r t r r - - PowerPoint PPT Presentation

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r t r r s

r t r r - - PowerPoint PPT Presentation

Jun 15, 2023 10 likes •552 views

r t r r s t s s rst r

slide-1

SLIDE 1

❆ ✉♥✐✜❡❞ ❛♣♣r♦❛❝❤ t♦ ❧✐♥❡❛r ♣r♦❜✐♥❣ ❤❛s❤✐♥❣

❙✈❛♥t❡ ❏❛♥s♦♥

❯♣♣s❛❧❛ ❯♥✐✈❡rs✐t②

❆❧❢r❡❞♦ ❱✐♦❧❛

❯♥✐✈❡rs✐❞❛❞ ❞❡ ❧❛ ❘❡♣ú❜❧✐❝❛

❆♦❢❆ ✷✵✶✹

❉❡❞✐❝❛t❡❞ t♦ P❤✐❧✐♣♣❡ ❋❧❛❥♦❧❡t

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SLIDE 2

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

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SLIDE 3

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-4

SLIDE 4

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

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SLIDE 5

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-6

SLIDE 6

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-7

SLIDE 7

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-8

SLIDE 8

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-9

SLIDE 9

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

slide-10

SLIDE 10

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

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SLIDE 11

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

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SLIDE 12

❚❤❡ ♣r♦❜❧❡♠✳

❚❤❡ t❛❜❧❡ ❤❛s ♠ ♣❧❛❝❡s ♦❢ ❝❛♣❛❝✐t② ❜ t♦ ❤❛s❤ ✭♣❛r❦✮ ❢r♦♠ ✵ t♦ ♠ ✶✱ ❛♥❞ ♥ ✐♥s❡rt❡❞ ❡❧❡♠❡♥ts ✭❝❛rs✮✳ ❊❛❝❤ ❡❧❡♠❡♥t ✐s ❣✐✈❡♥ ❛ ❤❛s❤ ✈❛❧✉❡ ✭♣r❡❢❡rr❡❞ ♣❛r❦✐♥❣ ❧♦t✮✳ ■❢ ♣❧❛❝❡ ✐s ♥♦t ❢✉❧❧✱ t❤❡♥ t❤❡ ❡❧❡♠❡♥t ✐s st♦r❡❞ t❤❡r❡✳ ❖t❤❡r✇✐s❡✱ ❧♦♦❦s s❡q✉❡♥t✐❛❧❧② ❢♦r ❛♥ ❡♠♣t② ♣❧❛❝❡✳ ■❢ ♥♦ ❡♠♣t② ♣❧❛❝❡ ✉♣ t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ t❛❜❧❡✱ t❤❡ s❡❛r❝❤ ❢♦❧❧♦✇s ❛t ❧♦❝❛t✐♦♥ ✵✳ ❙❡✈❡r❛❧ ❘✳❱✳ t♦ st✉❞②✱ ♠❛✐♥❧② r❡❧❛t❡❞ ✇✐t❤ ❝♦st ♦❢ ✐♥❞✐✈✐❞✉❛❧ s❡❛r❝❤❡s ❛♥❞ t♦t❛❧ ❝♦♥str✉❝t✐♦♥ ❝♦st✳ ❱❡r② ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❛s❡✿ P❛r❦✐♥❣ Pr♦❜❧❡♠✳ ■♥ ♣❛r❦✐♥❣ t❤❡ ❝❛r ✐s ❧♦st ✐❢ ♥♦ ❛✈❛✐❧❛❜❧❡ ♣❧❛❝❡ t♦ ♣❛r❦ ✉♣ t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ t❛❜❧❡✳ ▼❛✐♥ ❘✳❱✳ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❧♦st ❝❛rs✳

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SLIDE 13

▲✐♥❡❛r Pr♦❜✐♥❣ ❍❛s❤✐♥❣✳

slide-14

SLIDE 14

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❜❡❛✉t② ♦❢ ▲✐♥❡❛r Pr♦❜✐♥❣✦

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SLIDE 15

✶✾✻✷✿ ❙✉♠♠❡r ✇♦r❦ ❜② ❉♦♥ ❑♥✉t❤ ✳✳✳

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SLIDE 16

❖r✐❣✐♥❛❧ r❡s✉❧ts✳

▲❡t ❛ ❤❛s❤ t❛❜❧❡ ✇✐t❤ ♠ ♣♦s✐t✐♦♥s ❛♥❞ ♥ ✐♥s❡rt❡❞ ❡❧❡♠❡♥ts✳ ▲❡t P♠❀♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❧❛st ♣♦s✐t✐♦♥ ❜❡✐♥❣ ❡♠♣t②✳

P♠❀♥ ❂

✶ ♥

♠

✁ ✳

▲❡t ❈♠❀♥ t❤❡ ❘✳❱✳ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❝❡ss❢✉❧ s❡❛r❝❤❡s ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t✳

❊ ❬❈♠❀♥❪ ❂ ✶

✷ ✭✶ ✰ ◗✵✭✭♠❀ ♥ ✶✮✮✳

❊ ❬❈♠❀☛♠❪ ❂ ✶

✷

✏ ✶ ✰

✶ ✶☛

✑ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳ ❊ ❬❈♥❀♥❪ ❂ ♣ ✙♥

✽ ✰ ❖✭✶✮ ✭♣r♦✈❡❞ ♦♥ ♠❛② ✷✵✱ ✶✾✻✺✮✳

▲❡t ❯♠❀♥ t❤❡ ❘✳❱✳ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✉♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤❡s ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t✳

❊ ❬❯♠❀♥❪ ❂ ✶

✷ ✭✶ ✰ ◗✶✭✭♠❀ ♥ ✶✮✮✳

❊ ❬❯♠❀☛♠❪ ❂ ✶

✷

✏ ✶ ✰

✶ ✭✶☛✮✷

✑ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳

❚❤❡ ❘❛♠❛♥✉❥❛♥ ◗ ❢✉♥❝t✐♦♥ ✐s t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ◗✵✭♥❀ ♥✮ ♦❢ ◗r✭♠❀ ♥✮ ❂

♥

❳

❦❂✵

✥

❦ ✰ r ❦

✦

♥❦ ♠♥ ✿

slide-17

SLIDE 17

❚✇♦ ♠♦❞❡❧s t♦ ❛♥❛❧②③❡ t❤❡ ♣r♦❜❧❡♠✳

❊①❛❝t ✜❧❧✐♥❣ ♠♦❞❡❧✳

❆ ✜①❡❞ ♥✉♠❜❡r ♦❢ ❦❡②s ♥✱ ❛r❡ ❞✐str✐❜✉t❡❞ ❛♠♦♥❣ ♠ ❧♦❝❛t✐♦♥s✱ ❛♥❞ ❛❧❧ ♠♥ ♣♦ss✐❜❧❡ ❛rr❛♥❣❡♠❡♥ts ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧② t♦ ♦❝❝✉r✳

