˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/l0metricspaces.pdf˜nd4 m dn ˆ ˛ ˆqˆq8$85m 8...

44

Upload: others

Post on 29-Sep-2020

3 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

Page 2: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

!!"! #

$# % & ' (

)# ** & ' (

+#

) ""$,||$-)|| ./'||$-+||.&

0 "!1222

Page 3: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

2* 3,"""4#

||$||* .Σ $*

$ .* $5 .*

$ ≠ * $5 .

* ,2-1!

"" 6*1 67 2

Page 4: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

*

||$-)||*.898 $ ≠ ):8

,2" ! 22

6 5;

Page 5: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

55" 2*

< = !"2

< "= 2,"" "

< "3,4

< "= "!"!"

< ,2

Page 6: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

* !"! !"", 37!""

554#

>3?4 . , ""152 ?@ "

>3$4.>3$4>3$ 4A>3$4

B||$= )||* .C ||>3$4= >3)4|| @5,"D

Page 7: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

< B," "= 2"! !","

< 2"! 5 ""32 ""5 4 @, ""

< EF5"# 2,,", 25"

< B,"27"5 " 2

Page 8: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

< " !

< EF5"G 5 2 ,!","

< G 2 "",53?4

< 3?4 ? 3""14!

Page 9: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

!

!! , A5 "2 ,,"""5 3 42 !

> 55F3 4#

= * 2! #||||*||,||* A

= * 22 25#||= ,||*||= ||* A

= >12,#||H ,H ||*= I, ,#||H ,= H A||*

Page 10: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

"#< 55""52234

< B||||* ,2 @ . 6 1 2

< 6* . ||||* 2? 5!2""53?4

< J2?@ "" ,!= 2 5||= ,||*

< "26* 36"?" K(J-)2 L ! * ,,B5 * 4, @"!5

Page 11: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

$

< "","5? 2! ! 2555" ,

< 0 ""5? ""5

< 0 "" , !"22

< 6 ?* 2"! 5,",A

Page 12: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

%

55,"!"2! M J

||||* .Σ 88* ≤ Σ 88

5 ≤ Σ J5 88*≤ J5 ||||*

J5 .3H ε4

||||* ≤ ||||55 ≤ 3H ε4||||*5.ε N"J"" 55F2* , 5 = 5 1*

Page 13: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

$

$ ,!,", ,", ,",!55

$H $ H A $ .||3 A4||5$

2$ A $ ," ,"55

,," 5

,",F ,"2""5*M 5≤

,

Page 14: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

$

FAF ,!" 5"2,", 55

2""""*

" .< F" .F"H F" H A F"

0 3?4 !5 #

" " H ?OF"2".

Page 15: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

# %

> 55F*

5||P||*. 9885 88

5:

0 " ,||||5$

8"85 ,3||||558$854

.||||55 38$854

.(Nε "Nδ 5,5,,"5"!"22

Page 16: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

& "

,2,Q8$85M R.C -ε

2Q8$85M 8"85N||||55RM C - ε

8"85N||||55M "N

!"M Q,!R

)" .* 28"85N||||55 M "

)" 5Q) .R.C H ε E3Σ )"4.3C H ε4

J! , Σ)"M N

Page 17: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

"'

I55"22,# QΣ )" M N R

N .N 3C H ε4N3C H ε4.E3Σ )"4N3H ε4

3= ε4E3Σ )"4

QΣ )" M N RM F53-3C H ε4ε N 4

.F53-("Nδ 3C H ε4N 4

M F53-S "Nδ4M δN

Page 18: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

QQ8$85M 8"85RM C - ε RM δN

"δN ,23H ε4

0 63F4 2 "! , 2 28$85#

Q638?854∈ QC - εC H εRRT -δ

Q8?85 ∈ Q6-3C - ε)6-3C H ε)RRT -δ

Q8?8∈Q6-3C 43=U 3ε446-3C 43H U 3ε44RRT -δ

2!!26 ,

Page 19: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

"(

Q3-ε48$85 ≤ 8"85N||||55

≤ 3H ε) 8$85RT -δ

U !""5,," 3V ε4 ≥ - δ.U 3Nε "Nδ4 5

JA!"2," ,

""F" .U 34

Page 20: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

! $

5θ 2!,"Q*RQ-ππR

, "" W%#

,"3θ54 , ,", 55

pp

p rpp

prstable

−−=

1

/1 ln))1(cos(

cossin

),,(θ

θθθ

Page 21: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

)*

< F" ,2,", 55

< F" ,!

