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Multidimentional Scaling Trung Duc Tran Institute of Mathematics University of Silesia in KatowiceMultidimentional Scaling Trung Duc Tran Institute of Mathematics University of Silesia in KatowiceMultidimentional Scaling Trung Duc Tran Institute of Mathematics University of Silesia in KatowiceMultidimentional Scaling Trung Duc Tran Institute of Mathematics University of Silesia in KatowiceMultidimentional Scaling Trung Duc Tran Institute of Mathematics University of Silesia in Katowice
▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ❚r✉♥❣ ❉✉❝ ❚r❛♥ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❙✐❧❡s✐❛ ✐♥ ❑❛t♦✇✐❝❡ ❈♦♥t❡♥ts ✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ✜rst s♦❧✉t✐♦♥ ❜② ✉s✐♥❣ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ✷ ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ◆✉♠❡r✐❝❛❧ ❚❡❝❤♥✐q✉❡ ❢♦r ✷❉ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ✸❉ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷ ❈♦♠♣❛r❡ ❛❜♦✈❡ r❡s✉❧ts t♦ ❝❧❛ss✐❝❛❧ ▼❉❙ ✼ ✸ P❛❝❦❛❣❡s ❢♦r ♥♦♥✲♠❡tr✐❝ ▼❉❙✭◆▼❉❙✮ ✐♥ ❘ ✾ ✸✳✶ ❋✉♥❝t✐♦♥ ♠♦♥♦▼❉❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✷ ❋✉♥❝t✐♦♥ ♠❡t❛▼❉❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✹ ◆❡❣❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ◆▼❉❙ ✶✺ ✹✳✶ ❯s✐♥❣ ❈❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠✭❝♠❞s❝❛❧❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✹✳✷ ❯s✐♥❣ ◆♦♥♠❡tr✐❝ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣✭♠❡t❛▼❉❙✮ ✶✺ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ✭▼❉❙✮ ✐s ❛ ♠❡❛♥s ♦❢ ✈✐s✉❛❧✐③✐♥❣ t❤❡ ❧❡✈❡❧ ♦❢ s✐♠✐❧❛r✐t② ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❝❛s❡s ♦❢ ❛ ❞❛t❛s❡t ✳ ▼❉❙ s❝❛❧❡ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛ t♦ ♠✲❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛ ✭♠❁♥✮ ✳ ■♥ t❤✐s ♣❛♣❡r ✱ ✇❡ ✇✐❧❧ ❞♦ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❞❛t❛ ■r✐s ✐♥ ❘ ✇❤✐❝❤ ❤❛s ♥❂✹ ❛♥❞ ♠❂✷✱✸ ✳ ❇❡❝❛✉s❡ ✐♥ ✷❉ ❛♥❞ ✸❉ s♣❛❝❡✱ ✇❡ ❝❛♥ ✈✐s✉❛❧✐③❡ t❤❡ ❞✐ss✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ ♦❜❥❡❝ts ✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ✜rst s♦❧✉t✐♦♥ ❜② ✉s✐♥❣ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡r❡ ❛r❡ ♥ ♦❜❥❡❝ts 1, , , n✱ ❜❡t✇❡❡♥ ❡✈❡r② ✷ ♦❜❥❡❝ts✳ ❋♦r ❛ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ♣♦✐♥ts s♣❛❝❡✱ ✇✐t❤ ❡✈❡r② ✷ ♣♦✐♥ts ❞✐st❛♥❝❡s dij ❛r❡ ❞✐ss✐♠✐❧❛r✐t② ✐♥ t✲❞✐♠❡♥s✐♦♥❛❧ ✱ ✇❡ ❞❡✜♥❡❞ t❤❡ str❡ss ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❜② (dij − dˆij )2 d2ij S1 = T1 S= dˆij x1 , , , xn ❛♥❞ ✈❛❧✉❡s ✭✶✳✶✮ ❋✐rst✱ ✇❡ ❛ss✐❣♥ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ t♦ ❛ ♠❛tr✐①✿ ✶ ✷ ✸ ❛ ❂ ❞❛t❛ ✳ ♠❛tr✐① ✭ ✐r✐s ✮ ❜ ❂ s✉❜s❡t ✭❛ ✱ s❡❧❡❝t ❂ ✲ ❝ ✭ ❙♣❡❝✐❡s ✮ ✮ ♥ ❂ ♥r♦✇ ✭ ❜ ✮ ❲❡ ❞❡✜♥❡ t❤❡ str❡ss ❢✉♥❝t✐♦♥ ✐♥ ❘ ✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ str❡ss ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❞ ❂ ❞✐st ✭ ❜ ✮ ❡ ❂ ❞✐st ✭ ❝ ✮ ❢ ❂✭ ❡ ✲ ❞ ✮ ❫✷❀ s ❂ sqrt ✭ s✉♠ ✭ ❢ ✮ ✴ s✉♠ ✭ ❡ ❫✷✮ ✮ r❡t✉r♥ ✭ s ✮ ⑥ ❚❤❡ str❡ss ✐s ✐♥t❡♥❞❡❞ t♦ ❜❡ ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ✇❡❧❧ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♠❛t❝❤❡s t❤❡ ❞❛t❛ ❇② ❞❡✜♥✐t✐♦♥✱ t❤❡ ❜❡st✲✜tt✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ t✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✱ ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ t✱ ✐s t❤❛t ❝♦♥✜❣✉r❛t✐♦♥ ✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ str❡ss ❖❢ ♠❛❥♦r ✐♥t❡r❡st ✐s t❤❡ ♦r❞✐♥❛r② ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ ❞✐st❛♥❝❡s ❛r❡ ❊✉❝❧✐❞❡❛♥✳ ■❢ t❤❡ ♣♦✐♥t xi ✱ ❤❛s ✭ ♦rt❤♦❣♦♥❛❧✮ ❝♦♦r❞✐♥❛t❡s P②t❤❛❣♦r❡❛♥✮ ❞✐st❛♥❝❡ ❢r♦♠ xi ✱ t♦ xj xi1 , , xit t❤❡♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✭♦r ✐s ❣✐✈❡♥ ❜② l=t (xil − xjl )2 dij = l=1 ❲❡ ❞❡✜♥❡ ❛ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ✐♥ ❘ ✿ ✶ ✷ ✸ ✹ ❞✐st ❁✲ ❢✉♥❝t✐♦♥ ✭ ❜ ✮ ④ ❞ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ♥ ✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✷ ✭✶✳✷✮ ✺ ✻ ✼ ✽ ✾ ❞ ❬✐ ✱ ❥ ❪❂ ❞ ❬❥ ✱ ✐ ❪❂ sqrt ✭ s✉♠ ✭✭ ❜ ❬✐ ✱❪ ✲ ❜ ❬❥ ✱❪✮ ❫✷✮ ✮ ⑥⑥ r❡t✉r♥ ✭ ❞ ✮ ⑥ ❞ ❂ ❞✐st ✭ ❜ ✮ ❡ ❂ ❞✐st ✭ ❝ ✮ ✶✳✷ ◆✉♠❡r✐❝❛❧ ❚❡❝❤♥✐q✉❡ ❢♦r ✷❉ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ■♥ ♣r✐♥❝✐♣❧❡ t❤❡ ✐t❡r❛t✐✈❡ t❡❝❤♥✐q✉❡ ✇❡ ✉s❡ t♦ ♠✐♥✐♠✐③❡ t❤❡ str❡ss ✐s ♥♦t ❞✐✣❝✉❧t✳ ■t r❡q✉✐r❡s st❛rt✐♥❣ ❢r♦♠ ❛♥ ❛r❜✐tr❛r② ❝♦♥✜❣✉r❛t✐♦♥✱ ❝♦♠♣✉t✐♥❣ t❤❡ ✭♥❡❣❛t✐✈❡✮ ❣r❛❞✐❡♥t✱ ♠♦✈✐♥❣ ❛❧♦♥❣ ✐t ❛ s✉✐t❛❜❧❡ ❞✐st❛♥❝❡✱ ❛♥❞ t❤❡♥ r❡♣❡❛t✐♥❣ t❤❡ ❧❛st t✇♦ st❡♣s ❛ s✉✣❝✐❡♥t ♥✉♠❜❡r ♦❢ t✐♠❡s✳ ■❢ ❛ ❢❛✐r❧② ❣♦♦❞ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❝♦♥✈❡♥✐❡♥t❧② ❛✈❛✐❧❛❜❧❡ ❢♦r ✉s❡ ❛s t❤❡ st❛rt✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥✱ ✐t ♠❛② s❛✈❡ q✉✐t❡ ❛ ❢❡✇ ✐t❡r❛t✐♦♥s✳ ■❢ ♥♦t✱ ❛♥ ❛r❜✐tr❛r② st❛rt✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ ✐s q✉✐t❡ s❛t✐s❢❛❝t♦r②✳ ❖♥❧② t✇♦ ❝♦♥❞✐t✐♦♥s s❤♦✉❧❞ ❜❡ ♠❡t✿ ♥♦ t✇♦ ♣♦✐♥ts ✐♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡✱ ❛♥❞ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ s❤♦✉❧❞ ♥♦t ❧✐❡ ✐♥ ❛ ❧♦✇❡r✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ t❤❛♥ ❤❛s ❜❡❡♥ ❝❤♦s❡♥ ❢♦r t❤❡ ❛♥❛❧②s✐s✳ ■❢ ♥♦ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❝♦♥✈❡♥✐❡♥t❧② ❛✈❛✐❧❛❜❧❡✱ ❛♥ ❛r❜✐tr❛r② ❝♦♥✜❣✉r❛t✐♦♥ ♠✉st ❜❡ ❣❡♥❡r❛t❡❞✳ ❖♥❡ s❛t✐s❢❛❝t♦r② ✇❛② t♦ ❞♦ t❤✐s ✐s t♦ ✉s❡ t❤❡ ✜rst ♥ ♣♦✐♥ts ❢r♦♠ t❤❡ ❧✐st (1, 0, 0, , 0, 0), (0, 1, 0, , 0, 0), (0, 0, 0, , 0, 1), (2, 0, 0, , 0, 0), (0, 2, 0, , 0, 0), etc ❙♦ ✇❡ ❤❛✈❡ t❤❡ ❝♦❞❡ ✐♥ ❘ ✿ ✶ ✷ ✸ ✹ ✺ ❝ ❂ ♠❛tr✐① ✭✵ ✱ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✷✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✪✪ ✷❂❂✶✮ ④ ❝ ❬✐ ✱✶❪❂ ❝❡✐❧✐♥❣ ✭ ✐ ✴ ✷✮ ⑥ ✐❢ ✭ ✐ ✪✪ ✷❂❂✵✮ ④ ❝ ❬✐ ✱✷❪❂ ✐ ✴ ✷⑥ ⑥ x1 , , x n xi ❜❡ xi1 , , , xit ❲❡ s❤❛❧❧ ❝❛❧❧ ❛❧❧ t❤❡ ♥✉♠❜❡rs ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛rr✐✈❡❞ ❛t t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ①✱ ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ♥ ♣♦✐♥ts ✐♥ t ❞✐♠❡♥s✐♦♥s✳ ▲❡t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ xis ✱ ✇✐t❤ i = 1, , n ❛♥❞ s = 1, t❤❡ ✭♥❡❣❛t✐✈❡✮ ❣r❛❞✐❡♥t ♦❢ str❡ss ❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ① ✳ ❙✉♣♣♦s❡ x g ✱ ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s ❛r❡ gis ✳ ❚❤❡♥ x ❛♥❞ ♠♦✈✐♥❣ ❛❧♦♥❣ g ❛ ❞✐st❛♥❝❡ ♥❡✇ ❝♦♥✜❣✉r❛t✐♦♥ x ✐s ❣✐✈❡♥ ❜② ✐s ❣✐✈❡♥ ❜② ✇❡ ❢♦r♠ t❤❡ ♥❡①t ❝♦♥✜❣✉r❛t✐♦♥ ❜② st❛rt✐♥❣ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❛❧❧ t❤❡ st❡♣✲s✐③❡ α✳ ■♥ s②♠❜♦❧s✱ t❤❡ xis = xis + ❢♦r ❛❧❧ i ❛♥❞ s✳ ❍❡r❡ mag(g) α ∗ gis mag(g) ♠❡❛♥s t❤❡ r❡❧❛t✐✈❡ ♠❛❣♥✐t✉❞❡ ♦❢ mag(g) = ✸ i,s gis i,s x2is g ❛♥❞ ✐s ❣✐✈❡♥ ❜②✿ ❚❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ α ✇✐t❤ ❛♥ ❛r❜✐tr❛r② st❛rt✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ s❤♦✉❧❞ ❜❡ ❛❜♦✉t ✵✳✷✳ ❋♦r ❛ ❝♦♥✜❣✉r❛t✐♦♥ t❤❛t ❛❧r❡❛❞② ❤❛s ❧♦✇ str❡ss✱ ❛ s♠❛❧❧❡r ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ✉s❡❞✳ ✭❆ ♣♦♦r❧② ❝❤♦s❡♥ ✈❛❧✉❡ r❡s✉❧ts ♦♥❧② ✐♥ ❡①tr❛ ✐t❡r❛t✐♦♥s✳✮ ❲❡ ❤❛✈❡ ✿ T1 = d2ij S1 = (dij − dˆij )2 ❚♦ ❝❛❧❝✉❧❛t❡ t❤❡ ✭♥❡❣❛t✐✈❡✮ ❣r❛❞✐❡♥t ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s✳ ❞✐st❛♥❝❡ r=2 (σ ki − σ kj ) gkl = S i,j ✇❤❡r❡ σ ki ❛♥❞ ❋♦r ❊✉❝❧✐❞❡❛♥ ✿ σ kj dij dij − dˆij − S1 T1 (xil − xjl ) dij ✭✶✳✸✮ ❞❡♥♦t❡ t❤❡ ❑r♦♥❡❝❦❡r s②♠❜♦❧s ❋r♦♠ ✭✶✳✸✮✿ ✰ ✐❢ ✰ ✐❢ i=j i=j σ ki − σ kj = ki kj ❝❛♥ t❛❦❡ k = i ✱t❤❡♥ σ − σ = σ ii − σ ij = t❤❡♥ ✱✇❡ gil = S i,j dij − dˆij dij − S1 T1 ✱t❤✉s ✿ (xil − xjl ) dij ✭✶✳✹✮ ❲❡ ❞❡✜♥❡ t❤❡ ❣r❛❞✐❡♥t ❢✉♥❝t✐♦♥ ✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ❣r❛❞✐❡♥t ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❞ ❂ ❞✐st ✭ ❜ ✮ ❡ ❂ ❞✐st ✭ ❝ ✮ ❢ ❂✭ ❡ ✲ ❞ ✮ s ❂ sqrt ✭ s✉♠ ✭ ❢ ❫✷✮ ✴ s✉♠ ✭ ❡ ❫✷✮ ✮ s✶ ❂ s✉♠ ✭ ❢ ❫✷✮ ❀ t✶ ❂ s✉♠ ✭ ❡ ✮ ❣ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✷✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❣ ❬✐ ✱✶❪❂✵ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✦ ❂ ❥ ✮ ④ ❣ ❬✐ ✱✶❪❂ ❣ ❬✐ ✱✶❪✰✭✭ ❡ ❬✐ ✱ ❥ ❪ ✲ ❞ ❬✐ ✱ ❥ ❪✮ ✴ s✶ ✲ ❡ ❬✐ ✱ ❥ ❪ ✴ t✶ ✮ ✯ ✭ ❝ ❬✐ ✱✶❪ ✲ ❝ ❬❥ ✱✶❪✮ ✯ s ✴ ❡ ❬✐ ✱ ❥ ❪⑥ ⑥⑥ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❣ ❬✐ ✱✷❪❂✵ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✦ ❂ ❥ ✮ ④ ❣ ❬✐ ✱✷❪❂ ❣ ❬✐ ✱✷❪✰✭✭ ❡ ❬✐ ✱ ❥ ❪ ✲ ❞ ❬✐ ✱ ❥ ❪✮ ✴ s✶ ✲ ❡ ❬✐ ✱ ❥ ❪ ✴ t✶ ✮ ✯ ✭ ❝ ❬✐ ✱✷❪ ✲ ❝ ❬❥ ✱✷❪✮ ✯ s ✴ ❡ ❬✐ ✱ ❥ ❪⑥ ⑥⑥ r❡t✉r♥ ✭ ❣ ✮ ⑥ ❆♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r ♥❡①t ❝♦♥✜❣✉r❛t✐♦♥ ✿ ✶ ✷ ✸ ✹ ♥❡①t❝ ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❝✶ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✷✮ ❣ ❂ ❣r❛❞✐❡♥t ✭ ❝ ✮ ♠❛❣ ❂ sqrt ✭ s✉♠ ✭ ❣ ❫✷✮ ✴ s✉♠ ✭ ❝ ❫✷✮ ✮ ✹ ✺ ✻ ✼ ✽ ✾ ⑥ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❢♦r ✭ ❥ ✐♥ ✶✿✷✮ ④ ❝✶ ❬✐ ✱ ❥ ❪❂ ❝ ❬✐ ✱ ❥ ❪✰✵✳✷✷✺✼✶✵✻ ✴ ♠❛❣ ✯ ❣ ❬✐ ✱ ❥ ❪⑥⑥ r❡t✉r♥ ✭ ❝✶ ✮ ❍❡r❡ ✇❡ ❤❛✈❡ α = 0.2257106 ❜② ❛♣♣❧② t❤❡ ❢♦r♠✉❧❛s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r ❇② ❞♦✐♥❣ t❤❡ ❧♦♦♣ ♠❛♥② t✐♠❡ ✉♥t✐❧ t❤❡ str❡ss ✶ ✷ ✸ ✹ ✺ ❤ ❂ str❡ss ✭ ❝ ✮ ✇❤✐❧❡ ✭❤ ❃✵✳✵✻✮ ❝ ❂ ♥❡①t❝ ✭ ❝ ✮ ❤ ❂ str❡ss ✭ ❝ ✮ ⑥ < 6% ④ ❆❝❝♦r❞✐♥❣ t♦ ❑r✉s❦❛❧ ✱✇❡ ❝❛♥ s❛t✐s❢② ✇✐t❤ t❤✐s str❡ss ❋✐♥❛❧❧②✱ ✇❡ ♣❧♦t t❤❡ r❡s✉❧t ✶ ✷ ✸ ✹ ♣❧♦t ✭ ❝ ❬ ✱✷❪ ✱ ❝ ❬ ✱✶❪ ✱ ❝♦❧ ❂ ✧ r❡❞ ✧ ✮ ❧✐♥❡s ✭ ❝ ❬✺✶✿✶✵✵ ✱✷❪ ✱ ❝ ❬✺✶✿✶✵✵ ✱✶❪ ✱ ❝♦❧ ❂ ✧ ❜❧✉❡ ✧ ✮ ❧✐♥❡s ✭ ❝ ❬✶✿✺✵ ✱✷❪ ✱ ❝ ❬✶✿✺✵ ✱✶❪ ✱ ❝♦❧ ❂ ✧ ❜❧❛❝❦ ✧ ✮ ❧✐♥❡s ✭ ❝ ❬✶✵✶✿✶✺✵ ✱✷❪ ✱ ❝ ❬✶✵✶✿✶✺✵ ✱✶❪ ✱ ❝♦❧ ❂ ✧ ♣✉r♣❧❡ ✧ ✮ ❋✐❣✉r❡ ✶✿ ✷❉✲▼❉❙ ♣❧♦t ✉s✐♥❣ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ❲❡ ❝❛♥ ❡❛s✐❧② s❡❡ t❤❛t ✸ t②♣❡s ♦❢ ✢♦✇❡r ❛r❡ s❡♣❛r❛t❡❞ ✐♥ t❤❡ ❣r❛♣❤ ✶✳✸ ✶ ✷ ✸ ✹ ✸❉ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ❛ ❂ ❞❛t❛ ✳ ♠❛tr✐① ✭ ✐r✐s ✮ ❜ ❂ s✉❜s❡t ✭❛ ✱ s❡❧❡❝t ❂ ✲ ❝ ✭ ❙♣❡❝✐❡s ✮ ✮ ♥ ❂ ♥r♦✇ ✭ ❜ ✮ ❝ ❂ ♠❛tr✐① ✭✵ ✱ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✸✮ ✺ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✪✪ ✸❂❂✶✮ ④ ❝ ❬✐ ✱✶❪❂ ❝❡✐❧✐♥❣ ✭ ✐ ✴ ✸✮ ⑥ ✐❢ ✭ ✐ ✪✪ ✸❂❂✷✮ ④ ❝ ❬✐ ✱✷❪❂ ❝❡✐❧✐♥❣ ✭ ✐ ✴ ✸✮ ⑥ ✐❢ ✭ ✐ ✪✪ ✸❂❂✵✮ ④ ❝ ❬✐ ✱✸❪❂ ❝❡✐❧✐♥❣ ✭ ✐ ✴ ✸✮ ⑥ ⑥ ❞✐st ❁✲ ❢✉♥❝t✐♦♥ ✭ ❜ ✮ ④ ❞ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ♥ ✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ❞ ❬✐ ✱ ❥ ❪❂ ❞ ❬❥ ✱ ✐ ❪❂ sqrt ✭ s✉♠ ✭✭ ❜ ❬✐ ✱❪ ✲ ❜ ❬❥ ✱❪✮ ❫✷✮ ✮ ⑥⑥ r❡t✉r♥ ✭ ❞ ✮ ⑥ ❣r❛❞✐❡♥t ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❞ ❂ ❞✐st ✭ ❜ ✮ ❡ ❂ ❞✐st ✭ ❝ ✮ ❢ ❂✭ ❡ ✲ ❞ ✮ ❫✷❀ s ❂ sqrt ✭ s✉♠ ✭ ❢ ✮ ✴ s✉♠ ✭ ❡ ❫✷✮ ✮ s✶ ❂ s✉♠ ✭ ❢ ✮ ❀ t✶ ❂ s✉♠ ✭ ❡ ✮ ❣ ❂ ♠❛tr✐① ✭✵ ✱ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✸✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❣ ❬✐ ✱✶❪❂✵ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✦ ❂ ❥ ✮ ④ ❣ ❬✐ ✱✶❪❂ ❣ ❬✐ ✱✶❪✰✭✭ ❡ ❬✐ ✱ ❥ ❪ ✲ ❞ ❬✐ ✱ ❥ ❪✮ ✴ s✶ ✲ ❡ ❬✐ ✱ ❥ ❪ ✴ t✶ ✮ ✯ ✭ ❝ ❬✐ ✱✶❪ ✲ ❝ ❬❥ ✱✶❪✮ ✴ ❡ ❬✐ ✱ ❥ ❪ ✯ s ⑥ ⑥⑥ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❣ ❬✐ ✱✷❪❂✵ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✦ ❂ ❥ ✮ ④ ❣ ❬✐ ✱✷❪❂ ❣ ❬✐ ✱✷❪✰✭✭ ❡ ❬✐ ✱ ❥ ❪ ✲ ❞ ❬✐ ✱ ❥ ❪✮ ✴ s✶ ✲ ❡ ❬✐ ✱ ❥ ❪ ✴ t✶ ✮ ✯ ✭ ❝ ❬✐ ✱✷❪ ✲ ❝ ❬❥ ✱✷❪✮ ✴ ❡ ❬✐ ✱ ❥ ❪ ✯ s ⑥ ⑥⑥ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❣ ❬✐ ✱✸❪❂✵ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ✐❢ ✭ ✐ ✦ ❂ ❥ ✮ ④ ❣ ❬✐ ✱✸❪❂ ❣ ❬✐ ✱✸❪✰✭✭ ❡ ❬✐ ✱ ❥ ❪ ✲ ❞ ❬✐ ✱ ❥ ❪✮ ✴ s✶ ✲ ❡ ❬✐ ✱ ❥ ❪ ✴ t✶ ✮ ✯ ✭ ❝ ❬✐ ✱✸❪ ✲ ❝ ❬❥ ✱✸❪✮ ✴ ❡ ❬✐ ✱ ❥ ❪ ✯ s ⑥ ⑥⑥ r❡t✉r♥ ✭ ❣ ✮ ⑥ ✹✸ ✹✹ ✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ✺✶ str❡ss ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❞ ❂ ❞✐st ✭ ❜ ✮ ❡ ❂ ❞✐st ✭ ❝ ✮ ❢ ❂✭ ❡ ✲ ❞ ✮ ❀ s ❂ sqrt ✭ s✉♠ ✭ ❢ ❫✷✮ ✴ s✉♠ ✭ ❡ ❫✷✮ ✮ r❡t✉r♥ ✭ s ✮ ⑥ ✺✷ ✺✸ ✺✹ ✺✺ ✺✻ ✺✼ ♥❡①t❝ ❁✲ ❢✉♥❝t✐♦♥ ✭ ❝ ✮ ④ ❝✶ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ✷✮ ❣ ❂ ❣r❛❞✐❡♥t ✭ ❝ ✮ ♠❛❣ ❂ sqrt ✭ s✉♠ ✭ ❣ ❫✷✮ ✴ s✉♠ ✭ ❝ ❫✷✮ ✮ ✻ ✺✽ ✺✾ ✻✵ ✻✶ ✻✷ ✻✸ ✻✹ ✻✺ ✻✻ ✻✼ ⑥ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❢♦r ✭ ❥ ✐♥ ✶✿✷✮ ④ ❝✶ ❬✐ ✱ ❥ ❪❂ ❝ ❬✐ ✱ ❥ ❪✰✵✳✹✹✶✻✺✻✾✾✾✾✾✾✾✾✾ ✴ ♠❛❣ ✯ ❣ ❬✐ ✱ ❥ ❪⑥⑥ r❡t✉r♥ ✭ ❝✶ ✮ ✇❤✐❧❡ ✭❤ ❃✵✳✵✼✮ ❝ ❂ ♥❡①t❝ ✭ ❝ ✮ ❤ ❂ str❡ss ✭ ❝ ✮ ⑥ ④ ❚♦ ♣❧♦t t❤❡ r❡s✉❧t ✿ ✶ ✷ ✸ ✹ ❧✐❜r❛r② ✭ ✧ s❝❛tt❡r♣❧♦t✸❞ ✧ ✮ ❝♦❧♦rs ❁✲ ❝ ✭ ✧ ❜❧❛❝❦ ✧ ✱ ✧ ❜❧✉❡ ✧ ✱ ✧ ♣✉r♣❧❡ ✧ ✮ ❝♦❧♦rs ❁✲ ❝♦❧♦rs ❬ ❛s ✳ ♥✉♠❡r✐❝ ✭ ✐r✐s ✩ ❙♣❡❝✐❡s ✮ ❪ s❝❛tt❡r♣❧♦t✸❞ ✭ ❝ ❬ ✱✶✿✸❪ ✱ ♣❝❤ ❂ ✶✻ ✱ ❝♦❧♦r ❂ ❝♦❧♦rs ✮ ❋✐❣✉r❡ ✷✿ ✸❉✲▼❉❙ ♣❧♦t ✉s✐♥❣ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ✷ ❈♦♠♣❛r❡ ❛❜♦✈❡ r❡s✉❧ts t♦ ❝❧❛ss✐❝❛❧ ▼❉❙ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✉s❡ t❤❡ ❝♠❞s❝❛❧❡ ✇❤✐❝❤ ✐s ❛ ❜✉✐❧❞✲✐♥ ❢✉♥❝t✐♦♥ ✐♥ ❘ ❲❡ st✐❧❧ ✉s❡ t❤❡ ❝♦❞❡ ❢r♦♠ t❤❡ ✶st s❡❝t✐♦♥ ✇❤❡r❡ ✇❡ ❤❛✈❡ ❞ ✐s t❤❡ ❞✐st❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ❞❛t❛ ■r✐s ❚❤❡ ❝♦❞❡ ❢♦r ✷❉ ▼❉❙ ✿ ✶ ✷ ✸ ❃ ❝ ❁✲ ❝♠❞s❝❛❧❡ ✭❞ ✱ ❦ ❂✷✮ ❃ str❡ss ✭ ❝ ✮ ❬✶❪ ✵✳✵✹✷✷✼✵✼✹ ❆♥❞ ❢♦r ✸❉ ▼❉❙ ✿ ✼ ✶ ✷ ✸ ❃ ❝ ❁✲ ❝♠❞s❝❛❧❡ ✭❞ ✱ ❦ ❂✸✮ ❃ str❡ss ✭ ❝ ✮ ❬✶❪ ✵✳✵✶✷✸✷✾✹✺ ❲❡ ❝❛♥ ❡❛s✐❧② s❡❡ t❤❛t t❤❡ ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠ ❣✐✈❡ ❛ ❜❡tt❡r str❡ss t❤❛♥ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠✳ ■♥ ♣❛rt✐❝✉❧❛r ❞❛t❛s❡t ■r✐s✱ ❢♦r ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠✱ ✺✳✹% ✐♥ ✷❉✲▼❉❙ ❛♥❞ ✻✳✹% ✐♥ ✸❉✲▼❉❙ ❝♦♠♣❛r❡ t♦ ✹✳✷% ❛♥❞ ✶✳✷% r❡s♣❡❝t✐✈❡❧② ❢♦r ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠✳ ❆♥♦t❤❡r ❛❞✈❛♥❝❡ ♦❢ ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠ ✐s ✐t ❣✐✈❡ ❛ r❡s✉❧t ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❇✉t ✐♥ ❛♥♦t❤❡r s✐❞❡ ✱ t❤❡ ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠ ✐s ♣❛rt✐❝✉❧❛r ❢♦r ♠❡tr✐❝ ▼❉❙ ✱ ✐❢ ✇❡ ✉s❡ ❛♥♦t❤❡r ❞✐st❛♥❝❡ ♦♣❡r❛t♦r ❞✐✛❡r❡♥t t❤❛♥ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡✱ ✐t ✇✐❧❧ ❣✐✈❡ ❤✐❣❤❡r str❡ss ❋✐❣✉r❡ ✸✿ ✷❉✲▼❉❙ ♣❧♦t ✉s✐♥❣ ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠ ✽ ❋✐❣✉r❡ ✹✿ ✸❉✲▼❉❙ ♣❧♦t ✉s✐♥❣ ❝❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠ ✸ P❛❝❦❛❣❡s ❢♦r ♥♦♥✲♠❡tr✐❝ ▼❉❙✭◆▼❉❙✮ ✐♥ ❘ ✸✳✶ ❋✉♥❝t✐♦♥ ♠♦♥♦▼❉❙ ❋✐rst✱✇❡ ♥❡❡❞ t♦ ❝❛❧❧ ♦✉t t❤❡ ❧✐❜r❛r② ✶ ❃ ❧✐❜r❛r② ✭ ✈❡❣❛♥ ✮ ❋✉♥❝t✐♦♥ ♠♦♥♦▼❉❙ ✉s❡s ❑r✉s❦❛❧✬s ✭✶✾✻✹❜✮ ♦r✐❣✐♥❛❧ ♠♦♥♦t♦♥❡ r❡❣r❡ss✐♦♥ t♦ ♠✐♥✐♠✐③❡ t❤❡ str❡ss✳ ❚❤❡r❡ ❛r❡ t✇♦ ❛❧t❡r♥❛t✐✈❡s ♦❢ str❡ss✿ ❑r✉s❦❛❧✬s ✭✶✾✻✹❛✱❜✮ ♦r✐❣✐♥❛❧ ♦r ✏str❡ss ✶✑ ❛♥❞ ❛♥ ❛❧t❡r♥❛t✐✈❡ ✈❡rs✐♦♥ ♦r ✏str❡ss ✷✑ ✭❙✐❜s♦♥ ✶✾✼✷✮✳ ❇♦t❤ ♦❢ t❤❡s❡ str❡ss❡s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✇✐t❤ ❛ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ s2 = ˆ2 (d − d) (d − d0 )2 ❲❤❡r❡ ❞ ❛r❡ ❞✐st❛♥❝❡s ❛♠♦♥❣ ♣♦✐♥ts ✐♥ ♦r❞✐♥❛t✐♦♥ ❝♦♥✜❣✉r❛t✐♦♥ dˆ ❛r❡ t❤❡ ✜tt❡❞ d0 ❛r❡ t❤❡ ♦r❞✐♥❛t✐♦♥ ❞✐st❛♥❝❡s ✉♥❞❡r ♥✉❧❧ ♠♦❞❡❧✳ ❋♦r ✏str❡ss ✶✑ ✿d0 = ❛♥❞ ❢♦r ✏str❡ss ✷✑ d0 ✐s ♠❡❛♥ ❞✐st❛♥❝❡s 2 ✏❙tr❡ss ✷✑ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s s = − R ✇❤❡r❡ R ✐s sq✉❛r❡❞ ❝♦rr❡❧❛t✐♦♥ ♦r❞✐♥❛t✐♦♥ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ✜tt❡❞ ✈❛❧✉❡s ❛♥❞ ♦r❞✐♥❛t✐♦♥ ❞✐st❛♥❝❡s✱ ❛♥❞ s♦ r❡❧❛t❡❞ t♦ t❤❡ ✏❧✐♥❡❛r ✜t✑ ♦❢ str❡ss ♣❧♦t✳ ❋♦r ♣❛rt✐❝✉❧❛r ❞❛t❛ ■r✐s ❢♦r ✷❉ ◆▼❉❙ ✿ ✶ ✷ ✸ ✹ ✺ ✻ ❃ ♠♦♥♦▼❉❙ ✭❞ ✱ ❦ ❂✷✮ ❈❛❧❧ ✿ ♠♦♥♦▼❉❙ ✭ ❞✐st ❂ ❞ ✱ ❦ ❂ ✷✮ ◆♦♥ ✲ ♠❡tr✐❝ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✾ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺✵ ♣♦✐♥ts ✱ ❞✐ss✐♠✐❧❛r✐t② ✉♥❦♥♦✇♥ ❉✐♠❡♥s✐♦♥s ✿ ✷ ❙tr❡ss ✿ ✵✳✵✷✹✶✺✽✺✽ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ❙❝♦r❡s s❝❛❧❡❞ t♦ ✉♥✐t r♦♦t ♠❡❛♥ sq✉❛r❡ ✱ r♦t❛t❡❞ t♦ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❙t♦♣♣❡❞ ❛❢t❡r ✶✺✹ ✐t❡r❛t✐♦♥s ✿ ❙tr❡ss ♥❡❛r❧② ✉♥❝❤❛♥❣❡❞ ✭ r❛t✐♦ ❃ sr❛t♠❛① ✮ ❆♥❞ ✸❉ ◆▼❉❙ ✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ❃ ♠♦♥♦▼❉❙ ✭ ❞✐st ❂ ❞ ✱ ❦ ❂ ✸✮ ❈❛❧❧ ✿ ♠♦♥♦▼❉❙ ✭ ❞✐st ❂ ❞ ✱ ❦ ❂ ✸✮ ◆♦♥ ✲ ♠❡tr✐❝ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✶✺✵ ♣♦✐♥ts ✱ ❞✐ss✐♠✐❧❛r✐t② ✉♥❦♥♦✇♥ ❉✐♠❡♥s✐♦♥s ✿ ✸ ❙tr❡ss ✿ ✵✳✵✵✾✽✶✷✷✸✼ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ❙❝♦r❡s s❝❛❧❡❞ t♦ ✉♥✐t r♦♦t ♠❡❛♥ sq✉❛r❡ ✱ r♦t❛t❡❞ t♦ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❙t♦♣♣❡❞ ❛❢t❡r ✷✵✵ ✐t❡r❛t✐♦♥s ✿ ▼❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✭ ♠❛①✐t ✮ r❡❛❝❤❡❞ ❋✐❣✉r❡ ✺✿ ✷❉✲◆▼❉❙ str❡ss ♣❧♦t ✶✵ ❋✐❣✉r❡ ✻✿ ✸❉✲◆▼❉❙ str❡ss ♣❧♦t ❚❤❡ ✜❣✉r❡s ❛❜♦✈❡ s❤♦✇ str❡ss ❛❢t❡r ✐t❡r❛t✐♦♥s ❲❡ ❝❛♥ ❡❛s✐❧② s❡❡ t❤❛t ✐♥ t❤❡ ✜rst ❢❡✇ ✐t❡r❛t✐♦♥s✱t❤❡ str❡ss ✐♥ ✸❉ ◆▼❉❙ ❛r❡ ❜❡tt❡r t❤❛♥ ✷❉ ◆▼❉❙ ❛♥❞ ✐t ❝♦♥✈❡r❣❡s ❢❛st❡r ❛♥❞ t❤❡ ✜♥❛❧ str❡ss ✐♥ ✸❉ ◆▼❉❙ ✐s ❜❡tt❡r ✶✶ ✸✳✷ ❋✉♥❝t✐♦♥ ♠❡t❛▼❉❙ ❋✉♥❝t✐♦♥ ♠❡t❛▼❉❙ ♣❡r❢♦r♠s ◆♦♥♠❡tr✐❝ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✭◆▼❉❙✮✱ ❛♥❞ tr✐❡s t♦ ✜♥❞ ❛ st❛❜❧❡ s♦❧✉t✐♦♥ ✉s✐♥❣ s❡✈❡r❛❧ r❛♥❞♦♠ st❛rts✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐t st❛♥❞❛r❞✐③❡s t❤❡ s❝❛❧✐♥❣ ✐♥ t❤❡ r❡s✉❧t✱ s♦ t❤❛t t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ❛r❡ ❡❛s✐❡r t♦ ✐♥t❡r♣r❡t✱ ❛♥❞ ❛❞❞s s♣❡❝✐❡s s❝♦r❡s t♦ t❤❡ s✐t❡ ♦r❞✐♥❛t✐♦♥✳ ❚❤❡ ♠❡t❛▼❉❙ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❛❝t✉❛❧ ◆▼❉❙✱ ❜✉t ✐t ❝❛❧❧s ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ♣✉r♣♦s❡✳ ■♥ ♣❛rt✐❝✉❧❛r ❞❛t❛ ■r✐s✱❢♦r ✷❉ ◆▼❉❙ ✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ❃ ♠❡t❛▼❉❙ ✭❞ ✱ ❦ ❂✷✮ ❘✉♥ ✵ str❡ss ✵✳✵✷✹✶✺✽✻✺ ❘✉♥ ✶ str❡ss ✵✳✵✷✾✵✺✽✷✽ ❘✉♥ ✷ str❡ss ✵✳✵✷✹✶✺✼✺✶ ✳✳✳ ◆❡✇ ❜❡st s♦❧✉t✐♦♥ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✶✺✻✹✶✶✷ ♠❛① r❡s✐❞ ✵✳✵✵✶✻✸✽✻✷✺ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✸ str❡ss ✵✳✵✷✽✾✹✽✹✺ ❘✉♥ ✹ str❡ss ✵✳✵✸✺✻✸✷✻✸ ❘✉♥ ✺ str❡ss ✵✳✵✸✻✾✼✵✷✼ ❘✉♥ ✻ str❡ss ✵✳✵✸✶✹✷✹✻✺ ❘✉♥ ✼ str❡ss ✵✳✵✸✹✺✽✷✶✾ ❘✉♥ ✽ str❡ss ✵✳✵✸✷✾✸✺✶✶ ❘✉♥ ✾ str❡ss ✵✳✵✹✶✶✽✹✷✶ ❘✉♥ ✶✵ str❡ss ✵✳✵✷✽✾✹✾✶✶ ❘✉♥ ✶✶ str❡ss ✵✳✵✸✼✵✻✹✾✽ ❘✉♥ ✶✷ str❡ss ✵✳✵✸✺✾✼✽✽✼ ❘✉♥ ✶✸ str❡ss ✵✳✵✷✽✻✽✹ ❘✉♥ ✶✹ str❡ss ✵✳✵✷✹✶✶✽✸✶ ✳✳✳ ◆❡✇ ❜❡st s♦❧✉t✐♦♥ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✶✵✸✾✺✵✶ ♠❛① r❡s✐❞ ✵✳✵✶✶✹✸✶✼✹ ❘✉♥ ✶✺ str❡ss ✵✳✵✸✻✶✺✵✸✶ ❘✉♥ ✶✻ str❡ss ✵✳✵✸✹✵✸✷✼ ❘✉♥ ✶✼ str❡ss ✵✳✵✸✸✸✼✶✻✽ ❘✉♥ ✶✽ str❡ss ✵✳✵✸✼✺✵✸✺✷ ❘✉♥ ✶✾ str❡ss ✵✳✵✸✻✵✹✽✶✼ ❘✉♥ ✷✵ str❡ss ✵✳✵✸✹✼✹✷✽✼ ✯ ✯ ✯ ◆♦ ❝♦♥✈❡r❣❡♥❝❡ ✲✲ ♠♦♥♦▼❉❙ st♦♣♣✐♥❣ ❝r✐t❡r✐❛ ✿ ✹✿ ♥♦ ✳ ♦❢ ✐t❡r❛t✐♦♥s ❃❂ ♠❛①✐t ✶✺✿ str❡ss r❛t✐♦ ❃ sr❛t♠❛① ✶✿ s❝❛❧❡ ❢❛❝t♦r ♦❢ t❤❡ ❣r❛❞✐❡♥t ❁ s❢❣r♠✐♥ ❈❛❧❧ ✿ ♠❡t❛▼❉❙ ✭ ❝♦♠♠ ❂ ❞ ✱ ❦ ❂ ✷✮ ❣❧♦❜❛❧ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✉s✐♥❣ ♠♦♥♦▼❉❙ ❉❛t❛ ✿ ❞ ❉✐st❛♥❝❡ ✿ ✉s❡r s✉♣♣❧✐❡❞ ❉✐♠❡♥s✐♦♥s ✿ ✷ ❙tr❡ss ✿ ✵✳✵✷✹✶✶✽✸✶ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ◆♦ ❝♦♥✈❡r❣❡♥t s♦❧✉t✐♦♥s ✲ ❜❡st s♦❧✉t✐♦♥ ❛❢t❡r ✷✵ tr✐❡s ❙❝❛❧✐♥❣ ✿ ❝❡♥tr✐♥❣ ✱ P❈ r♦t❛t✐♦♥ ✶✷ ❆♥❞ ✸❉ ◆▼❉❙✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ✺✶ ✺✷ ✺✸ ✺✹ ✺✺ ❃ ♠❡t❛▼❉❙ ✭❞ ✱ ❦ ❂✸✮ ❘✉♥ ✵ str❡ss ✵✳✵✵✾✻✾✾✽✻✻ ❘✉♥ ✶ str❡ss ✵✳✵✵✾✼✵✵✽✷✻ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✶✽✸✷✻✸✹ ♠❛① r❡s✐❞ ✵✳✵✵✵✼✾✻✷✼✹✹ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✷ str❡ss ✵✳✵✶✵✻✾✾✽✼ ❘✉♥ ✸ str❡ss ✵✳✵✵✾✼✵✷✷✽✸ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✷✸✾✹✹✶✹ ♠❛① r❡s✐❞ ✵✳✵✵✶✵✹✸✽✹✷ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✹ str❡ss ✵✳✵✵✾✼✵✶✵✻✸ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✷✸✹✻✼✶✽ ♠❛① r❡s✐❞ ✵✳✵✵✵✾✶✾✼✸✾✾ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✺ str❡ss ✵✳✵✶✻✾✹✵✺✻ ❘✉♥ ✻ str❡ss ✵✳✵✵✾✼✵✶✶✵✹ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✶✾✹✹✽✺✷ ♠❛① r❡s✐❞ ✵✳✵✵✵✽✹✽✵✾✻✶ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✼ str❡ss ✵✳✵✶✵✻✾✾✷✸ ❘✉♥ ✽ str❡ss ✵✳✵✵✾✼✺✶✹✷✺ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✽✺✶✽✺✼✶ ♠❛① r❡s✐❞ ✵✳✵✵✸✺✸✻✻✽✸ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✾ str❡ss ✵✳✵✵✾✼✵✶✺✶✽ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✷✻✵✼✹✵✶ ♠❛① r❡s✐❞ ✵✳✵✵✵✾✽✺✼✾✼✼ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✶✵ str❡ss ✵✳✵✶✻✻✷✷✼ ❘✉♥ ✶✶ str❡ss ✵✳✵✶✵✻✹✸✷✾ ❘✉♥ ✶✷ str❡ss ✵✳✵✶✵✻✷✸✻✸ ❘✉♥ ✶✸ str❡ss ✵✳✵✶✵✻✶✽✻✷ ❘✉♥ ✶✹ str❡ss ✵✳✵✶✵✼✵✵✸✻ ❘✉♥ ✶✺ str❡ss ✵✳✵✶✽✽✶✹✵✺ ❘✉♥ ✶✻ str❡ss ✵✳✵✶✵✻✺✽✹✷ ❘✉♥ ✶✼ str❡ss ✵✳✵✵✾✻✾✾✽✾✹ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✶✻✻✺✵✷ ♠❛① r❡s✐❞ ✵✳✵✵✵✹✷✸✷✼✽✺ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✶✽ str❡ss ✵✳✵✶✵✻✾✾✶✾ ❘✉♥ ✶✾ str❡ss ✵✳✵✵✾✼✵✵✾✾✶ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✷✸✹✶✹ ♠❛① r❡s✐❞ ✵✳✵✵✵✽✽✽✺✷✸✾ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ❘✉♥ ✷✵ str❡ss ✵✳✵✵✾✼✵✶✻✺ ✳✳✳ Pr♦❝r✉st❡s ✿ r♠s❡ ✵✳✵✵✵✷✻✽✹✺✵✾ ♠❛① r❡s✐❞ ✵✳✵✵✵✾✽✸✸✽✼✾ ✳✳✳ ❙✐♠✐❧❛r t♦ ♣r❡✈✐♦✉s ❜❡st ✯ ✯ ✯ ❙♦❧✉t✐♦♥ r❡❛❝❤❡❞ ❈❛❧❧ ✿ ♠❡t❛▼❉❙ ✭ ❝♦♠♠ ❂ ❞ ✱ ❦ ❂ ✸✮ ❣❧♦❜❛❧ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✉s✐♥❣ ♠♦♥♦▼❉❙ ❉❛t❛ ✿ ❞ ❉✐st❛♥❝❡ ✿ ✉s❡r s✉♣♣❧✐❡❞ ❉✐♠❡♥s✐♦♥s ✿ ✸ ❙tr❡ss ✿ ✵✳✵✵✾✻✾✾✽✻✻ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ❚✇♦ ❝♦♥✈❡r❣❡♥t s♦❧✉t✐♦♥s ❢♦✉♥❞ ❛❢t❡r ✷✵ tr✐❡s ❙❝❛❧✐♥❣ ✿ ❝❡♥tr✐♥❣ ✱ P❈ r♦t❛t✐♦♥ ✶✸ ❚❤❡ str❡ss ❜② ✉s✐♥❣ t❤✐s ❢✉♥❝t✐♦♥ ❛r❡ 2.4% ❢♦r ✷❉ ▼❉❙ ❛♥❞ 0.097% s❧✐❣❤t❧② ❜❡tt❡r t❤❛♥ ❝❧❛ss✐❝❛❧ ▼❉❙ ❢♦r t❤❡ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡✳ ❋✐❣✉r❡ ✼✿ ✷❉✲◆▼❉❙ ♣❧♦t ❋✐❣✉r❡ ✽✿ ✸❉✲◆▼❉❙ ♣❧♦t ✶✹ ❢♦r ✸❉ ▼❉❙ ✱✐t✬s ✹ ◆❡❣❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ◆▼❉❙ ✹✳✶ ❯s✐♥❣ ❈❧❛ss✐❝❛❧ ❛❧❣♦r✐t❤♠✭❝♠❞s❝❛❧❡✮ ❋✐rst✱✇❡ ❛ss✐❣♥ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s ❢♦r s♦♠❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ ❞✐st❛♥❝❡ ♠❛tr✐① ❆♥❞ t❤❡ r❡st ♦❢ t❤❡ ❞✐st❛♥❝❡ ♠❛tr✐① ♦❜t❛✐♥ ❜② ✉s✐♥❣ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ❞✐st ❁✲ ❢✉♥❝t✐♦♥ ✭ ❜ ✮ ④ ❞ ❂ ♠❛tr✐① ✭ ♥r♦✇ ❂ ♥ ✱ ♥❝♦❧ ❂ ♥ ✮ ❢♦r ✭ ✐ ✐♥ ✶✿ ♥ ✮ ④ ❢♦r ✭ ❥ ✐♥ ✶✿ ♥ ✮ ④ ❞ ❬✐ ✱ ❥ ❪❂ ❞ ❬❥ ✱ ✐ ❪❂ sqrt ✭ s✉♠ ✭✭ ❜ ❬✐ ✱❪ ✲ ❜ ❬❥ ✱❪✮ ❫✷✮ ✮ ⑥⑥ ❞ ❬✶ ✱✸❪❂ ❞ ❬✸ ✱✶❪❂ ✲✶✵✵ r❡t✉r♥ ✭ ❞ ✮ ⑥ ❋♦r ✷❉ ❝❧❛ss✐❝❛❧ ▼❉❙✿ ✶ ✷ ✸ ❃ ❝ ❁✲ ❝♠❞s❝❛❧❡ ✭❞ ✱ ❦ ❂✷✮ ❃ str❡ss ✭ ❝ ✮ ❬✶❪ ✵✳✽✼✸✸✹✼✸ ❆♥❞ ✸❉ ❝❧❛ss✐❝❛❧ ▼❉❙✿ ✶ ✷ ✸ ❃ ❝ ❁✲ ❝♠❞s❝❛❧❡ ✭❞ ✱ ❦ ❂✸✮ ❃ str❡ss ✭ ❝ ✮ ❬✶❪ ✵✳✽✻✾✼✻✶✽ ❲❡ ❝❛♥ s❡❡ t❤❛t ❝❧❛ss✐❝❛❧ ▼❉❙ ✐s ♥♦t ❛ s✉✣❝✐❡♥t ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤✐s ❝❛s❡ ✹✳✷ ❯s✐♥❣ ◆♦♥♠❡tr✐❝ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣✭♠❡t❛▼❉❙✮ ❋♦r ✷❉ ◆▼❉❙✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ❃ ♠❡t❛▼❉❙ ✭❞ ✱ ❦ ❂✷✮ ❈❛❧❧ ✿ ♠❡t❛▼❉❙ ✭ ❝♦♠♠ ❂ ❞ ✱ ❦ ❂ ✷✮ ❣❧♦❜❛❧ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✉s✐♥❣ ♠♦♥♦▼❉❙ ❉❛t❛ ✿ ❞ ❉✐st❛♥❝❡ ✿ ✉s❡r s✉♣♣❧✐❡❞ ❉✐♠❡♥s✐♦♥s ✿ ✷ ❙tr❡ss ✿ ✵✳✵✷✹✶✼✾✷✼ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ❚✇♦ ❝♦♥✈❡r❣❡♥t s♦❧✉t✐♦♥s ❢♦✉♥❞ ❛❢t❡r ✷✵ tr✐❡s ❙❝❛❧✐♥❣ ✿ ❝❡♥tr✐♥❣ ✱ P❈ r♦t❛t✐♦♥ ✶✺ ❋♦r ✸❉ ◆▼❉❙✿ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ❃ ♠❡t❛▼❉❙ ✭❞ ✱ ❦ ❂✸✮ ❈❛❧❧ ✿ ♠❡t❛▼❉❙ ✭ ❝♦♠♠ ❂ ❞ ✱ ❦ ❂ ✸✮ ❣❧♦❜❛❧ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❛❧✐♥❣ ✉s✐♥❣ ♠♦♥♦▼❉❙ ❉❛t❛ ✿ ❞ ❉✐st❛♥❝❡ ✿ ✉s❡r s✉♣♣❧✐❡❞ ❉✐♠❡♥s✐♦♥s ✿ ✸ ❙tr❡ss ✿ ✵✳✵✵✾✼✻✷✸✾✸ ❙tr❡ss t②♣❡ ✶ ✱ ✇❡❛❦ t✐❡s ❚✇♦ ❝♦♥✈❡r❣❡♥t s♦❧✉t✐♦♥s ❢♦✉♥❞ ❛❢t❡r ✷✵ tr✐❡s ❙❝❛❧✐♥❣ ✿ ❝❡♥tr✐♥❣ ✱ P❈ r♦t❛t✐♦♥ ❙♣❡❝✐❡s ✿ s❝♦r❡s ♠✐ss✐♥❣ ■♥ t❤✐s ❝❛s❡✱✇❡ ❝❛♥ ❡❛s✐❧② s❡❡ t❤❛t ◆▼❉❙ st✐❧❧ ❣❛✈❡ ❛ ❣♦♦❞ str❡ss ❛♥❞ ✐t✬s ❛ ❣♦♦❞ ♠❡t❤♦❞ t♦ s♦❧✈❡ ❛ ❣❡♥❡r❛❧ ▼❉❙ ♣r♦❜❧❡♠ ✶✻ ❘❡❢❡r❡♥❝❡s ❬✶❪ ❋❛✐t❤✱ ❉✳P✳✱ ▼✐♥❝❤✐♥✱ P✳❘ ❛♥❞ ❇❡❧❜✐♥✱ ▲✳ ✶✾✽✼✳ ❈♦♠♣♦s✐t✐♦♥❛❧ ❞✐ss✐♠✐❧❛r✐t② ❛s ❛ r♦❜✉st ♠❡❛s✉r❡ ♦❢ ❡❝♦❧♦❣✐❝❛❧ ❞✐st❛♥❝❡✳ ❱❡❣❡t❛t✐♦ ✻✾✱ ✺✼✕✻✽✳ ●♦✇❡r✱ ❏✳❈✳ ✭✶✾✻✻✮✳ ❙♦♠❡ ❞✐st❛♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ ❧❛t❡♥t r♦♦t ❛♥❞ ✈❡❝t♦r ♠❡t❤♦❞s ✉s❡❞ ✐♥ ♠✉❧t✐✈❛r✐❛t❡ ❛♥❛❧②s✐s✳ ❇✐♦♠❡tr✐❦❛ ✺✸✱ ✸✷✺✕✸✷✽✳ ❬✷❪ ❑r✉s❦❛❧✱ ❏✳❇✳ ✶✾✻✹❛✳ ▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ❜② ♦♣t✐♠✐③✐♥❣ ❣♦♦❞♥❡ss✲♦❢✲✜t t♦ ❛ ♥♦♥♠❡tr✐❝ ❤②♣♦t❤❡s✐s✳ Ps②❝❤♦♠❡tr✐❦❛ ✷✾✱ ✶✕✷✽✳ ❬✸❪ ❑r✉s❦❛❧✱ ❏✳❇✳ ✶✾✻✹❜✳ ◆♦♥♠❡tr✐❝ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣✿ ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞✳ Ps②❝❤♦♠❡tr✐❦❛ ✷✾✱ ✶✶✺✕✶✷✾✳ ❬✹❪ ▼✐♥❝❤✐♥✱ P✳❘✳ ✶✾✽✼✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ r❡❧❛t✐✈❡ r♦❜✉st♥❡ss ♦❢ t❡❝❤♥✐q✉❡s ❢♦r ❡❝♦❧♦❣✲ ✐❝❛❧ ♦r❞✐♥❛t✐♦♥s✳ ❱❡❣❡t❛t✐♦ ✻✾✱ ✽✾✕✶✵✼✳ ❬✺❪ ❙✐❜s♦♥✱ ❘✳ ✶✾✼✷✳ ❖r❞❡r ✐♥✈❛r✐❛♥t ♠❡t❤♦❞s ❢♦r ❞❛t❛ ❛♥❛❧②s✐s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❘♦②❛❧ ❙t❛t✐st✐❝❛❧ ❙♦❝✐❡t② ❇ ✸✹✱ ✸✶✶✕✸✹✾✳ ✶✼