P♦✐ss♦♥ ♠♦❞❡❧✳

❊❛❝❤ ❧♦❝❛t✐♦♥ r❡❝❡✐✈❡s ❛ ♥✉♠❜❡r ♦❢ ❦❡②s t❤❛t ✐s P♦✐ss♦♥ ❞✐str✐❜✉t❡❞ ✇✐t❤ ♣❛r❛♠❡t❡r ❜☛✱ ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❦❡②s ❣♦✐♥❣ ❡❧s❡✇❤❡r❡✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦❡②s✱ ◆✱ ✐s ✐ts❡❧❢ ❛ P♦✐ss♦♥ ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡r ❜☛♠✿ Pr❬◆ ❂ ♥❪ ❂ ❡❜☛♠✭❜☛♠✮♥ ♥✦ ✿

slide-18

SLIDE 18

P♦✐ss♦♥ ❚r❛♥s❢♦r♠✳

❘❡s✉❧ts ✐♥ ♦♥❡ ♠♦❞❡❧ ❝❛♥ ❜❡ tr❛♥s❢❡r❡❞ ✐♥t♦ t❤❡ ♦t❤❡r ♠♦❞❡❧ ❜② t❤❡ P♦✐ss♦♥ ❚r❛♥s❢♦r♠✿ P♠❬❢♠❀♥❀ ❜☛❪ ❂

❳

♥✕✵

Pr❬◆ ❂ ♥❪❢♠❀♥ ❂ ❡❜☛♠ ❳

♥✕✵

✭❜☛♠✮♥ ♥✦ ❢♠❀♥✿ ■♥✈❡rs✐♦♥ ❚❤❡♦r❡♠✿ ❬●♦♥♥❡t ❛♥❞ ▼✉♥r♦ ✶✾✽✹❪ ■❢ P♠❬❢♠❀♥❀ ❜☛❪ ❂

❳

❦✕✵

❛♠❀❦✭❜♠☛✮❦ t❤❡♥ ❢♠❀♥ ❂

❳

❦✕✵

❛♠❀❦ ♥❦ ✭❜♠✮❦ ✿ ❚❤❡ P♦✐ss♦♥ ♠♦❞❡❧ ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡①❛❝t ✜❧❧✐♥❣ ♠♦❞❡❧ ✇❤❡♥ ♥❀ ♠ ✦ ✶ ✇✐t❤ ♥❂♠ ❂ ❜☛ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳

slide-19

SLIDE 19

▲✐♥❡❛r Pr♦❜✐♥❣ ❛♥❞ t❤❡ ❙②♠❜♦❧✐❝ ▼❡t❤♦❞ ✭■✮✳

slide-20

SLIDE 20

▲✐♥❡❛r Pr♦❜✐♥❣ ❛♥❞ t❤❡ ❙②♠❜♦❧✐❝ ▼❡t❤♦❞ ✭■✮✳

slide-21

SLIDE 21

❈♦♥❝❧✉s✐♦♥s ✭❑♥✉t❤✮✳

slide-22

SLIDE 22

▲✐♥❡❛r Pr♦❜✐♥❣ ❛♥❞ t❤❡ ❙②♠❜♦❧✐❝ ▼❡t❤♦❞ ✭■■✮✳

slide-23

SLIDE 23

❈♦♠❜✐♥❛t♦r✐❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✳

❆♥② ▲✐♥❡❛r Pr♦❜✐♥❣ ❍❛s❤ t❛❜❧❡ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡s ✭❛ s✉❜t❛❜❧❡ ✇✐t❤ ❛❧❧ ❜✉t t❤❡ ❧❛st ❜✉❝❦❡t ❢✉❧❧✮✳ ❊①❛♠♣❧❡✿ ❬✸✲✸❪✱❬✹✲✹❪✱❬✺✲✺❪✱❬✻✲✷❪✳ ❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❝❛♥ ❜❡ ♥✐❝❡❧② ❤❛♥❞❧❡❞ ❜② ❆♥❛❧②t✐❝ ❈♦♠❜✐♥❛t♦r✐❝s✱ s✐♥❝❡ ❢♦r ❡①❛♠♣❧❡✱ ✐t ✐♠♣❧✐❡s t❤❛t ✐t ✐s ❡♥♦✉❣❤ t♦ st✉❞② ❛❧♠♦st ❢✉❧❧ t❛❜❧❡s✱ ❛♥❞ t❤❡♥ ✉s❡ t❤❡ s❡q✉❡♥❝❡ ❝♦♥str✉❝t✐♦♥✳

slide-24

SLIDE 24

slide-25

SLIDE 25 CONSTR UCTIONS Dictionary (I) F 7! ff n g 7! f (z ) = X n f n z n n! : 1 1

f

= 1 + f + f 2 + f 3 +

exp

(f ) = 1 + f + 1 2! f 2 + 1 3! f 3 +

A[

B 7! A(z )+B (z ) AB 7! A(z )B (z ) Seq A 7! 1 1

A(z

) Set A 7! exp (A(z )) Cycle A 7! log 1 1

A(z

) 11

slide-26

SLIDE 26 4 L.P .H.: Generating functions Almost-full tables n = m

1

have tree decomp

sition.
|

<Full> := <Full> * <Last> * <Full> | | ^^^^^^ with Position |

Dictiona

ry: Pro ducts 7! Pro ducts , C n = P

n

k

A

k B nk C = A ? B 7! C (z ) = A(z )

B

(z ) Dictiona ry: Adding an element 7! Z , C n = A n1 C = Add (A) 7! C (z ) = Z z A(w ) dw : Dictiona ry: Cho

sing

a p

sition

7! @ , C n = (n + 1)A n C = P

s(A)

7! @ @ z ( z A(z )) : 19

slide-27

SLIDE 27 A nonlinear ODE translates Linear Probing T able = T able `?' T able F = Z (z F ) F T (1

1

T ) = 1 z with T = z F Lemma 1. F (z ) = 1 z T (z ) where T = z e T Lemma 2. [Lagrange + Eisenstein + Ca yley] T (z ) = z + 2 z 2 2! + 9 z 3 3! +

=

X n n n1 z n n! Denes the T ree function T (z ) Tree = Root * Set(Tree) [Knuth63] The numb er

f

almost full tables (n k eys) F n = (n + 1) (n1) 15

slide-28

SLIDE 28 Distributional equations (almost full tables)

construction

cost = total displacem en t

|

<Full> := <Full> * <Last> * <Full> | | ^^^^^^ with Position |

F

n (q ) = n1 X k =0

n
1

k

F

k (q )(1 + q +

+

q k )F n1k (q ):

q

{Calculus: The construction cost is w

rst-case

quadratic X n q n 2 z 2 22

slide-29

SLIDE 29 Q-calculus n 7! [n] = 1 + q + q 2 +

q

n1 = 1

q

n 1

q

n! 7! [n]! = [1]

[2]
[n]

P (n + 1)f n z n 7! P [n + 1]f n z n @ @ z (z f (z )) 7! H[f (z )] = F (z )

q

F (q z ) 1

q

Many \combinato ria l identities" survive in the q

w
rld,

with q ? 7! 1 Example. [Euler] (exp (z )) 1 = (exp (z ))

X

z n n!

