A Classical Perspective on Non-Diffractive Disorder
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- Μαργαρίτες Γεωργιάδης
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1 A Classical Perspective on Non-Diffractive Disorder The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms of Use Klales, Anna A Classical Perspective on Non-Diffractive Disorder. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. July 29, :28:27 AM EDT This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at (Article begins on next page)
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5 0
6 r 5 r 5
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9 R = 10MΩ
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13 1 itp2d
14 6B;m`2 RXR, M `` v Q7 2B;2Mbi i2b rbi? +QMb2+miBp2 2M2`;B2b BM i?2 MQBbBv [mbmib+ TQi2MiB HX h?2 T`272``2/ Q`Bi2Mi ibqmb Q7 i?2 bi i2b + M #2 b22m BM i?bb `` vx k
15 Q
16 0, t
17 h (t)i t M δp t = M δp 0. δx t δx 0 ψ G (x, t) =exp{(i/ )[(x x t ) A t (x x t )+p t (x x t )+γ t ]}
18 fully resolved third recurrence second recurrence first recurrence f A t = 1 2 δp t (δx t ) 1 γ t = φ t i [ln δx t (δx 0 ) 1 ] φ t = p t dx t Et φ t t =0 ψ G (0) ψ G (t)
19 exp{ nτλ/2} λ M n n ϵ T (ω) = 1 2 π T T exp{iωt} ψ G ψ G (t) dt. λ
20 λ/ω. ω/λ 1 t
21 r r 5 r 5
22 φ =0 r 0 r 5 ψ r,m (ρ, φ) =R r,m (ρ)e imφ ψ r,m ψ r, m ψ r,m = R(r)e imφ ψ r, m = R(r)e imφ. V r,±m = ψ r,m V ψ r,m ψ r,m V ψ r, m. ψ r, m V ψ r,m ψ r, m V ψ r, m ψ + ψ 2 2 ψ + V ψ =0 ψ + cos φ ψ sin φ
23 ψ + = 1 2 (ψ r,m + ψ r, m ) ψ = 1 2 (ψ r,m ψ r, m ). ψ + cos φ ψ sin φ φ =0
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27 , r 5
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29 E :5 resonant set 3:7 resonant set r r m E ω. r 5
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31 H = H 0 + λv H 0 = 2 2µ 2 + r 5 a ψ m V ψ m+a = = e imφ Ve i(m+a)φ dφ e iaφ Vdφ. V a
32 j i i ψ i V ψ j
33 W = ψi 0 V ψ 0 j. ψ 0 i V φ i. W φ 1 = ϵ 1 φ 1 W φ 2 = ϵ 2 φ 2 W φ n = ϵ n φ n. ϵ 1 > ϵ 2 >... > ϵ n ψ = i a i φ i i a i 2 =1.
34 ψ W ψ = i = i = i = i φ j a i a jw φ i j a i a j φ j W φ i j a i a j φ j ϵ i φ i j a i a jϵ i φ j φ i j i = j ψ W ψ = i a i 2 ϵ i. φ i ψ W ψ ψ φ ϵ φ i V,
35 itp2d 1:π,
36 state 3673 state 3678 state state state state state state state state state state B;m`2 RXN, ai i2b + H+mH i2/ pb /2;2M2` i2 T2`im`# ibqm i?2q`v QM bk HH bm#b2i Q7 bi i2b rbi?bm `2bQM Mi b2ix k9