!"2,",222,

>555" 5-,θ 2 2"

I22 """5

Page 22: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

! +

6"X" "2ε

3-ε4||||* ≤ 38"854N8$85 ≤ 3H ε4||||*

5,,"-δ

5 ""#* 2U 3Nε "Nδ4

B55 U 3Nε "Nδ4

Page 23: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

" * +

!"#

#

"$$

5"5" 7 """5

Page 24: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

&*

J"2 ""!2 5#

3H 4. 34H 34

3 - 4. 34- 34

,"25 55F||= ,||* ||H ,||* 3 V ε 5,,"-δ4

Page 25: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

& ,$*

< 5," !"2,",

< >"2 !"25" , !,

< 6,2"5" 52, """ ,,",

< "2 , Y2 Z2," , 55

Page 26: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

* -

< EF ""# , 2 2," , 55 266,

- 63F4H 63-F4.F-5 U 34

< >.>23-4).3>4O8>8-N5

< B) 2 2$354

< X5" $354 ) "55

Page 27: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

&

< I 55

< U 52 6"?"- 5,," 2 " 25

< 0 22!"" 5!

< U "5 ""2,"2"F,"

Page 28: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

+**

< [ 55" 2* 7"!!"5,"

< 2"

< X 5,"!"* 5""5

Page 29: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

* $

! """

2"5""2!",2 F ","2

EF5"#

= 2 " 222!"

= , " 222

= "5"2"5"5

Page 30: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

* $

< "2"5"" 5" 2

< .3Q?R4 Q?R!"2,? "

< I#Q?R ,,5""

< @? F5" 52!Q?R 5

< [,2"2""F5" "

Page 31: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

%

< B Y 2" Z222"

< 234. Σ F? 9Q?R:

< I2 2" 2F2"!"

< "2 255-!"52"

Page 32: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

%

< F5,""12

< I55"2 "55" " "

< B 2 ,"!

Page 33: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

%

< 55 Q?R *

< "2$?

34.8>? $?8

B ,"! "222"5"

",2 ,2

,"12,Q?RD

Page 34: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

+**.

3H ε43H ε4

3H ε4(

3H ε4'

3H ε4&

Page 35: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

+**.

3H ε43H ε4

3H ε4(

3H ε4'

3H ε4&

Page 36: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

+**.

3H ε43H ε4

3H ε4(

3H ε4'

3H ε4&

3H ε4&-3H ε4'

OQ3H ε4'-3H ε4(R

(OQ3H ε4(-3H ε4 R

'OQ3H ε4 -3H ε4R

'O3H ε4

Page 37: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

$*"

< "H ε 3F !" N !"4 "" .U 3"34Nε4

< 5 7U 35"-"34Nε 4 5 F 7

< B"5 U 35"-"34Nε(4

< , 5 5 Nε = ,,2* "

Page 38: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

+**. +

< " .* ""

< F ,!" 2,", 5 5.εN"

< 6!3Q?R4

1.1H Σ.Q?R F

< X55"5"""U 3Nε "Nδ4 5,,""-δ

3-ε434≤ 381854≤ 3H ε43438$854

Page 39: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

" *

< 5"Q?R!" 2!5

< J !" 2,",2,","

< >" ,2523H ε4 U 3"34Nε4!"

< 5U 3"34Nε(4

Page 40: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

#

< I55F V ε 5,,""-δ U 3Nε "3Nδ44

< B55U 3Nε( "3Nδ44

< ,"\, 22@22 2"= 5," ""

Page 41: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

< ["7#2 "5" ,"D

< B - #

34. Σ ? 9Q?R:

< $$ ,,29N :Q?R.⇔ ∈ $?

< B 34.8$∩ $ 8

< X7Ω34 5 55F!"" 5,,"!"5

Page 42: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

.

< U ,"22 = X"! ,"7"5

< I"7 5,"!2"5""D

< 0 7 2 ,"!D

Page 43: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

" & /

< Ω3Nε245 Q!0 22@"R,,D

< U 3N5"3ε44 U L2,5,"25

< [ 72 5!5U 34

< "26"?"- "!2

Page 44: ˘ ˇ ˆdimacs.rutgers.edu/~graham//pubs/slides/L0metricspaces.pdf˜Nd4 M dN ˆ ˛ ˆQˆQ8$85M 8 "85R M C -e R M dN ˛ " dN , 2 3˜He4 0 63F42 " ! , 2 28$85# ˆQ63 8 ?854 ˛ Q C -eˇ

*&

< B255" 2* " = 5 "" 2-5

< B255" 5!2,", 5 ,5,"

< B22 ",= 55F 5 " 7

< B2"