1

=

X

(z ) n n!

X

z n [n]!

1

=

X

q n(n1)=2 (z ) n [n]!

18

slide-30

SLIDE 30 @ z F (z ) = @ z (z F (z ))

F

(z ) @ z F (z ; q ) =

F

(z ; q )

q

F (q z ; q ) 1

q
F

(z ; q ) @ z F = H F

F

Moments result from | @ q , dierenti ati

n

w.r.t. q | U , setting q = 1 F

r

the r

th

moment, apply U @ r q to dierence-di ere nti al equation Lemma . Commutation rule with Z[f ] = z

f

U H = @ z Z U @ q H = U @ z Z@ q + 1 2 Z@ 2 z Z ; U @ 2 q H = @ z Z U @ 2 q + Z@ 2 z Z U @ q + 1 3 Z 2 @ 3 z Z U Pro

f.

@ q [n + 1], Leibniz, &c. 19

slide-31

SLIDE 31

❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s ❛♥❞ t❤❡ P♦✐ss♦♥ ❚r❛♥s❢♦r♠✳

▲❡t P♠❀♥✭q✮ ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❝✉♠✉❧❛t❡❞ ✈❛❧✉❡ ♦❢ ❛ ❘❱ ✤ ✐♥ ❛ ❤❛s❤ t❛❜❧❡ ♦❢ s✐③❡ ♠ ✇✐t❤ ♥ ❡❧❡♠❡♥ts✳ ▲❡t P✭③❀ ✇❀ q✮ ❂

❳

♠✕✵

✇❜♠ ❳

♥✕✵

P♠❀♥✭q✮ ♠♥ ✭❜♠③✮♥ ♥✦ ✿ ❚❤❡♥✱ ❢♦r ❛ ✜①❡❞ ✵ ❁ ☛ ❁ ✶✱ P✭☛❀ ②✶❂❜❡☛❀ q✮ ❂

❳

♠✕✵

②♠

✵ ❅❡❜♠☛ ❳

♥✕✵

P♠❀♥✭q✮ ♠♥ ✭❜♠☛✮♥ ♥✦

✶ ❆

❂

❳

♠✕✵

②♠P♠

✔P♠❀♥✭q✮

♠♥ ❀ ❜☛

✕

✿ ❘❡s✉❧ts ❢♦r t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ P♦✐ss♦♥ ▼♦❞❡❧ ✭♥❀ ♠ ✦ ✶❀ ✵ ✔ ♥❂❜♠ ❂ ☛ ❁ ✶✮ ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② s✐♥❣✉❧❛r✐t② ❛♥❛❧②s✐s ❢r♦♠ P✭☛❀ ②✶❂❜❡☛❀ q✮✳ ■♥ ♦✉r ♣r♦❜❧❡♠s✱ t❤❡ ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t② ✐s ❛t ② ❂ ✶✳

slide-32

SLIDE 32

❉❡❝♦♠♣♦s✐t✐♦♥✳

▲❡t ◗♠❀♥❀❞ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ♦❢ ✐♥s❡rt✐♥❣ ♥ ❡❧❡♠❡♥ts ✐♥t♦ ♠ ❜✉❝❦❡ts✱ ✇❤❡r❡ t❤❡ ❧❛st ❜✉❝❦❡t ❤❛s ♠♦r❡ t❤❛♥ ❞ ❡♠♣t② s❧♦ts✳ ❇② ❛ ❞✐r❡❝t ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ ❝♦♥str✉❝t✐♦♥ ✇❡ ❤❛✈❡ ✄❞✭❜③❀ ✇✮ ✿❂

❳

♠✕✵

❳

♥✕✵

◗♠❀♥❀❞ ✭❜③✮♥ ♥✦ ✇❜♠ ❂ ✶ ✰ ◆❞✭❜③❀ ✇✮ ✶ ◆✵✭❜③❀ ✇✮✿ ✄✵✭③❀ ✇✮ ✐s t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❝♦♥str✉❝t ❤❛s❤ t❛❜❧❡s s✉❝❤ t❤❛t t❤❡✐r ❧❛st ❜✉❝❦❡t ✐s ♥♦t ❢✉❧❧✳

slide-33

SLIDE 33

❉❡❝♦♠♣♦s✐t✐♦♥✳

▲❡t ❋❜✐✰❞ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❝♦♥str✉❝t ❛♥ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡ ♦❢ ❧❡♥❣t❤ ✐ ✰ ✶ ❛♥❞ s✐③❡ ❜✐ ✰ ❞ ✇✐t❤ ✵ ✔ ❞ ✔ ❜ ✶✳ ❚❤❡♥✱ ❋❞✭✉✮ ✿❂

❳

✐✕✵

❋❜✐✰❞ ✉❜✐✰❞ ✭❜✐ ✰ ❞✮✦❀ ◆❞✭③❀ ✇✮ ✿❂

❜✶❞

❳

s❂✵

✇❜s❋s✭③✇✮✿ ◆❞✭③❀ ✇✮ ✐s t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡s ✇✐t❤ ♠♦r❡ t❤❛♥ ❞ ❡♠♣t② ❧♦❝❛t✐♦♥s ✐♥ t❤❡ ❧❛st ❜✉❝❦❡t ❬❇❧❛❦❡ ❛♥❞ ❑♦♥❤❡✐♠ ✶✾✼✼❪✱ ❬❱✳ ✷✵✶✵❪✳

❜✶

❳

❞❂✵

❋❞✭❜③✮①❞ ❂ ①❜

❜✶

❨

❥❂✵

✥

① ❚✭✦❥③✮ ③

✦

❀ ◆❞✭❜③❀ ✇✮ ❂ ❬①♥❪

❜✶

❨

❥❂✶

✥

✶ ①❚✭✦❥③✇✮ ③

✦

❜✶

❨

❥❂✵

✥

✶ ❚✭✦❥③✇✮ ③

✦

❀ ✇❤❡r❡ ❚ ✐s t❤❡ ❚r❡❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✦ ✐s ❛ ❜✲t❤ r♦♦t ♦❢ ✉♥✐t②✳

slide-34

SLIDE 34

❉❡❝♦♠♣♦s✐t✐♦♥✳

▲❡t P ❜❡ ❛ ♣r♦♣❡rt② ✭❡✳❣✳ ❝♦st ♦❢ ❛ s✉❝❝❡ss❢✉❧ s❡❛r❝❤ ♦r ❜❧♦❝❦ ❧❡♥❣t❤✮✳ ▲❡t ♣❜✐✰❞✭q✮ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ P ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ❧❛st ❝❧✉st❡r✳ ❚❤❡♥ ♣♠❀♥✭q✮ ❂

