s♦ s✉♣❡r✐♠♣♦s❡❞ t♦ ❛•• t❤❡ ♣r❡✈✐♦✉s ✐♥ ❛ ✈❡r② s✐♠♣•❡ ✇❛②✱ ✇❤✐❝❤ r❡s✉•ts ✐♥ t❤❡ ♥♦t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s t❤❛t ❝❛♥ ❤❛✈❡ t❤❡ ❛❞❞✐t✐♦♥❛• r♦❜✉st♥❡ss t♦ ♦r✐❡♥t❛t✐♦♥ ✐ss✉❡s✳ ❆ ♠♦r❡ ❡①t❡♥s✐✈❡ r❡✢❡①✐♦♥ ♦♥ t❤❡ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞r❛✇❜❛❝❦s ♦❢ ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s ✈❡rs✉s ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts✱ ♥♦t❛❜•② ♦♥ t❤❡ q✉❡st✐♦♥ ♦❢ ♥♦✐s❡✱ ✐s st✐•• t♦ ❜❡ ❝♦♥❞✉❝t❡❞✳ ❆ ♥❛t✉r❛• ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ✇♦r❦✱ t❤❛t ✐s ❡ss❡♥t✐❛••② ❛ ❝♦♠♣✉t❛t✐♦♥❛• ✐ss✉❡✱ ✐s t♦ ❛❞❞ ♦t❤❡r ♣♦ss✐❜✐•✐t✐❡s ♦❢ s✐❣♥❛•s ❛♥❞ ❣r♦✉♣ ❛❝t✐♦♥s t♦ t❤❡ ♣r❡s❡♥t ❝♦❞❡✳ ❚❤✐s ❝♦♠❡s ❛•♦♥❣ ✇✐t❤ t❤❡ ❡✛♦rts ✐♥✐t✐❛t❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✸✳✸ t♦ ✐♠♣r♦✈❡ t❤❡ ♥✉♠❡r✐❝❛• ❝♦♠♣•❡①✐t② ❛♥❞ s♣❡❡❞ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❞✐✛❡r❡♥t t❡r♠s ✐♥✈♦•✈❡❞ ✐♥ ♦✉r ❛•❣♦r✐t❤♠s ❛♥❞ ❤♦♣❡❢✉••② ❡♥❛❜•❡ ♥✉♠❡r✐❝❛• s❝❤❡♠❡s t♦ ❛❞❞r❡ss ❛♣♣•✐❝❛t✐♦♥s ✇✐t❤ ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛• s✐❣♥❛•s✳ ▲❛st ❜✉t ♥♦t •❡❛st✱ ✇❡ ✇❛♥t t♦ ♣♦✐♥t ♦✉t t❤❛t t❤❡ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✭♦r ❢✉♥❝t✐♦♥❛• ✈❛r✐❢♦•❞s✮ s❡tt✐♥❣ ✐s ♥♦t •✐♠✐t❡❞ t♦ ✐ts ✉s❡ ❛s ❛♥ ✐♥t❡r♠❡❞✐❛t❡ t♦ ❞❡✜♥❡ ❞❛t❛ ❛tt❛❝❤♠❡♥t t❡r♠s ❢♦r r❡❣✐str❛t✐♦♥ ♦r ❛t•❛s ❡st✐♠❛t✐♦♥✳ ■♥❞❡❡❞✱ t❤❡ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡ t❤❛t ✇❡ ♦❜t❛✐♥ t❤r♦✉❣❤ ❦❡r♥❡•s ♦✛❡rs ♠❛♥② ♦t❤❡r ❛♣♣•✐❝❛t✐♦♥ ♣♦ss✐❜✐•✐t✐❡s✳ ❲❡ ❤❛✈❡ s❦❡t❝❤❡❞ s♦♠❡ ♦❢ t❤❡s❡ ✐♥ t❤✐s t❤❡s✐s✱ ❢♦r ✐♥st❛♥❝❡ ✇✐t❤ t❤❡ ▼❛t❝❤✐♥❣✲P✉rs✉✐t ♣r♦❝❡❞✉r❡ ♦❢ ✷✳✸✳✶ ❢♦r ❞❛t❛ ❝♦♠♣r❡ss✐♦♥✳ (❡t✱ ❛ st✐•• •❛r❣❡•② ✉♥❡①♣•♦r❡❞ r❛♥❣❡ ♦❢ ❛♣♣•✐❝❛t✐♦♥s ✇♦✉•❞ ❜❡ t♦ ❡①♣•♦✐t s✉❝❤ ♠❡tr✐❝s ❢♦r ❞♦✐♥❣ ❞✐r❡❝t st❛t✐st✐❝s ❛♥❞ ❝•❛ss✐✜❝❛t✐♦♥ ♦♥ ❣❡♦♠❡tr✐❝♦✲ ❢✉♥❝t✐♦♥❛• ❞❛t❛s❡ts ✭❛❢t❡r ❣r♦ss r❡❣✐str❛t✐♦♥ ♦❢ t❤❡ s✉❜•❡❝ts✮✱ ✉s✐♥❣ ❡✐t❤❡r ❞✐s❝r✐♠✐♥❛♥t ❛♥❛•②s✐s ❛s ✐♥ s❡❝t✐♦♥ ✹✳✸✳✶ ♦r s✉♣♣♦rt ✈❡❝t♦r ♠❛❝❤✐♥❡s✳ ❚❤✐s ✐s ❝•❡❛r•② ❛ •❡ss ❛❝❝✉r❛t❡ ❛♥❛•②s✐s t❤❛♥ t❤❡ ❛t•❛s✲❜❛s❡❞ ♠❡t❤♦❞s ❞❡✈❡•♦♣❡❞ ✐♥ ❝❤❛♣t❡r ✹ t❤❛t r❡❛••② ❡st✐♠❛t❡ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ✈❛r✐❛t✐♦♥ ❢r♦♠ ❛ t❡♠♣•❛t❡ ❢s❤❛♣❡ ❜✉t ✐t ❤❛s t❤❡ ✐♥t❡r❡st ♦❢ ❜❡✐♥❣ ❢❛st❡r t♦ ❝♦♠♣✉t❡ ❢♦r ✈❡r② ❜✐❣ ❞❛t❛s❡ts ❛♥❞ ❝♦✉•❞ t❤✉s s❡r✈❡ ❛s ❛ ♣r❡•✐♠✐♥❛r② ❛♥❛•②s✐s✳ ❖♥ t❤❡ ❡st✐♠❛t✐♦♥ ❛♥❞ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ ❛t•❛s❡s ❚❤❡ ✜♥❛• ♦❜•❡❝t✐✈❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛• ❛♥❛t♦♠② ✐s t♦ st✉❞② t❤❡ ✈❛r✐❛❜✐•✐t② ♦❢ str✉❝t✉r❡s ✇✐t❤✐♥ ❛ ❝❡rt❛✐♥ ♣♦♣✉•❛t✐♦♥ ♦❢ s✉❜•❡❝ts ❛♥❞ ❞❡❞✉❝❡ ❝•❛ss✐✜❝❛t✐♦♥ r✉•❡s t♦ ❞✐s❝r✐♠✐✲ ♥❛t❡ ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t ♣♦♣✉•❛t✐♦♥s✳ ❚❤✐s ✐s t❤❡ ♣r♦❜•❡♠ t❤❛t ✇❡ ✜♥❛••② ✐♥t❡♥❞❡❞ t♦ ❛❞❞r❡ss ✐♥ t❤❡ ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✐♥ ❝❤❛♣t❡r ✹✳ ❍❛✈✐♥❣ ♣r♦♣♦s❡❞ ❛ s❛t✐s❢②✐♥❣ s❡tt✐♥❣ t♦ ❝♦♠♣❛r❡ ❢s❤❛♣❡s t♦❣❡t❤❡r ❛♥❞ ❡st✐♠❛t❡ ❞❡❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t✇♦ s✉❜✲ •❡❝ts✱ ✇❡ ❣❡♥❡r❛•✐③❡❞ t❤✐s ❛♣♣r♦❛❝❤ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ ❛t•❛s ❡st✐♠❛t✐♦♥ ❛♥❞ st❛t✐st✐❝❛• ✐♥t❡r♣r❡t❛t✐♦♥✳ ❖✉r ✜rst t❛s❦ ✐♥ ❝❤❛♣t❡r ✹ ✇❛s t♦ ♣r♦♣♦s❡ ❛•❣♦r✐t❤♠s✱ ❛❞❛♣t❡❞ ❢r♦♠ t❤❡ ❢♦r✇❛r❞ ❣❡♥❡r❛t✐✈❡ ♠♦❞❡• ✉s❡❞ ❢♦r ✐♠❛❣❡s ❛♥❞ ❝✉rr❡♥ts✱ t♦ ❝♦♠♣✉t❡ ❛♥ ❛t•❛s ❢r♦♠ ❛ ♣♦♣✉•❛t✐♦♥ ✶✼✽ ♦❢ ❢s❤❛♣❡s✳ ■♥ t❤❡ s✐♠♣•❡st s❡tt✐♥❣✱ t❤❡s❡ ❛t•❛s❡s ❝♦♥s✐st ✐♥ ❛ t❡♠♣•❛t❡ ❢s❤❛♣❡ t♦❣❡t❤❡r ✇✐t❤ ❛ s❡t ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♠❛♣♣✐♥❣ t❤❡ t❡♠♣•❛t❡ t♦ t❤❡ s✉❜•❡❝ts✳ ❆s ✇❡ ❤❛✈❡ ❞❡t❛✐•❡❞ ✐♥ ✹✳✷✳✶✱ t❤❡r❡ ❛r❡ s❡✈❡r❛• ❛•❣♦r✐t❤♠✐❝ tr✐❝❦s t❤❛t ❝❛♥ ❜❡ ✐♠♣•❡♠❡♥t❡❞ t♦ ✐♠♣r♦✈❡ t❤❡ ❡st✐♠❛t✐♦♥ ❛•❣♦r✐t❤♠ ✐ts❡•❢✱ ♥♦t❛❜•② ❝♦♥❝❡r♥✐♥❣ r♦❜✉st♥❡ss t♦ s❝❛•❡ ❛♥❞ s❛♠♣•✐♥❣ ❛s ✇❡•• ❛s ❜♦✉♥❞❛r② ❡✛❡❝ts✳ ❚❤❡ ♠♦❞❡• ❝❛♥ ❜❡ t❤❡♥ ❡♥r✐❝❤❡❞ ❜② t❤❡ ❛❞❞✐t✐♦♥❛• ❡st✐♠❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ♦♥ t❤❡ t❡♠♣•❛t❡ t♦ ❞❡s❝r✐❜❡ s✐❣♥❛• ✈❛r✐❛❜✐•✐t② ✐♥ ❛♥ ✐♥t❡r♣r❡t❛❜•❡ ✇❛②✳ ❚❤✐s s❡tt✐♥❣ ✇❛s ✐♠♣•❡♠❡♥t❡❞ ❛s ✇❡•• ✐♥ ♦✉r ❝✉rr❡♥t ❝♦❞❡s✳ ❖♥ t❤❡ ❛•❣♦r✐t❤♠✐❝ s✐❞❡✱ t❤❡r❡ ❛r❡ ♠❛♥② ♦t❤❡r tr❛❝❦s t❤❛t ❝♦✉•❞ ❜❡ ❢♦••♦✇❡❞ t♦ ❡①t❡♥❞✱ s♣❡❡❞ ✉♣ ♦r s✐♠♣•② ✐♠♣r♦✈❡ t❤❡ ♣r❡s❡♥t ✈❡rs✐♦♥✱ s♦♠❡ ♦❢ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❛•r❡❛❞② ❡①♣❡r✐♠❡♥t❡❞ ✇❤✐•❡ ♦t❤❡rs ❛r❡ st✐•• ♠❡r❡ s✉❣❣❡st✐♦♥s✳ ❋♦r ✐♥st❛♥❝❡✱ ♠✉•t✐✲ s❤❛♣❡s ✬ s✐t✉❛t✐♦♥s ✇❤❡r❡ ❛•• t❤❡ ♦❜•❡❝ts t♦ ❛♥❛•②③❡ ❤❛✈❡ s❡✈❡r❛• ❞✐st✐♥❝t ❡•❡♠❡♥ts ✭❛s ❢♦r ❡①❛♠♣•❡ s❡✈❡r❛• s✉r❢❛❝❡ •❛②❡rs ✇✐t❤ ♦r ✇✐t❤♦✉t s✐❣♥❛•s✱ ♦r ♦❜•❡❝ts ♠✐①✐♥❣ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✮ ❛r❡ q✉✐t❡ ✉s✉❛• ✐♥ ❛♣♣•✐❝❛t✐♦♥s✳ ❲❡ ❤❛✈❡ ❡①t❡♥❞❡❞ ♦✉r ❛t•❛s ❡st✐♠❛t✐♦♥ s❝❤❡♠❡s t♦ ❛❝❝♦✉♥t ❢♦r t❤❡s❡✳ ❈♦♥❝❡r♥✐♥❣ s♣❡❡❞ ❛♥❞ s♣❛rs✐t② ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s✱ t❤❡ ❝♦♥tr♦• ♣♦✐♥ts s❡tt✐♥❣ ♦❢ ❬✷✾✱ ✸✹( ✐s ♣r♦❜❛❜•② ❛ ❣♦♦❞ ✐❞❡❛ t♦ r❡❞✉❝❡ ❝♦♠♣✉t❛t✐♦♥❛• t✐♠❡s r❡•❛t❡❞ t♦ t❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ ❢r❛♠❡✇♦r❦ ❜✉t ❤❛s ♥♦t ❜❡❡♥ ✐♥t❡❣r❛t❡❞ ✐♥ ♦✉r ❛•❣♦r✐t❤♠s ②❡t✳ ❆♥♦t❤❡r ♣♦ss✐❜✐•✐t② ✐s t♦ ✇♦r❦ r❛t❤❡r ♦♥ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t s❝❤❡♠❡ ♦♥ ✇❤✐❝❤ t❤❡ ❛•❣♦r✐t❤♠ ✐s ❜❛s❡❞✳ ❈♦♠♣✉t✐♥❣ ❢✉♥❝t✐♦♥❛•s ❛♥❞ ❣r❛❞✐❡♥ts ❛t ❡❛❝❤ st❡♣ ✐s ♦❜✈✐♦✉s•② •✐♥❡❛r ✐♥ t✐♠❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜•❡❝ts ✐♥ t❤❡ ♣♦♣✉•❛t✐♦♥ ❛♥❞ ✐t ✐s ❝•❡❛r t❤❛t✱ ❢♦r •❛r❣❡ ❞❛t❛s❡ts✱ ♣❛r❛••❡•✐③❛t✐♦♥ ✇✐•• ♥♦t ❜❡ ❡♥♦✉❣❤ t♦ •✐♠✐t ✐t✳ ❖♥❡ ♣♦t❡♥t✐❛• s♦•✉t✐♦♥ t❤❛t r❡♠❛✐♥s t♦ ❜❡ t❡st❡❞ ❢♦r t❤❡s❡ ❛♣♣•✐❝❛t✐♦♥s ✐s ❣✐✈❡♥ ❜② st♦❝❤❛st✐❝ ❣r❛❞✐❡♥t ❛•❣♦r✐t❤♠s✱ ✇❤✐❝❤ ♦♥•② r❡q✉✐r❡ s✉❝❤ ❝♦♠♣✉t❛t✐♦♥s ❢♦r ♦♥❡ ✐♥❞✐✈✐❞✉❛• ♣❡r ❞❡s❝❡♥t st❡♣ ❛t t❤❡ ♣r✐❝❡ ♦❢ ❛ s•♦✇❡r ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ ❝❤❛♣t❡r ✹✱ ✇❡ ❤❛✈❡ ❛❞❞r❡ss❡❞ t❤❡ ♥❡①t ✐ss✉❡ ♦❢ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ t❤❡ ✈❛r✐❛❜✐•✐t② ✐♥ s❤❛♣❡ ❛♥❞ ❢✉♥❝t✐♦♥✳ ❲❡ ♣r♦♣♦s❡❞ ❛♥ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ P❈❆ ❛♥❞ ▲❉❆ ✐♥ ❍✐•❜❡rt s♣❛❝❡s ❛♣♣•✐❡❞ t♦ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♠♦♠❡♥t❛ ❛♥❞ ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛• r❡s✐❞✉❛•s ♦❜t❛✐♥❡❞ ❜② ❛t•❛s ❛•❣♦r✐t❤♠s✱ t❤❛t ❡st✐♠❛t❡s ♣r✐♥❝✐♣❛• ♠♦❞❡s ♦❢ ✈❛r✐❛t✐♦♥s ❛♥❞ ❞✐s❝r✐♠✐♥❛♥t ❞✐r❡❝t✐♦♥s ❢♦r ❝•❛ss✐✜❝❛t✐♦♥✳ ❚❤❡r❡ ✐s s✉r❡•② r♦♦♠ ❢♦r ✐♠♣r♦✈❡♠❡♥t ❛t t❤✐s •❡✈❡• s✐♥❝❡ ✇❡ ❝❤♦s❡ t♦ ❢♦❝✉s ♦♥ s✐♠♣•❡ st❛t✐st✐❝❛• •❡❛r♥✐♥❣ t♦♦•s ✇❤❡r❡❛s ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ✐❞❡❛s ❢r♦♠ ♠❛❝❤✐♥❡ •❡❛r♥✐♥❣ ❝♦✉•❞ ❜❡ ❛•s♦ ✐♥✈❡st✐❣❛t❡❞✳ (❡t✱ ❛s ✇❡ ❡✈✐❞❡♥❝❡❞ ✇✐t❤ ❛ ❢❡✇ r❡s✉•ts ♦♥ t❤❡ r❡t✐♥❛ ❞❛t❛s❡t✱ t❤✐s ❢r❛♠❡✇♦r❦ s❡❡♠s q✉✐t❡ s✉❝❝❡ss❢✉• ✐♥ ✜♥❞✐♥❣ ♠❡❛♥✐♥❣❢✉• ✈❛r✐❛t✐♦♥ ♣❛tt❡r♥s ❜♦t❤ ♦♥ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ ❢✉♥❝t✐♦♥❛• •❡✈❡•✱ ❛♥❞ ♦✛❡rs s✐❣♥✐✜❝❛♥t ❝•❛ss✐✜❝❛t✐♦♥ r❛t❡s ♦♥ ❝r♦ss✲✈❛•✐❞❛t✐♦♥ t❡sts✳ ❯♥❞♦✉❜t❡❞•②✱ t❤❡s❡ r❡s✉•ts ♥❡❡❞ t♦ ❜❡ ❝♦♠❢♦rt❡❞ ✐♥ t❤❡ ❢✉t✉r❡ ❜② s✐♠✐•❛r st✉❞✐❡s ❝❛rr✐❡❞ ♦♥ ♦t❤❡r ❞❛t❛s❡ts✳ ❆ ❝♦♠♣•❡t❡ ❛♥❞ ✇❡••✲♣♦s❡❞ |✐❡♠❛♥♥✐❛♥ ❢r❛♠❡✇♦r❦ ❢♦r ❢s❤❛♣❡s✬ ❣r♦✉♣ ❛♥❛•②s✐s ❄ ❚♦ ❣✐✈❡ ❛ ✜♥❛• ❝♦♥❝•✉s✐♦♥ t♦ t❤❡ ♣r❡s❡♥t ✇♦r❦✱ ✇❡ ✇✐s❤ t♦ ❝♦♠❡ ❜❛❝❦ ♦♥ t❤❡ ♦r✐❣✐♥❛• ♣r♦❜•❡♠❛t✐❝ t❤❛t ✇❡ ❤❛❞ ✐♥✐t✐❛t❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✹✳✶ ♦❢ ❝❤❛♣t❡r ✶✳ ❲❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ✐❞❡❛ ♦❢ ❛ ♠❡t❛♠♦r♣❤♦s✐s ❢r❛♠❡✇♦r❦✱ ✐♥ ✇❤✐❝❤ t✇♦ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇✐t❤✐♥ ❛ ✜①❡❞ ❜✉♥❞•❡ ❛r❡ ❝♦♠♣❛r❡❞ ❜② t❤❡ ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡ ✐♥❞✉❝❡❞ ❢r♦♠ t❤❡ s♣❛❝❡ ♦❢ ❢s❤❛♣❡s ♠❡t❛♠♦r♣❤♦s❡s✳ ❆s ✇❡ ❡①♣•❛✐♥❡❞✱ t❤✐s ❣✐✈❡s ❛ |✐❡♠❛♥♥✐❛♥ ❢♦r♠✉•❛t✐♦♥ t♦ t❤❡ ♣r♦❜✲ •❡♠ ♦❢ ❡①❛❝t r❡❣✐str❛t✐♦♥✱ ✇❤✐❝❤ ❛♠♦✉♥ts ✐♥ ❝♦♠♣✉t✐♥❣ ❛ ❣❡♦❞❡s✐❝ ♣❛t❤ ✐♥ t❤❡ ♠❡t❛✲ ♠♦r♣❤♦s❡s s♣❛❝❡✳ ❙✐♠✐•❛r•②✱ ✇✐t❤✐♥ t❤✐s s❡tt✐♥❣✱ ❡st✐♠❛t✐♥❣ ❛t•❛s❡s ✐s s✐♠♣•② t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ♠❡❛♥ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤✐s ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡✳ ❍♦✇❡✈❡r✱ t❤✐s ❤♦•❞s ✐♥ t❤❡ ✐❞❡❛• s✐t✉❛t✐♦♥ ♦❢ s❤❛♣❡s ❜❡•♦♥❣✐♥❣ t♦ ❛ s✐♥❣•❡ ♦r❜✐t ❛♥❞ ✇❡ ❛r❣✉❡❞ ❢♦r t❤❡ ♥❡❝❡ss✐t② ♦❢ ❛♥ ✐♥❡①❛❝t ❢♦r♠✉•❛t✐♦♥ ✐♥✈♦•✈✐♥❣ ❞✐ss✐♠✐•❛r✐t② ♠❡❛✲ ✶✼✾ s✉r❡s ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t ♦r❜✐ts✳ ❉❡✜♥✐♥❣ s✉❝❤ ❞✐ss✐♠✐•❛r✐t② ♠❡❛s✉r❡s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ❤❛✈❡ ❜❡❡♥ ♦♥❡ ♦❢ t❤❡ ✐♠♣♦rt❛♥t ❝❤❛••❡♥❣❡ ❛❞❞r❡ss❡❞ ❤❡r❡✳ ■t ❛♣♣❡❛rs t❤❡r❡❢♦r❡ ❛s ❛♥ ❛tt❛✐♥❛❜•❡ ❣♦❛• t♦ ♣r♦✈✐❞❡ ❛ ❣❡♥❡r❛• ♠❡t❤♦❞♦•♦❣② ❢♦r ❣r♦✉♣ ❛♥❛•②s✐s ♦♥ ❢✉♥❝t✐♦♥❛• s❤❛♣❡s ✇❤✐❝❤ ✇♦✉•❞ ❝♦♠♣•❡t❡•② r❡•② ♦♥ ❛ ❝♦❤❡r❡♥t |✐❡♠❛♥♥✐❛♥ ❢r❛♠❡✇♦r❦✳ ❲❤❛t ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s t❤❡s✐s✱ ❛♥❞ ♥♦t❛❜•② ✐♥ t❤❡ •❛st ❝❤❛♣t❡r✱ ✐s ❛ ❜✐❣ st❡♣ ✐♥ t❤✐s ❞✐r❡❝t✐♦♥✳ ◆❡✈❡rt❤❡•❡ss✱ t❤❡ st♦r② ✐s ♥♦t ❝♦♠♣•❡t❡❞ ②❡t ✐♥ s♦ ❢❛r ❛s t❤❡ ❢r❛♠❡✇♦r❦ ❢♦r ❛t•❛s ❡st✐♠❛t✐♦♥ ✇❡ ❝♦♥s✐❞❡r❡❞ ❛♥❞ ✐♠♣•❡♠❡♥t❡❞ ✇❛s ❞♦♥❡ ✉♥❞❡r t❤❡ s✐♠♣•✐✜❡❞ ✬t❛♥❣❡♥t✐❛• ♠♦❞❡•✬ ✭❝❢ ✶✳✹✳✶✮ ♦❢ st❛t✐♦♥❛r② r❡s✐❞✉❛• ❢✉♥❝t✐♦♥s ♦♥ t❤❡ t❡♠♣•❛t❡✳ ❆ r❡♠❛✐♥✐♥❣ st❡♣ ✐s t♦ ❛❞•✉st ♦✉r ❝✉rr❡♥t ❢r❛♠❡✇♦r❦ t♦ t❤❡ r❡❛• ♠❡t❛♠♦r✲ ♣❤♦s✐s s❡tt✐♥❣✱ ✇❤✐❝❤ ✇♦✉•❞ ❛✛❡❝t ❡ss❡♥t✐❛••② t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❣❡♦❞❡s✐❝ s❤♦♦t✐♥❣ ❡q✉❛t✐♦♥s✳ ❆ •❛st ♣♦✐♥t ✇♦✉•❞ ❜❡ t❤❡♥ t♦ ❡①t❡♥❞ t❤❡ ❡①✐st❡♥❝❡ r❡s✉•t ♦❢ t❤❡♦r❡♠ ✹✳✷✳✶ t♦ t❤✐s s❡tt✐♥❣✳ ✶✽✵ ❆♣♣❡♥❞✐① ❆ Pr♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✷✳✶ ■♥ t❤✐s ❛♣♣❡♥❞✐①✱ ✇❡ ❣✐✈❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❣❡♦♠❡tr✐❝♦✲❢✉♥❝t✐♦♥❛• ❛t•❛s❡s ❡①♣r❡ss❡❞ ❜② t❤❡♦r❡♠ ✹✳✷✳✶✳ ❲❡ ❤❛✈❡ ❞❡❝♦♠♣♦s❡❞ ✐t ✐♥ s❡✈❡r❛• st❡♣s ❜② ✜rst s❤♦✇✲ ✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ t❡♠♣•❛t❡ ♣❛rt (X, f ) ❢♦r ❛ s✐♠♣•✐✜❡❞ ❢✉♥❝t✐♦♥❛• ❛♥❞ t❤❡♥ ❣❡♥❡r❛•✐③❡ t❤❡s❡ ❛r❣✉♠❡♥ts t♦ t❤❡ ❢✉•• ❛t•❛s ♠♦❞❡•✳ ❆✳✶ ❊①✐st❡♥❝❡ ♦❢ t❤❡ t❡♠♣•❛t❡ ❢s❤❛♣❡ ■♥ ❛•• t❤✐s ♣❛rt✱ ✇❡ ✇✐•• ❝♦♥s✐❞❡r t❤❡ s✐♠♣•❡r s✐t✉❛t✐♦♥ ✇❤❡r❡ ❛•• ✈❛r✐❛❜•❡s v i ❛♥❞ ζ i ✐♥ ❡q✉❛t✐♦♥ ✭✹✳✶✽✮ ❛r❡ ❢r❡❡③❡❞ ❛♥❞ s❤♦✇ t❤❛t ❛ ♠✐♥✐♠✉♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• ❡①✐sts ✐♥ t❤❡ ✈❛r✐❛❜•❡s X ❛♥❞ f ✳ ❲❡ st❛rt ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ ❢❡✇ ❛❞❞✐t✐♦♥❛• ❞❡✜♥✐t✐♦♥s✳ ❆✳✶✳✶ Pr❡•✐♠✐♥❛r② ❞❡✜♥✐t✐♦♥s ❋✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ❛♥❞ ♠❡❛s✉r❡s ♦♥ E × Λ d (E) × R ❋♦r t❤❡ s❛❦❡ ♦❢ t❤❡ ♣r♦♦❢✱ ✇❡ s❤❛•• ❤❛✈❡ t♦ ❝♦♥s✐❞❡r✱ ✐♥ t❤❡ ❢♦••♦✇✐♥❣✱ ❛ s•✐❣❤t•② ❜✐❣❣❡r s♣❛❝❡ t❤❛♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ✇❤✐❝❤ ✐s t❤❡ s❡t ♦❢ ♠❡❛s✉r❡s ♦♥ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ F . = E × Λ d (E) × R✳ ◆❛♠❡•②✱ ✇❡ ✇✐•• ❝❛•• M(F ) t❤❡ s❡t ♦❢ ✜♥✐t❡ ♣♦s✐t✐✈❡ ❇♦r❡• ♠❡❛s✉r❡s ♦♥ F ✳ ❋♦r ❛♥② f ∈ L 2 (X) ✇❡ ❞❡♥♦t❡ ν X,f t❤❡ ✜♥✐t❡ ❇♦r❡• ♠❡❛s✉r❡ ♦♥ F s✉❝❤ t❤❛t ❢♦r ❛♥② ❜♦✉♥❞❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s g ∈ C b (F, R) ✿ ν X,f (g) . = X g(x, ξ(x), f (x))dH d (x) ✭❆✳✶✮ ✇❤❡r❡ ξ(x) ✐s t❤❡ ✉♥✐q✉❡ ✉♥✐t ♥♦r♠❡❞ ❡•❡♠❡♥t ✐♥ Λ d (E) ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♦r✐❡♥t❡❞ t❛♥❣❡♥t s♣❛❝❡ T x