Lecture 12. Properties of Expanders
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1 Lecture 12. Properties of Expanders M2 Mitsuru Kusumoto Kyoto University 2013/10/29
2 Preliminalies G = (V, E) L G : A G : 0 = λ 1 λ 2 λ n : L G ψ 1,..., ψ n : L G µ 1 µ 2 µ n : A G ϕ 1,..., ϕ n : A G (Lecture note µ i α i... )
3 Preliminalies Expander Expander ε > 0 d- G µ i εd( i 2) G ε-expander
4 Overview : Expander Expander Expander Expander Vertex Expansion
5 Expander µ i εd( i 2)? ( ) H G x L H x x L G x( x) x L H x x L G x( x 1)
6 Expander G H ε- ε- (1 ε)h G (1 + ε)h. G d ε-expander Lemma H = d n K n G H ε-
7 Expander G d L G = di A G λ i = d µ i ε-expander : µ i εd( i 2) (1 ε)d λ i (1 + ε)d (i 2) Caurant-Fischer x 1 (1 ε)dx x x L G x (1 + ε)dx x
8 Expander G d L G = di A G λ i = d µ i ε-expander : µ i εd( i 2) (1 ε)d λ i (1 + ε)d (i 2) Caurant-Fischer x 1 (1 ε)dx x x L G x (1 + ε)dx x
9 Expander K n L Kn = ni 1 1 x 1 x L Kn x = nx x H = d n K n x L H x = dx x (1 ε)dx x x L G x (1 + ε)dx x G H ε-
10 Expander K n L Kn = ni 1 1 x 1 x L Kn x = nx x H = d n K n x L H x = dx x (1 ε)dx x x L G x (1 + ε)dx x G H ε-
11 Expander K n L Kn = ni 1 1 x 1 x L Kn x = nx x H = d n K n x L H x = dx x (1 ε)dx x x L G x (1 + ε)dx x G H ε-
12 Expander (1 ε)h G (1 + ε)h. εx T L H x x (L G L H )x εx T L H x εl H 0 d L G L H εd.
13 Expander (1 ε)h G (1 + ε)h. εx T L H x x (L G L H )x εx T L H x εl H 0 d L G L H εd.
14 Quasi-Randomness G H = d n K n ε- E(S, T ) := {(u, v) u S, v T, (u, v) E} Theorem G d ε-expander S V, T V S, T disjoint S = αn, T = βn E(S, T ) αβdn εdn (α α 2 )(β β 2 ). α, β > ε αβdn (S, T disjoint?)
15 Quasi-Randomness G H = d n K n ε- E(S, T ) := {(u, v) u S, v T, (u, v) E} Theorem G d ε-expander S V, T V S, T disjoint S = αn, T = βn E(S, T ) αβdn εdn (α α 2 )(β β 2 ). α, β > ε αβdn (S, T disjoint?)
16 Quasi-Randomness G H = d n K n ε- E(S, T ) := {(u, v) u S, v T, (u, v) E} Theorem G d ε-expander S V, T V S, T disjoint S = αn, T = βn E(S, T ) αβdn εdn (α α 2 )(β β 2 ). α, β > ε αβdn (S, T disjoint?)
17 Quasi-Randomness G H = d n K n ε- E(S, T ) := {(u, v) u S, v T, (u, v) E} Theorem G d ε-expander S V, T V S, T disjoint S = αn, T = βn E(S, T ) αβdn εdn (α α 2 )(β β 2 ). α, β > ε αβdn (S, T disjoint?)
18 Quasi-Randomness χ S L G χ T = d S T E(S, T ) G H L H χ S L H χ T = χ S (di d n 11T )χ T = d S T αβdn. E(S, T ) αβdn = χ S L G χ T χ S L H χ T.
19 Quasi-Randomness χ S L G χ T = d S T E(S, T ) G H L H χ S L H χ T = χ S (di d n 11T )χ T = d S T αβdn. E(S, T ) αβdn = χ S L G χ T χ S L H χ T.
20 Quasi-Randomness χ S L G χ T = d S T E(S, T ) G H L H χ S L H χ T = χ S (di d n 11T )χ T = d S T αβdn. E(S, T ) αβdn = χ S L G χ T χ S L H χ T.
21 Quasi-Randomness bound... χ S (L G L H )χ T χ S L G L H χ T αn (εd) βn = εdn αβ.
22 Quasi-Randomness bound... χ S (L G L H )χ T χ S L G L H χ T αn (εd) βn = εdn αβ.
23 Quasi-Randomness 1 χ S (L G L H )χ T = (χ S α1) (L G L H )(χ T β1). χ S α1 χ T β1 = n (α α 2 )(β β 2 ). E(S, T ) αβdn εdn (α α 2 )(β β 2 ). S, T disjoint d
24 Quasi-Randomness 1 χ S (L G L H )χ T = (χ S α1) (L G L H )(χ T β1). χ S α1 χ T β1 = n (α α 2 )(β β 2 ). E(S, T ) αβdn εdn (α α 2 )(β β 2 ). S, T disjoint d
25 Quasi-Randomness 1 χ S (L G L H )χ T = (χ S α1) (L G L H )(χ T β1). χ S α1 χ T β1 = n (α α 2 )(β β 2 ). E(S, T ) αβdn εdn (α α 2 )(β β 2 ). S, T disjoint d
26 Quasi-Randomness 1 χ S (L G L H )χ T = (χ S α1) (L G L H )(χ T β1). χ S α1 χ T β1 = n (α α 2 )(β β 2 ). E(S, T ) αβdn εdn (α α 2 )(β β 2 ). S, T disjoint d