❜✶

❳

❞❂✵

❳

✐✕✵

✥

♥ ❜✐ ✰ ❞

✦

◗♠✐✶❀♥❜✐❞❀✵ ❋❜✐✰❞ ♣❜✐✰❞✭q✮✿

slide-35

SLIDE 35

❉❡❝♦♠♣♦s✐t✐♦♥✳

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ P✭③❀ ✇❀ q✮ ✿❂

❳

♠❀♥✕✵

♣♠❀♥✭q✮ ✇❜♠ ③♥ ♥✦ ❂ ❫ ◆✵✭③❀ ✇❀ q✮ ✶ ◆✵✭③❀ ✇✮❀ ✇✐t❤ ❫ ◆✵✭③❀ ✇❀ q✮ ✿❂

❜✶

❳

❞❂✵

✇❜❞ ❳

✐✕✵

❋❜✐✰❞ ✭③✇✮❜✐✰❞ ✭❜✐ ✰ ❞✮✦ ♣❜✐✰❞✭q✮❀ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❞✐r❡❝t❧② ❞❡r✐✈❡❞ ✇✐t❤ t❤❡ s❡q✉❡♥❝❡ ❝♦♥str✉❝t✐♦♥✳ ❫ ◆✵✭③❀ ✇❀ ✶✮ ❂ ◆✵✭③❀ ✇✮✱ P✭③❀ ✇❀ ✶✮ ❂ ✄✵✭③❀ ✇✮ ✶✳ ▼♦r❡♦✈❡r✱ ❜② s✐♥❣✉❧❛r✐t② ❛♥❛❧②s✐s ♦❢ ✄✵✭☛❀ ②✶❂❜❡☛❀ q✮ ❛t ② ❂ ✶✱ ❧✐♠

♠✦✶ P♠❬◗♠❀♥❀✵❂♠♥❀ ❜☛❪ ❂ ❚✵✭❜☛✮ ❂

❜✭✶ ☛✮

◗❜✶

❥❂✶

✏

✶ ❚✭✦❥☛❡☛✮

☛

✑✿

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ❧✐♠

♠✦✶ P♠❬♣♠❀♥✭q✮❂♠♥❀ ❜☛❪

❂ ❚✵✭❜☛✮ ❫ ◆✵✭❜☛❀ ❡☛❀ q✮✿

slide-36

SLIDE 36

❙✉♠♠❛r② ♦❢ t❤❡ q✲❝❛❧❝✉❧✉s✳

❆❞❞✐♥❣ ❛♥ ❡❧❡♠❡♥t ✼✦

❘

❈♥ ❂ ❆♥✶ ❈ ❂ ❆❞❞✭❆✮ ❈✭③✮ ❂

❘ ③

✵ ❆✭✇✮❞✇

❈❤♦♦s✐♥❣ ❛ ♣♦s✐t✐♦♥ ✼✦ ❅ ❈♥ ❂ ✭♥ ✰ ✶✮❆♥ ❈ ❂ P♦s✭❆✮ ❈✭③✮ ❂

❅ ❅③✭③❆✭③✮✮

❆✈❡r❛❣✐♥❣ ✼✦ ✶

❩

❘

❈♥ ❂

❆♥ ♥✰✶

❈ ❂ ❆✈❡✭❆✮ ❈✭③✮ ❂ ✶

③

❘ ③

✵ ❆✭✇✮❞✇

❆❞❞✐♥❣ ❛ ❜✉❝❦❡t ✼✦ ❡①♣ ❈♥ ❂ ✶ ❈ ❂ ❇✉❝❦❡t✭❩✮ ❈✭③✮ ❂ ❡①♣✭③✮ ♥ ✼✦ ❬♥❪ ❂ ✶ ✰ q ✰ q✷ ✰ ✿ ✿ ✿ ✰ q♥✶ ❂ ✶ q♥ ✶ q ✿

❳

✭♥ ✰ ✶✮❢♥③♥ ✼✦

❳

❬♥ ✰ ✶❪❢♥③♥✿ ❅ ❅③ ✭③❆✭③✮✮ ✼✦ ❍❬❢✭③✮❪ ❂ ❋✭③✮ q❋✭q③✮ ✶ q ✿

slide-37

SLIDE 37

❚❤❡ ♦✈❡r✢♦✇ ✭♣❛r❦✐♥❣ ♣r♦❜❧❡♠✮✳

▲❡t ◆♠❀♥❀❦ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ t❛❜❧❡s ♦❢ ❧❡♥❣t❤ ♠ ✇✐t❤ ♥ ❡❧❡♠❡♥ts ❛♥❞ ♦✈❡r✢♦✇ ❦ ❛♥❞ ✡✭③❀ ✇❀ q✮ ✿❂