X ❞❡✜♥❡❞ ❛t ❛•♠♦st ❡✈❡r② •♦❝❛t✐♦♥ x ♦♥ X s✐♥❝❡ X ✐s r❡❝t✐✜❛❜•❡✳ ◆♦t❡ t❤❛t ν X,f (|ξ|) = H d (X) < ∞ . ✭❆✳✷✮ ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② ν ∈ M(F ) s✉❝❤ t❤❛t ν(|ξ|) < ∞✱ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ν ✇✐t❤ ❛ ❢✉♥❝✲ t✐♦♥❛• ❝✉rr❡♥t C ν ∈ W ′ s✉❝❤ t❤❛t ❢♦r ❛♥② ω ∈ W ✱ ✇❡ ❤❛✈❡ C ν (ω) = ω (x,f ) (ξ)dν(x, ξ, f ) . ✭❆✳✸✮ ❖♥❡ ❝❛♥ ❝❤❡❝❦ ✐♥❞❡❡❞ t❤❛t |C ν (ω)| ≤ |ω| ∞ ν(|ξ|) ≤ K ω W ν(|ξ|) ❢♦r ❛ ✜①❡❞ K > 0 s♦ t❤❛t C ν W ′ ≤ Kν(|ξ|) .

  ✭❆✳✹✮ ❚❤✉s✱C ν ✐s ✐♥❞❡❡❞ ❛ ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ ♦♥ W ✳ ✶✽✶ ■♥ ❛❞❞✐t✐♦♥✱ ✐❢ φ ✐s ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ ♦❢ E✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡✱ ❢♦r ν ∈ M(F )✱ ✐ts tr❛♥s♣♦rt φ * ν ❜② t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✱ ❢♦r ❛•• g ∈ C b (F, R) ✿ φ * ν(g) . = |d x φ.ξ|.g(φ(x), [d x φ.ξ], f )dν(x, ξ, f ) . ✭❆✳✺✮ ✇❤❡r❡✱ ❢♦r ξ = ξ 1 ∧ ... ∧ ξ d ✱ dφ x .ξ ✐s✱ ❛s ✉s✉❛•✱ t❤❡ tr❛♥s♣♦rt❡❞ d✲✈❡❝t♦r d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ d ) ❛♥❞ [d x φ.ξ] ✐ts ♥♦r♠❡❞ ✈❡rs✐♦♥✳ ❖♥❡ ❝❛♥ t❤❡♥ ❡❛s✐•② ❝❤❡❝❦ t❤❛t t❤✐s ❛❝t✐♦♥ ❝♦♠♠✉t❡s ✇✐t❤ t❤❡ ✉s✉❛• ❛❝t✐♦♥ ♦♥ ❢✉♥❝t✐♦♥❛• ❝✉rr❡♥ts ♦❢ ❡q✳✭✷✳✶✶✮ ✐✳❡✳ φ * C ν = C φ * ν ❛♥❞ t❤❛t φ * ν X,f = ν φ(X),f •φ -1 . ✭❆✳✻✮ X s✉♣♣♦rt❡❞ ♠❡❛s✉r❡s ❋♦r ❛ ✜①❡❞ ✜♥✐t❡✲✈♦•✉♠❡✱ ❜♦✉♥❞❡❞ ❛♥❞ ♦r✐❡♥t❡❞ r❡❝t✐✜❛❜•❡ s✉❜s❡t X✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♣❛rt✐❝✉•❛r s✉❜s❡t ♦❢ M(F ) t❤❛t ✇❡ ❞❡✜♥❡ ❛s t❤❡ s❡t M X (F ) ♦❢ ❛•• ❇♦r❡• ✜♥✐t❡ ♠❡❛s✉r❡s ν ♦♥ E × Λ d E × R s✉❝❤ t❤❛t ✿ h(x, ξ)dν(x, ξ, f ) = X h(x, ξ(x))dH d (x) . ✭❆✳✼✮ ❢♦r ❛•• ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❛♣♣•✐❝❛t✐♦♥s h ♦♥ E × Λ d (E)✳ ❚❤❡s❡ ❛r❡ ♠❡❛s✉r❡s t❤❛t ❤❛✈❡ ❛ ♣❛rt✐❝✉•❛r ♠❛r❣✐♥❛• ♦♥ E ×Λ d (E)✳ ◆♦t❡ t❤❛t ❛♥② ν X,f ❢♦r f ❛ L 2 ❢✉♥❝t✐♦♥ ♦♥ X ❜❡•♦♥❣s t♦ M X (F )✳ ■♥ ❛❞❞✐t✐♦♥✱ M X (F ) ✐s ❛ ❝•♦s❡❞ s✉❜s❡t ❢♦r t❤❡ ✉s✉❛• ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠❡❛s✉r❡s ❞❡✜♥❡❞ ❜② ν n ⇀ ν ∞ ✐❢ ❢♦r ❛♥② g ∈ C b (F, R) ν n (g) → ν ∞ (g) . ✭❆✳✽✮ P❡rt✉r❜❛t✐♦♥ ❲❡ ✐♥tr♦❞✉❝❡ ❛•s♦ ❛ ♣❡rt✉r❜❛t✐♦♥ ♣r♦❝❡ss ♦♥ ❛♥② ♠❡❛s✉r❡ ν ∈ M(F ) t❤❛t s❤❛•• ❜❡ ✉s❡❢✉• ❢♦r t❤❡ ❢♦••♦✇✐♥❣✳ ▲❡t a > 0 t♦ ❜❡ ✜①❡❞ •❛t❡r ❛♥❞ ❝♦♥s✐❞❡r ❢♦r ❛♥② t ∈ R t❤❡ ❢✉♥❝t✐♦♥ ρ t : R → R s✉❝❤ t❤❛t ρ t (z) = z + t(s❣♥(z)az)1 |z|>a ✭❆✳✾✮ ✇❤❡r❡ s❣♥(z) ✐s t❤❡ s✐❣♥ ♦❢ z✳ ❲❡ ❤❛✈❡ ρ 0 = ■❞ R ❛♥❞ ρ 1 ✐s ❛ s②♠♠❡tr✐❝ t❤r❡s❤♦•❞ ❛t •❡✈❡• a✳ ◆♦✇ ❢♦r ❛♥② t ∈ R✱ ✇❡ ❞❡♥♦t❡ ν t t❤❡ ♥❡✇ ♠❡❛s✉r❡ ❞❡✜♥❡❞ ❢♦r ❛♥② g ∈ C b (F, R) ❛s✿ ν t (g) = F g(x, ξ, ρ t (f ))dν(x, ξ, f ) . ✭❆✳✶✵✮ ❖❜✈✐♦✉s•② ν 0 = ν ❛♥❞ ν 1 ✐s s✉❝❤ t❤❛t ν 1 (|f | > a) = 0 s♦ t❤❛t t → ν t ✐s ❛♥ ❤♦♠♦t♦♣② ❢r♦♠ ν t♦ ❛ ♠❡❛s✉r❡ ✉♥❞❡r ✇❤✐❝❤ t❤❡ s✐❣♥❛• ✐s ❛✳❡✳ ❜♦✉♥❞❡❞ ❜② a✳ ❆✳✶✳✷ ❊①✐st❡♥❝❡✇✐t❤ X ✜①❡❞ ❚❤✐s s✉❜s❡❝t✐♦♥ ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ✐♥t❡r♠❡❞✐❛t❡ ♣r♦♣♦s✐t✐♦♥ ✿ ✶✽✷ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✶✳ ❆ss✉♠❡ t❤❛t W ✐s ❝♦♥t✐♥✉♦✉s•② ❡♠❜❡❞❞❡❞ ✐♥ C 2 0

