David Burns and Matthias Flach. t y
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- Colleen Stewart
- 4 years ago
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1 4 Dv B F v 4 b Dv 4 B Dv B F v b F bk bk R F R q v q b R F L L - b q q - Eq: L T C O b q ˆ bk bk * T C O b TΩ bk + ψ T Ω bk T Ω T C T O bk bk + ˆ O w w ψ * R ˆ R T bk Ω bk T O ˆ w R k R bk bk R T C O bk b bk bk R w T Ω Eq: ˆ L T C + δ O /k + RΩ ˆ bk bk R L R q T H O L ˆ E bk bk k R b C O bk b δ k w b H L T bk b q O ˆ T Ω T O ð ℶð q T Ω E b + / k -- R O - bk bk - w bk b k I b -- w v - H b -- Rw Rw Wbkw v - T C O -O - H L ˆ - bk bk T C E w bk bk b R -O bk q -bk ˆ w-bk k ˆ -bk - - bk I - - q - w v w - - H- - -b v ˆ ˆ b T - O - T -w O ˆ w- - L -I T -- O w -- - ˆ w bk -- --v z - v- -- bk- - --bk R --- O -w ˆ b W - k - ˆ -- b - w - k ----b -- wb --- k - -- L -- b w -x- k b-- w w --- ˆ q L I v -- b -- - T w - w -- T H - - b -I v -- C v ˆ Wb - q- k - w - v-v Rw v Rw w- T w -W w w w w T w - - ˆ T- - w - - ˆ w b I w w b k v z bk b k ˆ b k b x b ˆ w ˆ ˆ v k b & & v q vv b w & T & w W w ˆ w w ˆ T w T ˆ
2 qv Tw b 5 q v B Tw b w v b Hb 5qv W Tw b 5 bk 6 bk B C w v bk Hb w v W b 6 C q F k w v b B : w v b Hb bk W bk 6 F C w k δ w v b : R x q v bk q bk - C I w x q w v q C F : O k I O / k w R w O bk C bk bk I w x w k x q v C k C k bk I C- w C -O - b O C b -- / k- -I O- -- -k- w- -O- ˆ k - - O b-k- Cˆ b -- / ˆ k v w O ˆ x-- w - / P I -- - k - x w ˆ k w - C - bk ˆ ˆ k k k - -bk ˆ : /k b W k - - b - z b C - b bk - - b C b ˆ / ˆ C δ ψ T Ω b Ω/kw / k v P F - I P - - C δ Rw L Rw ˆ b w - bk ˆ - + δ / - - v ˆ / P : I k ˆ b W b - ˆ ˆ ˆ δ O * T C O bk 4w bk ˆ : / k b W q W bk C O k R w w z ˆ Ωb T b x Rw k F - P T 5 - b - bk w - / L O bk ˆ -bk ˆ w / k w w b /k z C / H k b - O b v w k T b - bk- bk bk w 4 w - -bk C O -R C O W R F w - w P ˆ C O b T b L x C T / Rw / I z P Rw 5 w b - w - v O k -w - - k z- - C O H b b / k Ω ˆ O k v ˆ b w k - T - b w b k v ˆ E w ˆ b 4 w b k k - ˆ + O C -T R - ˆ O - j q Eq /T O I-z-P -- bk --w - E - E b v -O -- k- bkvq O- - F F T W W R w k-- TObkbbk C w O -- T q k b --- -w -- - v-e T - ℸ b -- bkbkww ˆ b j--q - -E q- O T- Rw Rw b - - v - -E- v -- - B E--- Ω -bk-v --q x -- k -- b qz -- w -- C F-F-T-W b b b -T- O-- - bk-b w w -bk -- T--q-bk / -T-- w -O-v-x ˆ -- w P / k z bk C --- H ---bk b bk-- b -w -w ˆ v - -T x-o-w T-- -O -- w - k b ˆ v T v b - - B-- w q C O -- - x-k - - z- -w C v -w b -b bk w- bk C T / IzP ˆ ˆ w ˆ v O ˆ k ˆ O ˆ / k ˆ O ˆ k b w ve j q E q / T O b E E b ˆ b k v q ˆ ˆ FFTW T O b k Tqb k T O v x P b k bk w w T xow ˆ ˆ T O b v