37 state 3673 state 3678 state state state state state state state state state state B;m`2 RXRy, ai i2b + H+mH i2/ U HKQbiV 2t +ihv pb /B ;QM HBx ibqm QM p2`v H `;2 bm#b2i Q7 # bbb bi i2bx k8
38 y x x x V
39 itp2d itp2d
40
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43 2
44 dp/dx
45 p / x
46 y p x x x
47
48
49 p n+1 = p n dv n(x) dx x n+1 = x n + p n+1 p x V n (x) x
50 V n (x) =0, V n (x)v n (x )=v 2 0e x x 2 /ξ 2. t c v 2/3 0
51 transverse position longitudinal position log( ) transverse position longitudinal position log( )
52 p x p x p x a p x p a x 2 V a x 2 a ( p x 1+ p ) 1 p a x a x. a 1 ( 1, 0]
53
54 1 0 δs = δp 2 + δx 2 = δp ( p 1+( x p) 2 = 2 δx 1+ x) δp δx p x p δs > x 0 q 1+2 p x +2 p S drift = q x 1+ p x 2 q 2 q 2 < p x < 0
55 S kick = 1+ p x + 2 V n 2 q dx 2 p 2 q x 1+ p x 2 q 2 V n q dx 2 q. αp = p βx = x p x = α β p 0 < x < 2 α β α = β =1
56 r(t) = log M(x 0,x t ) d(x 0 ) d(x 0 ) x 0 M(x 0,x t ) x 0 x t
57 p x log( ) v 0 = 10 6 ξ =0.1 x =0 x =1 λ c λ c v 0 λ c v 2/3 o p(x) =0
58 v 0 =2 20 v 0 =2 18 v 0 =2 16 v 0 =2 14 v 0 =2 12 r t/t c v 0 t c v 0 =2 10 v 0 =2 12 v 0 =2 14 v 0 =2 16 v 0 =2 18 v 0 = c λ c v 0
59 p n+1 = p n dv n(x) dx x n+1 = x n + p n+1 p x
60 V n (x)v n (x )=v 2 0e x x 2 /ξ 2 x v 0 ξ 2 c 0 v 0 ξ 2 m = mv m 2 V 1+m V m p x V 2 V x 2 m c V c 0 V m c = c 0 c c 0 2 c 0 m c = c 1/2 0. V m
61 V,m<<1 m =(mv m 2 V )(1 m + V ) m<<m c m 2 << V m = V mm c m 2 V m = m 2 V m>>m c m 2 >> V m = m 2 m c m n (n n 0 ) 1 n 0
62 L n+1 L n = 2 = 2 (( V 1 2 dx dp) dx2 dx 2 + dp 2 ( = 2 ( V dx ) dp dx dp 2 dx V 1 2 dp dx 1+ dp 2 dx ) ) 1 2 r = r n+1 r n =ln(l n+1 /dx 0 ) ln(l n /dx 0 ) = ln(l n+1 /L n ) r = ln ( 2 = 1 2 ln V 1 2 dp dx ( 2 1+ dp 2 dx V 1 2 dp dx 1+ dp 2 dx ) )
63 dp dx = m ( r = 1 ( 2 V 2 ln 1 2 m) ) m 2 ln(x) x 1 ( r 1 ( 2 V 1 2 m) ) m 2 1 ( 1 ( 2 V 1 2 m) ) m2 2 1+m 2 ( ( 2 V V mv 1 2 V m mv m + m m2) ) m 2 ( V V mv 1 2 V m mv m + m ) 1 2 m2 1+m 2 ( ) V 2 V + m 2mV m2 1+m 2 (1 + m 2 ) 1 1 m 2 r V 2 V + m 2mV m2 r m V
64 m m 1 t t c t c r ln(t t c ) r = m +ṁ m<<m crit α = (x dx 1,p dp) β = (x, p) γ = ((x + dx 2,p+ dp)) α = (x + p dx 1 dp V 1,p dp V 1) β = (x + p V,p V ) γ = (x + p + dp + dx 2,p+ dp V 2)
65 p (x + dx 2,p+ dp) (x dx 1,p dp) (x, p) x p (x + p dx 1 dp V1 0,p dp V 0 1 ) x (x + p + dp + dx 2,p+ dp V 0 2 ) (x + p V 0,p V 0 )