❳

♠✕✵

❳

♥✕✵

❳

❦✕✵

◆♠❀♥❀❦✇❜♠ ③♥ ♥✦ q❦✿

❚❤❡♦r❡♠ ✭❬❙❡✐t③ ❛♥❞ P❛♥❤♦❧③❡r ✷✵✵✾❪✮

✡✭❜③❀ ✇❀ q✮ ❂ ✶ q❜ ✇❜❡q❜③ ✁

◗❜✶

❥❂✵

✏

q ❚✭✦❥③✇✮

③

✑ ◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇✮

③

✑✿

Pr♦♦❢✳

❬❙❦❡t❝❤❪ ✡✭③❀ ✇❀ q✮ ❂ ✶ ✰ ✡✭③❀ ✇❀ q✮✇❜❡③q q❜ ✰

❜✶

❳

s❂✵

✭✶ qs❜✮❖s✭③❀ ✇✮❀ ✇✐t❤ ❖s✭③❀ ✇✮ ❂ ❋s✭③✇✮✇❜s ✶ ◆✵✭③❀ ✇✮✿

slide-38

SLIDE 38

P❛r❦✐♥❣✱ r❛♥❞♦♠ ❣r❛♣❤s ❛♥❞ r❛♥❞♦♠ tr❡❡s✳

❬❙♣❡♥❝❡r ✶✾✾✼❪✳ ❇❋❙ tr❛✈❡rs❛❧ ♦❢ ❛ r❛♥❞♦♠ ❣r❛♣❤ ✌ ✇✐t❤ ✈❡rt✐❝❡s ④✵✱ ✶✱ ✳ ✳ ✳ ✱ ♥⑥✳ ■♥❞✉❝❡s ❛ q✉❡✉❡ ✭❍✐✭✜✮✮✶✔✐✔♥ ❛♥❞ ❛ s♣❛♥♥✐♥❣ tr❡❡ ✜✳ ❇❋❙ ✐♥❞✉❝❡s ❛ ♣❛r❦✐♥❣ s❡q✉❡♥❝❡ ✭❳✐✭✜✮✮✶✔✐✔♥✳ ❊①✿ ✭❳✐✭✜✮✮ ❂ ❢❢✻❀ ✽❣❀ ❢✷❀ ✸❣❀ ✣❀ ❢✼❣❀ ❢✶❀ ✹❣❀ ❢✺❣❀ ❢✾❣❀ ✣❀ ✣❣✳ ①✐✭✜✮ ❂ ❥❳✐✭✜✮❥✳ ❊①✿ ✭①✐✭✜✮✮ ❂ ❢✷❀ ✷❀ ✵❀ ✶❀ ✷❀ ✶❀ ✶❀ ✵❀ ✵❣✳ ②✐✭✜✮ ❂ ①✶✭✜✮ ✰ ①✷✭✜✮ ✰ ✿ ✿ ✿ ✰ ①✐✭✜✮ ✐ ✰ ✶✱ s✐③❡ ♦❢ q✉❡✉❡ ✭❍✭✜✮✮ ❜❡❢♦r❡ st❡♣ ✐✳ ❊①✿ ✭②✐✭✜✮✮ ❂ ❢✷❀ ✸❀ ✷❀ ✷❀ ✸❀ ✸❀ ✸❀ ✷❀ ✶❣✳ ②✶✭✜✮ ✰ ✿ ✿ ✿ ②♥✭✜✮ ♥ ✐s t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t✳ ❊①✿ ✶✷✳ ❇❋❙ ✐♥❞✉❝❡s ❛ r❛♥❞♦♠ ✇❛❧❦ ❡①❝✉rs✐♦♥✳ ❊①✿ ❜✳

slide-39

SLIDE 39

■❞❡❛s ❛❜♦✉t t❤❡ ♣r♦❜❛❜✐❧✐st✐❝ ❛♣♣r♦❛❝❤✳

▲❡t ❳✐ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts t❤❛t ❤❛✈❡ ❤❛s❤ ❛❞❞r❡ss ✐✳ ▲❡t ❍✐ ❜❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts t❤❛t tr② ❜✉❝❦❡t ✐✳ ❧❡t ◗✐ ❜❡ t❤❡ ♦✈❡r✢♦✇ ❢r♦♠ ❜✉❝❦❡t ✐✳ ❲❡ t❤✉s ❤❛✈❡ t❤❡ ❡q✉❛t✐♦♥s ❍✐ ❂ ❳✐ ✰ ◗✐✶❀ ◗✐ ❂ ✭❍✐ ❜✮✰✿ ✭✶✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts st♦r❡❞ ✐♥ ❜✉❝❦❡t ✐ ✐s ❨✐ ❂ ♠✐♥✭❍✐❀ ❜✮✳ ❚❤❡ ❜✉❝❦❡t ✐s ❢✉❧❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❍✐ ✕ ❜✳

slide-40

SLIDE 40

❋✐♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡ ❤❛s❤ t❛❜❧❡s✳

▲❡♠♠❛

▲❡t ❳✐✱ ✐ ✷ T✱ ❜❡ ❣✐✈❡♥ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✳ ■❢ T ❂ ❢✶❀ ✿ ✿ ✿ ❀ ♠❣ ♦r N✱ t❤❡♥ t❤❡ ❡q✉❛t✐♦♥s ✭✶✮✱ ❢♦r ❛❧❧ ✐ ✷ T✱ ❤❛✈❡ ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❣✐✈❡♥ ❜②✱ ❝♦♥s✐❞❡r✐♥❣ ❥ ✕ ✵✱ ❍✐ ❂ ♠❛①

❥❁✐ ✐

❳

❦❂❥✰✶

✭❳❦ ❜✮ ✰ ❜❀ ◗✐ ❂ ♠❛①

❥✔✐ ✐

❳

❦❂❥✰✶

✭❳❦ ❜✮ ✭✷✮ Pr♦♣❡r❧② ❞❡✜♥❡❞✱ t❤❡ r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ✐♥✜♥✐t❡ t❛❜❧❡s✳

slide-41

SLIDE 41

❈♦♥✈❡r❣❡♥❝❡ t♦ ❛♥ ✐♥✜♥✐t❡ ❤❛s❤ t❛❜❧❡✳

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❤❛s❤✐♥❣ ♦♥ ❩♠ ✇✐t❤ ♥ ❡❧❡♠❡♥ts ❤❛✈✐♥❣ ✐♥❞❡♣❡♥❞❡♥t ✉♥✐❢♦r♠❧② r❛♥❞♦♠ ❤❛s❤ ❛❞❞r❡ss❡s✳ ❳✶❀ ✿ ✿ ✿ ❀ ❳♠ ❤❛✈❡ ❛ ♠✉❧t✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ♥ ❛♥❞ ✭✶❂♠❀ ✿ ✿ ✿ ❀ ✶❂♠✮✳ ✭❲❡ ❞❡♥♦t❡ t❤❡s❡ ❳✐ ❜② ❳♠❀♥❀✐✳✮ ❲❡ ❞❡♥♦t❡ t❤❡ ♣r♦✜❧❡ ♦❢ t❤✐s ❤❛s❤ t❛❜❧❡ ❜② ❍♠❀♥❀✐✱ ✇❤❡r❡ ❛s ❛❜♦✈❡ ✐ ✷ ❩♠ ❜✉t ✇❡ ❛❧s♦ ❝❛♥ ❛❧❧♦✇ ✐ ✷ Z ✐♥ t❤❡ ♦❜✈✐♦✉s ✇❛②✳ ❲❡ ❝♦♥s✐❞❡r ❛ ❧✐♠✐t ✇✐t❤ ♠❀ ♥ ✦ ✶ ❛♥❞ ♥❂❜♠ ✦ ☛ ✷ ✭✵❀ ✶✮✳ ❚❤❡ ❛♣♣r♦♣r✐❛t❡ ❧✐♠✐t ♦❜❥❡❝t ✐s ❛♥ ✐♥✜♥✐t❡ ❤❛s❤ t❛❜❧❡ ♦♥ Z ✇✐t❤ ❳✐ ❂ ❳☛❀✐ t❤❛t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✭✐✳✐✳❞✳✮ ✇✐t❤ t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ❳✐ ✘ P♦✭☛❜✮✳ ❲❡ ❞❡♥♦t❡ t❤❡ ♣r♦✜❧❡ ♦❢ t❤✐s ❤❛s❤ t❛❜❧❡ ❜② ❍☛❀✐✳

▲❡♠♠❛

▲❡t ♠❀ ♥ ✦ ✶ ✇✐t❤ ♥❂❜♠ ✦ ☛ ❢♦r s♦♠❡ ☛ ✇✐t❤ ✵ ❁ ☛ ❁ ✶✳ ❚❤❡♥ ✭❍♠❀♥❀✐✮✶

✐❂✶ ❞

✦ ✭❍☛❀✐✮✶

✐❂✶✳

slide-42

SLIDE 42

❚❤❡ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✳

❚❤❡♦r❡♠

▲❡t ✵ ❁ ☛ ❁ ✶✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✥❍✭③✮ ❛♥❞ ✥◗✭③✮ ❡①t❡♥❞ t♦ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❣✐✈❡♥ ❜② ✥❍✭③✮ ❂ ❜✭✶ ☛✮✭③ ✶✮ ③❜❡☛❜✭✶③✮ ✶