27 Quasi-Randomness S V N(S) S S (closed neighbors) Expander S N(S) Theorem (Tanner 84) S = αn N(S) S ε 2 (1 α) + α.
28 Quasi-Randomness S V N(S) S S (closed neighbors) Expander S N(S) Theorem (Tanner 84) S = αn N(S) S ε 2 (1 α) + α.
29 Quasi-Randomness R := N(S), T = V \ R S T 1 T = βn S, T αβdn εdn (α α 2 )(β β 2 ). R = γn γ = 1 β (pdf ) N(S) S (open neighbors)
30 Quasi-Randomness R := N(S), T = V \ R S T 1 T = βn S, T αβdn εdn (α α 2 )(β β 2 ). R = γn γ = 1 β (pdf ) N(S) S (open neighbors)
31 Quasi-Randomness R := N(S), T = V \ R S T 1 T = βn S, T αβdn εdn (α α 2 )(β β 2 ). R = γn γ = 1 β (pdf ) N(S) S (open neighbors)
32 Quasi-Randomness R := N(S), T = V \ R S T 1 T = βn S, T αβdn εdn (α α 2 )(β β 2 ). R = γn γ = 1 β (pdf ) N(S) S (open neighbors)
33 Good expanders ε Expander ε? : S = v (singleton) d ε 2 (1 1/n) + 1/n ε = Ω(1/ d) ε 2 d 1 d
34 Good expanders ε Expander ε? : S = v (singleton) d ε 2 (1 1/n) + 1/n ε = Ω(1/ d) ε 2 d 1 d
35 Good expanders ε Expander ε? : S = v (singleton) d ε 2 (1 1/n) + 1/n ε = Ω(1/ d) ε 2 d 1 d
36 Good expanders ε Expander ε? : S = v (singleton) d ε 2 (1 1/n) + 1/n ε = Ω(1/ d) ε 2 d 1 d
37 Good expanders ( Lecture note ε upper-bound... ) (Expander ) Theorem (Nilli 91) G (u 0, u 1 ) (v 0, v 1 ) 2k + 2 { λ 2 d 2 d d 1 1 k+1 λ n d + 2 d 1 2 d 1 1 k+1 λ 2, λ n bound?
38 Good expanders ( Lecture note ε upper-bound... ) (Expander ) Theorem (Nilli 91) G (u 0, u 1 ) (v 0, v 1 ) 2k + 2 { λ 2 d 2 d d 1 1 k+1 λ n d + 2 d 1 2 d 1 1 k+1 λ 2, λ n bound?
39 Good expanders ( Lecture note ε upper-bound... ) (Expander ) Theorem (Nilli 91) G (u 0, u 1 ) (v 0, v 1 ) 2k + 2 { λ 2 d 2 d d 1 1 k+1 λ n d + 2 d 1 2 d 1 1 k+1 λ 2, λ n bound?
40 Good expanders : (λ 2 ) U i (u 0, u 1 ) i (i k) V i (v 0, v 1 ) i (i k) 1/(d 1) i/2 if a U i x(a) = β/(d 1) i/2 if a V i 0 otherwise β x 1 pdf+...
41 Good expanders : (λ 2 ) U i (u 0, u 1 ) i (i k) V i (v 0, v 1 ) i (i k) 1/(d 1) i/2 if a U i x(a) = β/(d 1) i/2 if a V i 0 otherwise β x 1 pdf+...
42 Good expanders : (λ 2 ) U i (u 0, u 1 ) i (i k) V i (v 0, v 1 ) i (i k) 1/(d 1) i/2 if a U i x(a) = β/(d 1) i/2 if a V i 0 otherwise β x 1 pdf+...
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79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arct
27 6 2 1 2 2 5 3 8 4 13 5 16 6 19 7 23 8 27 N Z = {, ±1, ±2,... }, R =, R + = [, + ), R = [, ], C =. a b = max{a, b}, a b = mi{a, b}, a a, a a. f : X R [a < f < b] = {x X; a < f(x) < b}. X [f] = [f ],
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
現代物理化学 1-1(4)16.ppt
(pdf) pdf pdf http://www1.doshisha.ac.jp/~bukka/lecture/index.html http://www.doshisha.ac.jp/ Duet -1-1-1 2-a. 1-1-2 EU E = K E + P E + U ΔE K E = 0P E ΔE = ΔU U U = εn ΔU ΔU = Q + W, du = d 'Q + d 'W
LLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
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1 / 74 ( ) 2019 3 8 URL: http://www.math.kyoto-u.ac.jp/ ichiro/ 2 / 74 Contents 1 Pearson 2 3 Doob h- 4 (I) 5 (II) 6 (III-1) - 7 (III-2-a) 8 (III-2-b) - 9 (III-3) Pearson 3 / 74 Pearson Definition 1 ρ
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
koji07-01.dvi
2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?
