Multidimentional 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, 2

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

2
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 = 0
ki
kj
❝❛♥ t❛❦❡ k = i ✱t❤❡♥ σ − σ
= σ ii − σ ij = 1

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 = 0 ❛♥❞ ❢♦r ✏str❡ss ✷✑ d0 ✐s ♠❡❛♥ ❞✐st❛♥❝❡s
2
2
2
✏❙tr❡ss ✷✑ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s s = 1 − 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 ❇ ✸✹✱ ✸✶✶✕✸✹✾✳

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