66 m 2 = dp dx 2 m = dp dx 1 c = dp dx 2 dp dx 1 dx 2 = m 2 m dx 2. c = m 2 m dx 2 c = ( ) ( ) dp V 2 +V dp+v dp+dx 2 V 2 +V 1 V dx 1 +dp+v 1 V dp + dx 2 V 2 + V = 1 dx 2 m 2 V 2 m 2 +1 V 2 m V 1+m V m 2 +1 V 2 c c = 1 dx 2 = 1 dx 2 ( ( ) ( ) m2 V 2 m 2 m V +1 V 2 1+m V m 2 +1 V 2 (m 2 m) m 2 m V 2 + V (m 2 +1 V 2 )2 (1 + m V ) m 2 + m ) c c 1 ( (m2 m V 2 + V )( 2m V 2 )(1 m + V ) dx 2 m 2 + m )
67 dx 2 (c c) V V 2 +4m 2 V 2 2m 2 2 mv 2 + mm 2 2V V 2 V V m 2 2mV + m 2 + V 2 t t =6
68 d periodic potential Intensity 5.0 y y time time B;m`2 kxrk, S2`BQ/B+ #` M+?2/ ~Qr Q7 K MB7QH/ Q7 +H bbb+ H i` D2+iQ`B2b 8e
69 6B;m`2 kxrj, S2`BQ/B+ #` M+?2/ ~Qr Q7 [m MimK K2+? MB+ H TH M2 r p2 7Q` p `B2iv Q7 `2T2 i H2M;i?bX 8d
70 6B;m`2 kxr9, +QKT `BbQM Q7 i?2 /2MbBiv Q7 +H bbb+ H i` D2+iQ`B2b UH27i T M2HV rbi? r p2t +F2i H mm+?2/ BM i?2 b K2 TQi2MiB H M/ KmHiBTHB2/ #v e ie0 t/! iq ;2i i?2 `2bmHiBM; [m MimK bi i2 rbi? 2M2`;v E0 U`B;?i T M2HVX h?2 v2hhqr ``Qr BM/B+ i2b #` M+? 2pB/2Mi BM i?2 +H bbb+ H /2MbBiv i? i Bb #b2mi BM i?2 +Q``2bTQM/BM; [m MimK bi i2x "Qi? T M2Hb b?qrm 7mHH irq@/bk2mbbqm H 2pQHmiBQM 83
71 e ie0t/ E 0 E 0. 0.
72 3
73 V I R R = 10MΩ V
74 R T = V I.
75 n m = πn/v F
76 Ω n 0 n (r) n n(r) =n 0 + n (r)+n (r). L = 1 2 m(r)ṙ2 L d L =0 q i dt q i [ ] 1 2 m(r)ṙq2 t + r (m(r)ṙq) = m(r)ṙ2 = m(r) r + ṙ (ṙ m(r))
77 m(r) = π v F n(r) r = [ ṙ2 1 ) (ˆṙ m(r) 2 m(r) m(r) ˆṙ]. r B = eṙb z /m(r) r = [ ṙ2 1 ) (ˆṙ m(r) 2 m(r) m(r) ˆṙ] + eṙb z /m(r).
78 ] [ 2 + ω2 c 2 n2 (r) ψ =0. ψ = e iωs(r)/c S(r) S S = n 2 (r). n(r) dr ds = S [ d n(r) dr ] ds ds = n(r). d/ds = ˆk dr/ds = ˆk ] [ˆk n(r) ˆk + n(r) [ˆk ˆk] = n(r). ˆk = n(r)v/c F 2n(r) 2 [v n(r)] v + n(r)[v v] =c 2 F n(r)
79 2 1 y x x x x x d dt = t + v v dv dt =(v )v d 2 r dt 2 = c2 F ( ) n(r) n(r) 3 2 ( v n(r) n(r) ) v. m(r) n(r).
80 1 1 y x n 3
81 2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) x(µm) x(µm) x(µm)
82 2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) x(µm) x(µm) x(µm)
83 2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) x(µm) x(µm) x(µm)
84 n 3 n 0 R = n 3 n 0 N N n tot = N i e l i/l l i i σ(ρ) = qa 2π (ρ 2 + a 2 ) 3/2 q a ρ 2ρ =2a 2 2/3 1 σ = q 2πa 2
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86 y(µm) x(µm) B =0.12T n =
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88 y(µm) y(µm) B = 0.00 T B = 0.06 T B = 0.07 T B = 0.08 T B = 0.09 T B = 0.10 T B = 0.11 T B = 0.12 T B = 0.13 T T/T y(µm) x(µm) x(µm) x(µm)
89 1.0 B = 0.00 T B = 0.06 T B = 0.07 T y(µm) y(µm) B = 0.08 T B = 0.09 T B = 0.10 T B = 0.11 T B = 0.12 T B = 0.13 T T/T y(µm) x(µm) x(µm) x(µm)