◗❜✶

❵❂✶

③ ❚ ✦❵☛❡☛✁❂☛ ✁ ◗❜✶

❵❂✶

✶ ❚ ✦❵☛❡☛✁❂☛ ✁❀

✭✸✮ ✥◗✭③✮ ❂ ❜✭✶ ☛✮✭③ ✶✮ ③❜ ❡☛❜✭③✶✮

◗❜✶

❵❂✶

③ ❚ ✦❵☛❡☛✁❂☛ ✁ ◗❜✶

❵❂✶

✶ ❚ ✦❵☛❡☛✁❂☛ ✁✿

✭✹✮

slide-43

SLIDE 43

❚❤❡ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✳

❈♦r♦❧❧❛r②

❋♦r ❦ ❂ ✵❀ ✿ ✿ ✿ ❀ ❜ ✶✱ Pr✭❨✐ ❂ ❦✮ ❂ Pr✭❍✐ ❂ ❦✮ ❂ ❜✭✶ ☛✮❬③❦❪ ◗❜✶

❵❂✵

③ ❚ ✦❵☛❡☛✁❂☛ ✁ ◗❜✶

❵❂✶

✶ ❚ ✦❵☛❡☛✁❂☛ ✁

✿ ✭✺✮ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ❜✉❝❦❡t ✐s ♥♦t ❢✉❧❧ ✐s ❣✐✈❡♥ ❜② Pr✭❨✐ ❁ ❜✮ ❂ Pr✭❍✐ ❁ ❜✮ ❂ ❚✵✭❜☛✮ ❂ ❜✭✶ ☛✮

◗❜✶

❵❂✶

✶ ❚ ✦❵☛❡☛✁❂☛ ✁

✭✻✮ ❛♥❞ t❤✉s Pr✭❨✐ ❂ ❜✮ ❂ Pr✭❍✐ ✕ ❜✮ ❂ ✶ ❚✵✭❜☛✮✿ ✭✼✮

slide-44

SLIDE 44

❘♦❜✐♥ ❍♦♦❞✿ ❛♥ ❡①❛♠♣❧❡ ✭❜❂✷✮✳

❑❡②s ✐♥s❡rt❡❞✿ ✸✻✱ ✼✼✱ ✷✹✱ ✼✾✱ ✺✻✱ ✻✾✱ ✹✾✱ ✶✽✱ ✸✽✱ ✾✼✱ ✼✽✱ ✶✵✱ ✺✽✳ ❍❛s❤ ❢✉♥❝t✐♦♥ ❤✭①✮ ❂ ① ♠♦❞ ✶✵✳

❛ ✹✾ ✼✾ ✷✹ ✸✻ ✼✼ ✶✽ ✺✽ ✻✾ ✶✵ ✺✻ ✾✼ ✸✽ ✼✽ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✷✾ ✐s ✐♥s❡rt❡❞❄

❛ ✷✾ ✻✾ ✶✵ ✷✹ ✸✻ ✼✼ ✶✽ ✺✽ ✹✾ ✼✾ ✺✻ ✾✼ ✸✽ ✼✽ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾

slide-45

SLIDE 45

Pr♦♣❡rt✐❡s ♦❢ ❘♦❜✐♥ ❍♦♦❞ ❍❛s❤✐♥❣✳

❛ ✷✾ ✻✾ ✶✵ ✷✹ ✸✻ ✼✼ ✶✽ ✺✽ ✹✾ ✼✾ ✺✻ ✾✼ ✸✽ ✼✽ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾

❆t ❧❡❛st ♦♥❡ r❡❝♦r❞ ✐s ✐♥ ✐ts ❤♦♠❡ ❜✉❝❦❡t✳ ❚❤❡ ❦❡②s ❛r❡ st♦r❡❞ ✐♥ ♥♦♥❞❡❝r❡❛s✐♥❣ ♦r❞❡r ❜② ❤❛s❤ ✈❛❧✉❡✱ st❛rt✐♥❣ ❛t s♦♠❡ ❧♦❝❛t✐♦♥ ❦ ❛♥❞ ✇r❛♣♣✐♥❣ ❛r♦✉♥❞✳ ■♥ ♦✉r ❡①❛♠♣❧❡✱ ❦ ❂ ✺ ✭t❤❡ ✜rst s❧♦t ♦❢ t❤❡ t❤✐r❞ ❜✉❝❦❡t✮✳ ■❢ ❛ ✜①❡❞ r✉❧❡ ✐s ✉s❡❞ t♦ ❜r❡❛❦ t✐❡s ❛♠♦♥❣ t❤❡ ❝❛♥❞✐❞❛t❡s t♦ ♣r♦❜❡ t❤❡✐r ♥❡①t ♣r♦❜❡ ❜✉❝❦❡t ✭❡❣✿ ❜② s♦rt✐♥❣ t❤❡s❡ ❦❡②s ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✮✱ t❤❡♥ t❤❡ r❡s✉❧t✐♥❣ t❛❜❧❡ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ r❡❝♦r❞s ✇❡r❡ ✐♥s❡rt❡❞✳ ❚❤❡♥✱ ✇❡ ♠❛② ✐♥s❡rt t❤❡ ❡❧❡♠❡♥ts ✐♥ ❛♥② ♦r❞❡r✱ ❛♥❞ st✉❞② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❧❛st ♦♥❡ ✐♥s❡rt❡❞✦✳

slide-46

SLIDE 46

❘♦❜✐♥ ❍♦♦❞ ❞✐s♣❧❛❝❡♠❡♥t✳

❲✳❧✳♦✳❣✳ ✇❡ s❡❛r❝❤ ❢♦r ❛ r❡❝♦r❞ t❤❛t ❤❛s❤❡s t♦ ❜✉❝❦❡t ✵✳ ❲❡ ❤❛✈❡ t♦ ♣r♦❜❡ ❜✉❝❦❡ts ♦❝❝✉♣✐❡❞ ❜② t❤❡ ❡❧❡♠❡♥ts t❤❛t ✇♦✉❧❞ ❤❛✈❡ ❣♦♥❡ t♦ t❤❡ ♦✈❡r✢♦✇ ❛r❡❛✳ ❚❤❡♥ ❝♦♥s✐❞❡r ❝♦❧❧✐s✐♦♥s ✇✐t❤ ❛❧❧ t❤❡ ❡❧❡♠❡♥ts t❤❛t ❤❛s❤ t♦ ✵✳ ▲❡t ❉RH ❜❡ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❛ ❣✐✈❡♥ ❡❧❡♠❡♥t ①✳ ▲❡t ❈RH ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts t❤❛t ✇✐♥ ♦✈❡r ① ✐♥ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❢♦r s❧♦ts ✐♥ t❤❡ ❜✉❝❦❡ts✳ ❚❤❡♥ ❉RH ❂ ❜❈RH❂❜❝✳ ❚❤❡ s♣❡❝✐✜❝❛t✐♦♥ ✐s ❈RH✭❜③❀ ✇❀ q✮ ❂ ✡✭❜③❀ ✇❀ q✮ ❆✈❡✭P♦s✭❇✉❝❦❡t✭❜③❀ ✇❀ q✮✮✮ ❂ ✡✭❜③❀ ✇❀ q✮✇❜ ❡❜③ ❡q❜③ ❜③✭✶ q✮ ❂ ✭✇❡③✮❜✭✶ ❡❜③✭q✶✮✮ ❜③✭✶ q✮✭q❜ ✇❡q③✮