  ❛❝❤✐❡✈❡s ✐ts ♠✐♥✐♠✉♠ ♦♥ L 2 (X) ❛♥❞ ❛♥② ♠✐♥✐♠✐③❡r f * ✐s s✉❝❤ t❤❛t f * ∈ L ∞ (X)✳ ▼♦r❡♦✈❡r✱ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t X ✐s ❛ C p s✉❜♠❛♥✐❢♦•❞ ❛♥❞ t❤❛t W ֒→ C m 0 (E ×R, Λ d E * ) ✇✐t❤ m ≥ max{p, 2}✱ t❤❡♥ f * ∈ C p-1 (X)✳ ❊①✐st❡♥❝❡ ♦❢ ❛ ♠❡❛s✉r❡ ♠✐♥✐♠✐③❡r ■♥ t❤❡ ✜rst ♣•❛❝❡✱ ✇❡ ✇✐•• s❤♦✇ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r ✐♥ t❤❡ s♣❛❝❡ M(F ) ♦❢ ♠❡❛s✉r❡s✱ ❛♥❞ t❤✉s ❝♦♥s✐❞❡r t❤❡ ❡①t❡♥❞❡❞ ❢✉♥❝t✐♦♥❛• J ❞❡✜♥❡❞ ❢♦r ν ∈ M X (F ) ❜② J(ν).

  ♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭❆✳✶✺✮ ✐s t❤❛t ♦♥❡ ❝❛♥ r❡str✐❝t t❤❡ s❡❛r❝❤ ♦❢ ❛ ♠✐♥✐♠✉♠ ❢♦r J ♦♥ ν ∈ M X (F ) s✉❝❤ t❤❛t ν(1 |f |>a ) = 0 ✭❆✳✶✻✮ ✇✐t❤ a = K γ W γ f N i=1 H d (X i ) + H d (X) ✳ ■♥ ♣❛rt✐❝✉•❛r✱ s✐♥❝❡ ν ∈ M X (F )✱ ✇❡ ✇✐•• ❤❛✈❡ x ∈ X, |ξ| ≤ 1 ❛♥❞ |f | ≤ a ν ❛✳❡✳ ✭❆✳✶✼✮ ❙✐♥❝❡ X ✐s ❜♦✉♥❞❡❞✱ ✇❡ ❝❛♥ r❡str✐❝t t❤❡ s❡❛r❝❤ ♦❢ ❛ ♠✐♥✐♠✉♠ ♦♥ ❛ ♠❡❛s✉r❡ s✉♣♣♦rt❡❞ ♦♥ ❛ ❝♦♠♣❛❝t s✉❜s❡t C ⊂ F s♦ t❤❛t ✇❡ ✐♥tr♦❞✉❝❡ ✿ M X,C (F ) . = { ν ∈ M X (F ) | (x, ξ, f ) ∈ C ν ❛✳❡✳ } . ✭❆✳✶✽✮ ❖♥❡ ❝❛♥ t❤❡♥ ❝❤❡❝❦ ❡❛s✐•② t❤❛t J ✐s •♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ s❡t M X,C (F ) ❢♦r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦•♦❣②✳ ■♥ ❛❞❞✐t✐♦♥✱ M X,C (F ) ✐s s❡q✉❡♥t✐❛••② ❝♦♠♣❛❝t✳ ■♥✲ ❞❡❡❞✱ ✐❢ ν n ✐s ❛ s❡q✉❡♥❝❡ ✐♥ M X,C (F ) t❤❡♥ ❛•• ν n ❛r❡ s✉♣♣♦rt❡❞ ❜② t❤❡ ❝♦♠♣❛❝t C ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r (ν n ) ✐s t✐❣❤t✳ ❆•s♦✱ ❛s ✇❡ ❛•r❡❛❞② ♥♦t✐❝❡❞✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t ❝t❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ n s✉❝❤ t❤❛t ν n (F ) ≤ ❝t❡H d (X) ❛♥❞ t❤✉s t❤❡ s❡q✉❡♥❝❡ ✐s ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ❢♦r t❤❡ t♦t❛• ✈❛r✐❛t✐♦♥ ♥♦r♠✳ ■t r❡s✉•ts✱ t❤❛♥❦s t♦ Pr♦❦❤♦r♦✈ t❤❡♦r❡♠✱ t❤❛t t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ (ν n ) ❝♦♥✈❡r❣✐♥❣ ❢♦r t❤❡ ✇❡❛❦ t♦♣♦•♦❣②✳ ❚❤❡s❡ ❝♦♠♣❛❝t✲ ♥❡ss ❛♥❞ •♦✇❡r s❡♠✐❝♦♥t✐♥✉✐t② ♣r♦♣❡rt✐❡s ❣✉❛r❛♥t❡❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r ν * ♦❢ J ✇✐t❤ ν * ∈ M X,C (F ) ❛♥❞ J(ν * ) ≤ inf f ∈L 2 (X) J X (f ) . ✭❆✳✶✾✮ ❊①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③✐♥❣ ❢✉♥❝t✐♦♥ f ❆t t❤❛t ♣♦✐♥t ✇❡ ❞♦ ♥♦t ❤❛✈❡ ②❡t ❛ ♠✐♥✐♠✐③❡r ♦❢ J X ✳ ❚❤❡ ♣r♦❜•❡♠ ✐s t❤❛t ✐❢ t❤❡ ♠❛r❣✐♥❛• ♦♥ E × Λ d (E) ♦❢ ν * ✐s t❤❡ tr❛♥s♣♦rt ♦❢ H d |X ✉♥❞❡r t❤❡ ❛♣♣•✐❝❛t✐♦♥ x → (x,ξ(x))✱ ✇❡ ❝❛♥♥♦t ❣✉❛r❛♥t❡❡ t❤❛t ν * ❞♦❡s ♥♦t ✇❡✐❣❤t ♠✉•t✐♣•❡ s✐❣♥❛• ✈❛•✉❡s ✐♥ t❤❡ ✜❜❡r ❛❜♦✈❡ ❛ •♦❝❛t✐♦♥ (x, ξ(x))✳ ❲❡ ✇✐•• s❤♦