90 2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) x(µm) x(µm) x(µm) x =1µ y =0
91 B =0.10T,
92 2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) x(µm) x(µm) x(µm)
93
94 x
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96 4
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98 Ψ β (q, t) = dq G (q, q,t)ψ β (q, 0), G (q, q,t)= 1 2πi 2 S(q, q ) q q 1/2 [ ] exp is (q, q,t). S(q, q ) q 0 q t
99 ( ) ( ) S S S (q, q ) = S(q t,q 0 )+ (q q t )+ (q q 0 ) q t q 0 q 0 q t + 1 ( 2 ) S 2 qt 2 (q q t ) ( 2 ) S q 0 2 q0 2 (q q 0 ) 2 q t ( 2 ) S + (q q 0 )(q q t ) q 0 q t + 1 ( 3 ) S 6 q0 3 (q q 0 ) ( 3 ) S q t 6 qt 3 (q q t ) 3 q ( 3 ) S 2 q0 2 q (q q t )(q q 0 ) ( 3 ) S t 2 q 0 qt 2 (q q t ) 2 (q q 0 ). Ψ β (q, t) = 1 N 2πi ( 3 ) S + q 0 qt 2 dq 2 ( S 3 S + q 0 q t q0 2 q t 1/2 (q q t ) exp { i [ Ξq 3 +Υq 2 +Ωq +Λ ]} ) (q q 0 )
100 Ξ = 1 ( 3 ) S 6 q0 3, q [ t ( Υ = 1 2 ) ( S 3 ) ( S 3 ) ( S 3 ) ] S 2 q0 2 + q t q0 2 q q t q0 3 q 0 q t q0 2 q q t + A, t ( ) ( S 2 ) ( S 2 ) ( S 3 ) S Ω = + (q q t ) q 0 q t q 0 q t q0 2 q 0 + q t q0 2 q (q 0 q t qq 0 ) t + 1 ( 3 ) ( S 2 q0 3 q0 2 3 )( S 1 + q t q 0 qt 2 2 q2 t + 1 ) 2 q2 qq t 2Aq β + ξ, ( ) S Λ = q + 1 ( 2 ) S q t q 0 2 qt 2 q ( 3 ) ( ) S S q 0 6 qt 3 q 3 q 0 q 0 q 0 q t ( 2 ) S qq 0 1 ( 3 ) S q 0 q t 2 q 0 qt 2 q 2 q ( 2 ) S 2 q0 2 q0 2 q t + 1 ( 3 ) S 2 q0 2 q qq0 2 t 1 ( 3 ) ( ) ( S S 6 q0 3 q0 3 2 ) S q t q t q t q 0 qt 2 qq t 1 ( 3 ) S q 0 2 qt 3 q 2 q t q 0 ( 2 ) ( S 3 ) S + q 0 q t + q 0 q t q 0 qt 2 qq 0 q t 1 ( 3 ) S 2 q0 2 q q0q 2 t t ( 2 ) S qt 2 qt 2 q ( 3 ) S 2 qt 3 qqt 2 1 ( 3 S 2 q 0 qt 2 +Aq 2 β + S + γ ξq β. ) q 0 q 2 t 1 6 ( 3 ) S q 3 t q 0 q 3 t
101 φ Ψ 1 β (q, t) =N 2πi i dq A(q, t)exp{iφ} : φ = Ξq 3 +Υq 2 +Ωq +Λ ( = Ξ q 3 + Υ Ξ q 2 + Ω Ξ q + Λ ) Υ [ ( = Ξ q + 1 ) 3 Υ 1 3 Ξ 3 [( = Ξ q + 1 ) ( 3 Υ Ω + 3 Ξ Ξ Υ 3 Ξ ( Ω Ξ 1 3 ( Υ Ξ ( ) 2 Υ q 1 Ξ 27 ) 2 ) ( Υ Ξ 1 27 ( ) 3 Υ + Ω Ξ Ξ q + Λ Ξ ) ) 2 ( q + 1 ) Υ 3 Ξ ( ) 3 Υ + Λ ]. Ξ Ξ ] t = ( q + 1 ) Υ 3 Ξ = φ =Ξt 3 +Ξ ( Ω Ξ 1 3 ( ) ) ( Υ 2 t 13 Ξ Υ Ω Ξ 1 3 ( ) ) Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ. Ψ β (q, t) = N 1 2πi i exp { i [ Ξq 3 +Υq 2 +Ωq +Λ ]} = N 1 2πi exp { i [ i + 1 Υ 3 Ξ [ dq 2 ( S 3 ) ( S + q 0 q t q0 2 q (q 3 ) ] 1/2 S q 0 )+ t q 0 qt 2 (q q t ) + 1 Υ 3 Ξ Ξt 3 +Ξ ( [ 2 ( S 3 S dt + q 0 q t q0 2 q t Ω Ξ 1 3 ( Υ Ξ ) 2 ) t 13 Υ ( Ω Ξ 1 3 ) ( (q 3 S q 0 )+ q 0 qt 2 ( Υ Ξ ) 2 ) ) ] 1/2 (q q t ) 1 Υ 3 27 Ξ 2 +Λ ]}.