◗❜✶

❥❂✵

✏

q ❚✭✦❥③✇✮

③

✑ ◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇✮

③

✑✿

slide-47

SLIDE 47

❘♦❜✐♥❣ ❍♦♦❞ ❞✐s♣❧❛❝❡♠❡♥t✳

❲❡ ❤❛✈❡ ❈RH ❂ ◗✶ ✰ ❱ ✱ ✇❤❡r❡ ◗✶ ❛♥❞ ❱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ❆ s✐♠♣❧❡ ❝❛❧❝✉❧❛t✐♦♥ s❤♦✇s t❤❛t ❱ ❤❛s ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✥❱ ✭q✮ ❂

✶ ❡❜☛✭q✶✮✁❂❜☛✭✶ q✮✳ ❚❤❡♥

✥❈✭q✮ ❂ ✥◗✭q✮✥❱ ✭q✮ ❂ ✶ ☛ ☛ ✶ ❡❜☛✭q✶✮ ❡❜☛✭q✶✮ q❜

◗❜✶

❵❂✶

q ❚ ✦❵☛❡☛✁❂☛ ✁ ◗❜✶

❵❂✶

✶ ❚ ✦❵☛❡☛✁❂☛ ✁✿

✭✽✮ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ❉RH ❂ ❜❈RH❂❜❝ t❤❡♥ ❡q✉❛❧s✱ ✥RH✭q✮ ❂ ✶ ❜

❜✶

❳

❥❂✵

✥❈

✦❥q✶❂❜✁

✶ q✶ ✶ ✦❥q✶❂❜ ✿ ✭✾✮

slide-48

SLIDE 48

❇❧♦❝❦ ❧❡♥❣t❤✳

▲❡t ❋❜✐✰❞ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t♦ ❝♦♥str✉❝t ❛♥ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡ ♦❢ ❧❡♥❣t❤ ✐ ✰ ✶ ❛♥❞ s✐③❡ ❜✐ ✰ ❞ ✇✐t❤ ✵ ✔ ❞ ✔ ❜ ✶✳ ❚❤❡♥✱ ❋❞✭✉✮ ✿❂

❳

✐✕✵

❋❜✐✰❞ ✉❜✐✰❞ ✭❜✐ ✰ ❞✮✦❀ ◆✵✭③❀ ✇✮ ✿❂

❜✶

❳

❞❂✵

✇❜❞❋❞✭③✇✮✿ ■♥ ❛♥ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❜❧♦❝❦ ✐s ♠❛r❦❡❞ ❜② ✇ ✐♥ ◆✵✭❜③❀ ✇✮✳ ❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❇✭③❀ ✇❀ q✮ ❢♦r t❤❡ ❜❧♦❝❦ ❧❡♥❣t❤ ✐s ❇✭❜③❀ ✇❀ q✮ ❂ ✄✵✭❜③❀ ✇✮◆✵✭❜③❀ ✇q✶❂❜✮ ❂ ✶ ◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇q✶❂❜✮

③

✑ ◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇✮

③

✑

✿

slide-49

SLIDE 49

❇❧♦❝❦ ❧❡♥❣t❤✳

▲❡t ❇ ❜❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✜rst ❜❧♦❝❦✱ ✐✳❡✳✱ ❇ ✿❂ ♠✐♥❢✐ ✕ ✶ ✿ ❨✐ ❁ ❜❣ ❂ ♠✐♥❢✐ ✕ ✶ ✿ ❍✐ ❁ ❜❣✿ ✭✶✵✮ ❍❡♥❝❡✱ ❇ ✐s t❤❡ ✜rst ♣♦s✐t✐✈❡ ✐♥❞❡① ✐ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ❙✐ ❂ ❳✶ ✰ ✁ ✁ ✁ ✰ ❳✐ ❤❛s❤❡❞ t♦ t❤❡ ✐ ✜rst ❜✉❝❦❡ts ✐s ❧❡ss t❤❛♥ t❤❡ ❝❛♣❛❝✐t② ❜✐ ♦❢ t❤❡s❡ ❜✉❝❦❡ts✱ ✐✳❡✳✱ ❇ ❂ ♠✐♥❢✐ ✕ ✶ ✿ ❙✐ ❁ ❜✐❣✿ ✭✶✶✮

❚❤❡♦r❡♠

❚❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✥❇✭③✮ ✿❂ E ③❇ ♦❢ ❇ ✐s ❣✐✈❡♥ ❜② ✥❇✭③✮ ❂ ✶

❜✶

❨

❵❂✵

✏

✶ ❚

✦❵☛❡☛③✶❂❜✁❂☛ ✑

✿ ✭✶✷✮

slide-50

SLIDE 50

❯♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤✳

■♥ ❛ ❝❧✉st❡r ✇✐t❤ ♥ ❦❡②s✱ t❤❡ ♥✉♠❜❡r ♦❢ ✈✐s✐t❡❞ ❜✉❝❦❡ts ✐♥ ❛ ✉♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤✱ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♦♥❡ ♥❡❡❞❡❞ t♦ ✐♥s❡rt t❤❡ ✭♥ ✰ ✶✮st ❡❧❡♠❡♥t✳ ❚❤❡♥✱ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ P♦s✭◆✵✮ ✭♠❛r❦✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤✐s ✐♥s❡rt❡❞ ❡❧❡♠❡♥t✮ ❧❡❛❞s t♦ ❯✭❜③❀ ✇❀ q✮ ❂

❳

♠✕✶

✇❜♠ ❳

♥✕✵

✭❜♠③✮♥ ♥✦ P♠❀♥✭q✮ ❂ ✄✵✭❜③❀ ✇✮◆✵✭❜③❀ ✇✮ ◆✵✭❜③❀ ✇q✶❂❜✮ ✶ q ❂

◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇q✶❂❜✮

③

✑

◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇✮

③

✑

✭✶ q✮ ◗❜✶

❥❂✵

✏

✶ ❚✭✦❥③✇✮

③

✑

❀ ✇❤❡r❡ P♠❀♥✭q✮ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦❢ t❤❡ ✭♥ ✰ ✶✮st ✐♥s❡rt❡❞ ❡❧❡♠❡♥t✳

slide-51

SLIDE 51

❋❈❋❙ ❞✐s♣❧❛❝❡♠❡♥t✳

❚❤❡ ♣❣❢ ❢♦r t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t ✇❤❡♥ ❤❛✈✐♥❣ ♥ ✰ ✶ ❡❧❡♠❡♥ts ✐s ❋❈♠❀♥✭q✮ ✿❂