  ξ) δf (x, ξ, f ) , ✶✽✹ s♦ t❤❛t ✇❤❡♥ ν = ν * ✇❡ ❣❡t • ❞❡r✐✈❛t✐✈❡ ♦❢ f → γ f f + γ W A(x, ξ, f ) ✇✐t❤ r❡s♣❡❝t t♦ f ❡q✉❛•s γ f + γ W ∂A ∂f (x, ξ, f )✳ ❯s✐♥❣ ❛s ♣r❡✈✐♦✉s•② t❤❡ ❝♦♥t✐♥✉♦✉s ❡♠❜❡❞❞✐♥❣ W ֒→ C 2 0 (E×R, Λ d E * )✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t ✇❡ ❤❛✈❡ ♦♥❝❡ ❛❣❛✐♥ ❛ ❝❡rt❛✐♥ ❝♦♥st❛♥t K s✉❝❤ t❤❛t ∂A ∂f (x, ξ, f )

  ξ, f )dν * = g(x, ξ, f (x, ξ))dν * ❛♥❞ ✐❢ ✇❡ ❞❡♥♦t❡ ❜② g(x, ξ) . = g(x, ξ, f (x, V )) ✇❤✐❝❤ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ♦♥ E × Λ d E✱ ✇❡ ❤❛✈❡ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ M X (F ) ✭❡q✳✭❆✳✼✮✮ ✿ ν * (g) = ν * (g) = X g(x, ξ(x), f (x, ξ(x)))dH d (x) . ✭❆✳✷✸✮ ❚❤❡r❡❢♦r❡✱ s❡tt✐♥❣ f * (x) = f (x, ξ(x)) ❢♦r x ∈ X✱ ✇❡ s❡❡ t❤❛t |f * | ≤ a s♦ t❤❛t f * ∈ L ∞ (X) ❛♥❞ ✇✐t❤ ❡q✳✭❆✳✷✸✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ν * = ν (X,f * ) ✇❤✐❝❤ s❤♦✇s t❤❛t t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ✐s t❤❡ ❢❝✉rr❡♥t ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❢s❤❛♣❡ (X, f * )✳ ▼♦r❡♦✈❡r✱ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t W ֒→ C m 0 (E × R, Λ d E * ) ✇✐t❤ m ≥ max{p, 2} t❤❡♥ A ❛♥❞ f ❛r❡ C p-1 ❢✉♥❝t✐♦♥s ❛♥❞ ✐❢ X ✐s ❛ C p s✉❜♠❛♥✐❢♦•❞ t❤❡♥ x → ξ(x) ✐s ❛ C p-1 ❢✉♥❝t✐♦♥ ♦♥ X s♦ t❤❛t f * ✐s ❛•s♦ C p-1 ✱ ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ❆✳✶✳✶✳ ❆✳✶✳✸ ❊①✐st❡♥❝❡ ✇✐t❤ ♥♦♥✲✜①❡❞ X ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ t❡♠♣•❛t❡ ✇❤❡♥ X ✐s ♥♦ ♠♦r❡ ✜①❡❞ ❛♥❞ ✐s ❡st✐♠❛t❡❞ ❛s ✇❡••✳ ■♥ t❤✐s ❝❛s❡✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞ ✐♥ ✹✳✷✳✹✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ✐♥tr♦❞✉❝❡ ❛ |❑❍❙ ❍✐•❜❡rt s♣❛❝❡

  ✇❡r s❡♠✐✲ ❝♦♥t✐♥✉♦✉s ❢♦r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦•♦❣②✳ ❚❤❡r❡❢♦r❡✱ ❢♦r ❛•• ✜①❡❞ f 0 ∈ L 2 (X 0 )✱ v 0 → J 3 (v 0 , f 0 ) ✐s ✇❡❛❦•② •♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ♦♥ L 2 V 0 ✳ ■t r❡s✉•ts t❤❛t ✇❡ ♦❜t❛✐♥ ❡①✲ ✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③✐♥❣ ✈❡❝t♦r ✜❡•❞ v 0 ❛♥❞✱ r❡❛s♦♥✐♥❣ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥✱ ♦♥❡ ❞❡❞✉❝❡s ❡❛s✐•② t❤❡ ❝•❛✐♠ ♦❢ ♣r♦♣♦s✐t✐♦♥ ❆✳✶✳✷✳ ✶✽✻ ❆✳✷ ❊①✐st❡♥❝❡ ♦❢ ❢✉•• ❢s❤❛♣❡ ❛t•❛s❡s ❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ ❣❡♥❡r❛•✐③❡ t❤❡ ♣r❡✈✐♦✉s r❡s✉•ts t♦ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❝♦♠♣•❡t❡ ❛t•❛s❡s ♦❢ ❢s❤❛♣❡s✬ ❞❛t❛s❡ts t❤❛t ✐s t❤❡ ❝•❛✐♠ ♦❢ t❤❡♦r❡♠ ✹✳✷✳✶✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ t❡♠♣•❛t❡ (X, f )✱ ♦♥❡ ✇❛♥ts t♦ ❡st✐♠❛t❡ s✐♠✉•t❛♥❡♦✉s•② tr❛♥s❢♦r♠❛t✐♦♥s ❢r♦♠ t❤❡ t❡♠♣•❛t❡ t♦ ❡❛❝❤ s✉❜•❡❝t✳ ❚❤❡s❡ ❝♦♥s✐st ✐♥ ❞❡❢♦r♠❛t✐♦♥s (φ v i ) i=1,..,N ♦❜t❛✐♥❡❞ ❛s ✢♦✇s ♦❢ t✐♠❡ ✈❛r②✐♥❣ ✈❡❝t♦r ✜❡•❞s (v i ) ❛♥❞ r❡s✐❞✉❛• s✐❣♥❛•s (ζ i ) t❤❛t ❛r❡ L 2 ❢✉♥❝t✐♦♥s ♦♥ X✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❞♦♥❡ ❜② •❡❛♥✐♥❣ ❡ss❡♥t✐❛••② ♦♥ t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts ❛s ❞❡t❛✐•❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s✉❜s❡❝t✐♦♥s✳ ❲❡ s❤❛•• ❣♦ r❛♣✐❞•② t❤r♦✉❣❤ ❡❛❝❤ ♦❢ t❤❡♠ ❜② r❡❢❡rr✐♥❣ ❛•s♦ t♦ ✇❤❛t ✇❛s ♣r♦✈❡❞ ♣r❡✈✐♦✉s•②✳ ❋✐rst ♦❢ ❛••✱ ❡①❛❝t•② ❛s ✐♥ ❆✳✶✳✸✱ ❡①✐st❡♥❝❡ ♦❢ ❛ t❡♠♣•❛t❡ s❤❛♣❡ X ✐s ❣✉❛r❛♥t❡❡❞ ✇✐t❤ t❤❡ s❛♠❡ ❝♦♠♣❛❝✐t② ❛♥❞ •♦✇❡r s❡♠✐❝♦♥t✐♥✉✐t② ❛r❣✉♠❡♥ts✳ ❚❤✉s ✇❡ ♠❛② ❛ss✉♠❡ t❤❛t X ✐s ✜①❡❞ ❛♥❞ ✇❡ ♦♥•② ❤❛✈❡ t♦ s❤♦✇ ❡①✐st❡♥❝❡ ♦❢ ♠✐♥✐♠✐③❡rs t♦ t❤❡ s✐♠♣•✐✜❡❞ ❢✉♥❝t✐♦♥❛• ✿

  ν(h) = h(x, V )dν(x, ξ, f, (ζ i )) = X h(x, ξ(x))dH d (x) . ❋♦r ❛ ♠❡❛s✉r❡ ν ♦♥ F ❛♥❞ ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ φ✱ ✇❡ ❞❡♥♦t❡ ❜② φ * ν t❤❡ tr❛♥s♣♦rt ♦❢ ν ❜② φ ❞❡✜♥❡❞✱❜② ❛♥❛•♦❣② ✇✐t❤ ❡q✳✭❆✳✺✮ ❜② ✿ (φ * ν)(g) = |d x φ.ξ|g(φ(x), [d x φ.ξ], f, (ζ i ))dν(x, ξ, f, (ζ i )) .

  ❚❤❡r❡❢♦r❡✱ ♦♥❡ ♠❛② r❡str✐❝t t❤❡ s❡❛r❝❤ ♦❢ ❛ ♠✐♥✐♠✉♠ ♦♥ ❛ s❡t ♦❢ ♠❡❛s✉r❡s ν t❤❛t ❛r❡ s✉♣♣♦rt❡❞ ♦♥ ❛ ❝♦♠♣❛❝t s✉❜s❡t C ♦❢ E × Λ d E × R × R N ✱ ✇❤✐❝❤ s♣❛❝❡ ✇❡ s❤❛•• ❞❡♥♦t❡ ❛❣❛✐♥ M X,C (F )✳ ❚❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ✐s ♥♦✇ ✈❡r② ❝•♦s❡ t♦ t❤❡ ♦♥❡ ♦❢ ❆✳✶✳✷✳ ❉✉❡ t♦ •♦✇❡r s❡♠✐✲❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛• ❛♥❞ t❤❡ ❝♦♠♣❛❝✐t② ♦❢ M X,C (F ) ❛♥❞ B ❢♦r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦•♦❣✐❡s ✭r❡s♣❡❝t✐✈❡•② ♦♥ t❤❡ s♣❛❝❡ ♦❢ ♠❡❛s✉r❡s ❛♥❞ ♦♥ L 2 V ✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r (ν * , (v i ) * ) ❢♦r t❤❡ ❢✉♥❝t✐♦♥❛• J✳ ❚❤❡ •❛st st❡♣ ✐s t♦ ♣r♦✈❡ t❤❛t ν * ✱ ✇❤✐❝❤ ❜❡•♦♥❣s ❛ ♣r✐♦r✐ t♦ t❤❡ ♠❡❛s✉r❡ s♣❛❝❡ M X,C (F )✱ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✉♥❞❡r t❤❡ ❢♦r♠ ν * = ν X,f * ,(ζ i * ) ✱ ✐✳❡ t❤❛t t❤❡r❡ ❡①✐sts ❢✉♥❝✲ t✐♦♥s f * ❛♥❞ ζ i * ♦♥ X s✉❝❤ t❤❛t✱ ❢♦r ❛•• ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ g ♦♥ F ✿ ν * (g) = X g(x, ξ(x), f * (x), (ζ i * (x)))dH d (x) ✭❆✳✸✶✮ ❍❡♥❝❡✱ ✇❡ ❝♦♥s✐❞❡r ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ s✐❣♥❛•s (δf, (δζ i )) ❛•• ❜❡•♦♥❣✐♥❣ t♦ t❤❡ s♣❛❝❡ C b (F, R) ❛♥❞ t❤❡ ♣❛t❤ t → ν t ❞❡✜♥❡❞ ❜② ✿ ν t (g) = g(x, ξ, f + tδf (x, ξ, f, (ζ i )), (ζ i + tδζ i (x, ξ, f, (ζ i ))))dν * (x, ξ, f, (ζ i )) .

◆

  ♦✇✱ ✐❢ J t . = J(ν t , v i * )✱ ❡①♣r❡ss✐♥❣ t❤❛t J ′ t | t=0 = 0 ❢♦r ❛•• δf ❛♥❞ (δζ i ) ❣✐✈❡s t❤❡ ❢♦••♦✇✐♥❣ s❡t ♦❢ ❡q✉❛t✐♦♥s ✿ (γ f f, (γ ζ ζ i )) = -A(x, V, f, (ζ i )) ν * ✲❛✳❡ ✭❆✳✸✷✮ ✇✐t❤ A(x, ξ, f, (ζ i )) v i * (x), d x φ v i * .ξ, f + ζ i ), ( ∂ω i ∂f (φ v i * (x), d x φ v i