102 q q 0 Ψ β (q, t) = N Υ 3 Ξ 2πi [ 2 S + q 0 q t { [ exp i + 1 Υ 3 Ξ t ( 3 S Ξt 3 +Ξ = N Υ 3 Ξ 2πi [ 2 S + q 0 q t { [ exp i q 0 q 2 t + 1 Υ 3 Ξ ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) n=0 ( Ω Ξ 1 3 t ( 3 S Ξt 3 +Ξ = N Υ 3 Ξ 2πi [ 2 S + q 0 q t n n m { [ m=0 exp i q 0 q 2 t + 1 Υ 3 Ξ n=0 ( Υ Ξ ) n (q q 0 ) n q0 2 q t ) ) ( 2 t 13 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) ( Ω Ξ 1 3 t ( 3 S n=0 ( Υ Ξ ) n ( t 1 Υ 3 q0 2 q t ) ) ( 2 t 13 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) q0 2 q t q 0 qt 2 ( 1 ) Υ n m 3 Ξ q 0 t m Ξt 3 +Ξ ( Ω Ξ 1 3 ) n ( ) ) ( Υ 2 t 13 Ξ Υ Ω Ξ 1 3 ( ) ) ]} Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ ( Υ Ξ Ξ q 0 ) ) 2 ) n 1 Υ 3 27 Ξ 2 +Λ ]} ( ) ) ]} Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ
103 { [ ( = N 1 1 exp i 2πi 3 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) n=0 n n m=0 m { [ 1 6 ( 1)m (iξ) m 3 1 ( ( 1 ) Υ n m 3 Ξ q 0 ( ) ) Υ 2 Ξ [ 2 ( S 3 S + q 0 q t q 0 qt Υ 3 27 Ξ 2 Λ ) (q q t ) ]} ] 1 2 n ( 3 S q 2 0 q t ( ) ( m +1 m 2(iΞ) 2/3 Γ 1F ; 1 3, 2 3 ; Z 3 ) 27(iΞ) + Z 2 3 ( ) ( m +2 m (iξ)γ 1F ; 2 3, 4 3 ; Z 3 ) 27(iΞ) ( m ) ( m + ZΓ F ; 4 3, 5 3 ; Z 3 ) )] 27(iΞ) + 1 ( ) ( 6 ( (iξ)) m m +1 m 3 [2( (iξ)) 1 2/3 Γ 1F ; 1 3, 2 3 ; Z 3 ) 27(iΞ) ( ( m ) ( m + Z ZΓ F ; 4 3, 5 3 ; Z 3 ) 27(iΞ) 2 3 ( ) ( m +2 m (iξ)γ 1F ; 2 3, 4 3 ; Z 3 ) )]}. 27(iΞ) ) n Z = Ξ ( Ω Ξ 1 3 ( ) ) Υ 2, Ξ
104 x n e αx3 e px dx = 0 x n e αx3 e px dx + 0 ( x) n e α( x)3 e p( x) dx [ = L x x n e αx3] (p)+l x [( x) n e α( x)3] ( p) = 1 ( ) ( 6 ( 1)n α n n +1 n 3 {2α 1 2/3 Γ 1F ; 1 3, 2 3 ; p 3 ) 27α [ ( ) ( +p 2 3 n +2 n αγ 1F ; 2 3, 4 3 ; p 3 ) 27α ( n ) ( n +pγ F ; 4 3, 5 3 ; p 3 ) ]} 27α + 1 ( ) ( 6 ( α) n n +1 n 3 {2( α) 1 2/3 Γ 1F ; 1 3, 2 3 ; p 3 ) 27α [ ( n ) ( n +p pγ F ; 4 3, 5 3 ; p 3 ) 27α 2 3 ( ) ( n +2 n αγ 1F ; 2 3, 4 3 ; p 3 ) ]}. 27α M(t) = p t p 0 q0 q t p 0 q0 p t q 0 p0 q t q 0 p0.