P♥

✐❂✵ P♠❀✐✭q✮

♥✰✶

✳ ❲❡ ♥❡❡❞ t❤❡♥ ❛ tr❛♥s❢♦r♠ ✇❜♠ ✭❜♠③✮♥

♥✦

P♠❀♥✭q✮ ✼✦ ✇❜♠③♥

P♥

✐❂✵ P♠❀✐✭q✮

♥✰✶

✳ ❚❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❧❡❛❞s t♦ t❤❡ ♦r❞✐♥❛r② ❣❢

❩ ✶

✵

❯✭❜②t❀ ✇❡t❀ q✮ ❞t ❂

❳

♠✕✶

✇❜♠ ❜♠

❳

♥✕✵

②♥P♠❀♥✭q✮✿ ❲❡ t❤❡♥ ❤❛✈❡ t❤❡ ♦r❞✐♥❛r② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❋❈❋❙✭❜③❀ ✇❀ q✮ ❂

❳

♠✕✶

✇❜♠ ❳

♥✕✵

❋❈♠❀♥✭q✮③♥ ❂ ✇❅✇ ③

❩ ③

✵

✒❩ ✶

✵

❯✭❜②t❀ ✇❡t❀ q✮ ❞t

✓

❞② ✶ ②✿

slide-52

SLIDE 52

❙♣❡❝✐✜❝❛t✐♦♥ ❢♦r ❋❈❋❙ ❞✐s♣❧❛❝❡♠❡♥t ✭❜❂✶✮✳

▲❡t ❋❈✭③❀ ✇❀ q✮ ❜❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❝♦st ♦❢ ❛ s✉❝❝❡ss❢✉❧ s❡❛r❝❤ ✐♥ ❛♥ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡ ✇❤❡♥ ♥ ✰ ✶ ❡❧❡♠❡♥ts ❛r❡ ✐♥s❡rt❡❞ ❛♥❞ ♦♥❡ ❡❧❡♠❡♥t ✎ ✐s ♠❛r❦❡❞✳ ❆❋✭③✮ ❂ ❚✭③✮

③

✇❤❡r❡ ❚✭③✮ ✐s t❤❡ tr❡❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥✱ ❁❋❈❃ ❂ ❁❆❞❞✭❆❋✮❃✯❁❆❋❃ ✰ ❁P♦s✭❋❈✮❃✯❁❆❋❃ ✰ ❁P♦s✭❆❋✮❃✯❁❋❈❃ ❧❡❛❞s t♦ ❅③✭❋❈✮ ❂ ❍❬❆❋❪ ✄ ❆❋ ✰ ❅ ❅③ ✭③❋❈✮ ✄ ❆❋ ✰ ❅ ❅③ ✭③❆❋✮ ✄ ❋❈✿ ❋❈❋❙✭③❀ ✇❀ q✮ ❂ ✭✭✶ ❚✭③✇q✮✮✷ ✭✶ ❚✭③✇✮✮✷✮❚✭③✇✮ ✷③✭✶ q✮✭✶ ❚✭③✇✮✮ ✿

slide-53

SLIDE 53

❯♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤ ❛♥❞ ❋❈❋❙ ❞✐s♣❧❛❝❡♠❡♥t✳

❚❤❡♦r❡♠

❚❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✥❯✭③✮ ✿❂ E ③❯✐ ♦❢ ❯✐ ✐s ❣✐✈❡♥ ❜② ✥❯✭③✮ ❂ ❚✵✭❜☛✮ ✶ ③

❜✶

❨

❵❂✵

✏

✶ ❚

✦❵☛❡☛③✶❂❜✁❂☛ ✑

✿ ✭✶✸✮

❚❤❡♦r❡♠

❚❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✥FC✭③✮ ✿❂ E ③❉FC

✐

♦❢ ❉FC

✐

✐s ❣✐✈❡♥ ❜② ✥FC✭③❀ ☛✮ ❂ ✶ ☛

❩ ☛

✵

✥❯✭③❀ ☞✮ ❞☞ ❂ ✶ ☛

❩ ☛

✵

✜✭☞✮ ✶ ③

❜✶

❨

❵❂✵

✶ ✏❵✭③❀ ☞✮ ✁ ❞☞

❂ ✶ ☛

❩ ☛

✵

❜✭✶ ☞✮ ◗❜✶

❵❂✵

✶ ✏❵✭③❀ ☞✮ ✁

✭✶ ③✮ ◗❜✶

❵❂✶✭✶ ✏❵✭✶❀ ☞✮✮

❞☞✿ ✭✶✹✮

slide-54

SLIDE 54

❙♦♠❡ ✜♥❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s✳

Pr♦❜❧❡♠ ✇✐t❤ ❛ ✈❡r② r✐❝❤ ❤✐st♦r②✳ P❛r❛❞✐❣♠ ♦❢ ❛ ♣r♦❜❧❡♠ t❤❛t ♥✐❝❡❧② ✐♥t❡❣r❛t❡s ❛♥❛❧②t✐❝❛❧✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♥❞ ♣r♦❜❛❜✐❧✐st✐❝ ❛♣♣r♦❛❝❤❡s✳ ❚❤✐s ✐♥t❡❣r❛t✐♦♥ ✭t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✉s❡ ♦❢ s②♠❜♦❧✐❝ ♠❡t❤♦❞s✦✮ ❤❛s ❛❧❧♦✇❡❞ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❞❡❡♣ r❡❧❛t✐♦♥s ✇✐t❤ ♦t❤❡r ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠✳ ❆ ✉♥✐✜❡❞ ❛♥❛❧②s✐s ♦❢ s❡✈❡r❛❧ ✐♠♣♦rt❛♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s r❡❧❛t❡❞ ✇✐t❤ ❧✐♥❡❛r ♣r♦❜✐♥❣✿ s②♠❜♦❧✐❝ ♠❡t❤♦❞s ✰ r❛♥❞♦♠ ✇❛❧❦s ✭❧✐♥❦❡❞ ❜② t❤❡ P♦✐ss♦♥ ❚r❛♥s❢♦r♠✮✳ ❙♦♠❡ ♦♥❣♦✐♥❣ ✇♦r❦✳ ❚♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✇✐t❤ ❜✉❝❦❡ts✳ ❘❡❧❛t✐♦♥ ✇✐t❤ ♦t❤❡r ♣r♦❜❧❡♠s ❛s ❢♦r ❜❂✶❄ ◆✉♠❜❡r ♦❢ ♠♦✈❡♠❡♥ts ✐♥ ❞❡❧❡t✐♦♥ ❛❧❣♦r✐t❤♠✳