105 Ṁ(t) =K(t) M(t) K(t) = 2 H q p 2 H p 2 2 H q 2 2 H p q. H = p2 2m + V (q) K(t) = 0 2 V (q) q 2 1 m 0. 2 S = 1, q 0 q t M 21 2 S = M 22, q 0 q 0 M 21 2 S = M 11, q t q t M 21 M(0) = I 2 2 Γ(t) = K(t) (p, q) ijk M(t) kl M(t) jm + K(t) ij Γ(t) jlm,
106 Γ(t) 2N 2N 2N M (p 0,q 0 ) K(t) p = 3 H q p 2 3 H p 3 3 H q 2 p 3 H q p 2, K(t) q = 3 H q 2 p 3 H p 2 q 3 H q 3 3 H q 2 p. H = p2 2m + V (q) K(t) p =0 I 2 2, K(t) q = 0 3 V (q) q S q0 3 3 S q0 2 q t 3 S q 0 qt 2 = = = = 3 S q t q t q t = 1 M 21 1 (M 21 ) 2 1 M 21 Γ 211 ( Γ 222 M 22 ) Γ 221 M ( 22 M 21 (M 21 ) 2 Γ 212 M ) 22 Γ 211, M 21 ) Γ 211, M 21 ) M ( 11 (M 21 ) 2 Γ 212 M ) 22 Γ 211, M 21 ( Γ 212 M 22 ( Γ 112 M 22 M 21 Γ 111 (M 21 ) 3, ( 1 Γ 111 M ) 11 Γ 211. M 21 M 2 21 Γ(0) = 0 I 2 2 2
107 p q
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Defects in Hard-Sphere Colloidal Crystals
Defects in Hard-Sphere Colloidal Crystals The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms
Gradient Descent for Optimization Problems With Sparse Solutions
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Diamond platforms for nanoscale photonics and metrology
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m i N 1 F i = j i F ij + F x
N m i i = 1,..., N m i Fi x N 1 F ij, j = 1, 2,... i 1, i + 1,..., N m i F i = j i F ij + F x i mi Fi j Fj i mj O P i = F i = j i F ij + F x i, i = 1,..., N P = i F i = N F ij + i j i N i F x i, i = 1,...,
Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α
Α Ρ Χ Α Ι Α Ι Σ Τ Ο Ρ Ι Α Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α Σ η µ ε ί ω σ η : σ υ ν ά δ ε λ φ ο ι, ν α µ ο υ σ υ γ χ ω ρ ή σ ε τ ε τ ο γ ρ ή γ ο ρ ο κ α ι α τ η µ έ λ η τ ο ύ
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Α Ρ Ι Θ Μ Ο Σ : 6.913
Α Ρ Ι Θ Μ Ο Σ : 6.913 ΠΡΑΞΗ ΚΑΤΑΘΕΣΗΣ ΟΡΩΝ ΔΙΑΓΩΝΙΣΜΟΥ Σ τ η ν Π ά τ ρ α σ ή μ ε ρ α σ τ ι ς δ ε κ α τ έ σ σ ε ρ ι ς ( 1 4 ) τ ο υ μ ή ν α Ο κ τ ω β ρ ί ο υ, η μ έ ρ α Τ ε τ ά ρ τ η, τ ο υ έ τ ο υ ς δ
2. Α ν ά λ υ σ η Π ε ρ ι ο χ ή ς. 3. Α π α ι τ ή σ ε ι ς Ε ρ γ ο δ ό τ η. 4. Τ υ π ο λ ο γ ί α κ τ ι ρ ί ω ν. 5. Π ρ ό τ α σ η. 6.
Π Ε Ρ Ι Ε Χ Ο Μ Ε Ν Α 1. Ε ι σ α γ ω γ ή 2. Α ν ά λ υ σ η Π ε ρ ι ο χ ή ς 3. Α π α ι τ ή σ ε ι ς Ε ρ γ ο δ ό τ η 4. Τ υ π ο λ ο γ ί α κ τ ι ρ ί ω ν 5. Π ρ ό τ α σ η 6. Τ ο γ ρ α φ ε ί ο 1. Ε ι σ α γ ω
Lifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
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16 0 17 0 17 0 18 0 18 0 19 0 20 A A = A 1 î + A 2 ĵ + A 3ˆk A (x, y, z) r = xî + yĵ + zˆk A B A B B A = A 1 B 1 + A 2 B 2 + A 3 B 3 = A B θ θ A B = ˆn A B θ A B î ĵ ˆk = A 1 A 2 A 3 B 1 B 2 B 3 W = F
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