3 6 Dv B F 6 Dv 6 B Dv bk B F bk F R w b R k /k bk R R R Γ R - v I bk b R v w T Ω w 4 C bk 6 R C w bk T Ω R / v k I R v T C O R w w bk 4 C R 6 bk T C v I v T O ˆ w D- D - b- 4 -k-e keq - --q C z -z bk T -bk O bk ˆ T- z-o q--v -- z-bk D q-bk C z b T ke - R--- O q - w--b --- z ---T -O- H - -b ˆ b k ˆ - z- bk q- L-O z-- -B --- T bk- -T q T --b- v v ˆ zb -- Uw k -- ˆ x bkv -- qb -- k - --T -O wb T O b k z q LO z wb B --C-z-T ----O H D- -- -b-- bk b ---T- -O- w -- T-- V -v U-w -x ----v -- T --- O -- ˆ -- -w T- --O- - -v - --q-q- z k w C C q- - qb C w b Cz T O H D w -w V C- - v b SL T T O T---- v -- T --v- v w-- --z z- -x B Uw x ˆ bk -- C C - -b k -- v -- wv -w -- x Ω q q-- - B-- 6 L w R -- B T O qq w- -- z - -- v b T & C & C q x-v B ˆ LP bf bk bk w ˆ v ˆ w v w x C O ˆ V q C ˆ T 6 w L z Bv v R B z ˆ x R O ˆ L-P b F- b - q v & T v bk x - - bk- Rw Rw - -- & -- b - q- - q- - - b b C- k w -- q - B bk 5 -w -- - bk -z - C ˆ w v T - - v bk T B-v - w v- ˆ b - - bk T - - x - -R ˆ -O bk- - bk -q-v & T- - v- -- xv bk- --- B ----x ˆ q--- - w-- -H- - -C b b - - -P- b-l Rw 4 Rw ˆ ---bk- v - ˆ w - T v -v T w b x ˆ - O - ˆ x - q ˆ 6 L R B LP b F b k ˆ q ˆ b C ˆ ˆ x 5 w z Bv v R O ˆ b k ˆ bk qv T v ˆ bk x q w H C v b b R b k ˆ bk ˆ T v T w b x
4 qv Tw b 4 F b q v w Tw z b : 4 qv Tw b b 4 F x b b w ζ : π/ z b σ F F x b b w ζ w z : ˆ b b b - F x b η w b η C F C F qv C j C qv χ w x b z χ C η χ η j C b χ χ v χ T - χ χ F b x Y w b χ χy v T Y W k v ˆ Y C F C Y W j k v C x-- x ---- z-- z -- x-- -- x --H -- -R - HR T- -O- -w bk - - I w b T v T ΣH : H v I T C w b w T v T k w b σ WE Y Y S Y z C S H : b H S X q b w b z vw bk Y W k v w - - k b w b P 4 WY S-- z- - b - - -S X T - -q b- x b - bk bxbbxbb - z x -z-v ɛ : w- χ ɛ H R χ -bk b - k-w -w T b O - ˆ - - χ ˆ b k P 4 W ɛ bχ bv b bv bb w I Rw χ Rw ˆ b χ ˆ > T w T L ˆ T L v ˆ b - ϑ S H : ˆ H ˆ w b w q- T k w b v W E Y S ˆ b ˆ S X T q b z v b k ˆ w kw w ˆ b ˆ P 4 W ˆ b bx b bx b b ˆ : b b b ˆ ˆ ˆ ˆ ˆ > T v ˆ
5 4 Dv B F 4 Dv 4 B Dv bk B 6F C F 4 bk T b v 6 6 C 4 T b v 6 η E 6 C 4 T b v S C E η C T C T E ˆ ˆ C C ρη δ η v η b b Γ π Rw τη π δ ; ρ η Rw C - C H w v ˆ ˆ b H w v b ˆ E ; b b & bb v b w & E C C -B v b η b bbb πη B δ τ η η τη δ x π η B x ρ x w H w v x τητ η η η b w v b v b - w E ˆ ˆ τ ηπ b b b E η x δv η ˆ ˆ b & & ρη+ ηπ η x δ b w ˆ ˆ E L 4 T Y + ˆ ˆ B ˆ ΣC H BC + q H B R Y ΣR C 5 v b ˆ ˆ x x α σ+ v C ˆ ˆ Rα + πiα σ - - / < > w x w L b 4 T Y C S C H B b w C k q-h B R S R 5 E v b ˆ ˆ b & b L ˆ ˆ w- b x R v & I ˆ + ˆ b ˆ b ˆ L x L 4 T Y ˆ + S C H B C ˆ + q H B R Y S R 5 v b b ˆ ˆ R + ˆ I ˆ ˆ ˆ ˆ / < >
6 qv Tw b 4 w q v v Tw b 4 qv Tw b 4 Fw v / /± F D η η D w q v / ˆ η σ - b q b F D L π R π/ Rη q b / / ± L x L π π/ + π/ η b ˆ R ˆ / ˆ 4 / / L ± 4 x L 5 / ˆ π / L π/ 6 η ˆ ˆ / + ˆ / ˆ 4 / η v δ/ v ˆ b / L ˆ ˆ / ˆ 4 I π π I π/ η L / ˆ x L / / ± x π / ± π π/ π/ η v b L 4 ˆ / L 5 b π π/ ˆ η I ˆ ˆ / -X - ˆ -- / / T- - -k -q- η Rw -w δ / - X w-- T w- - - k - - / q - ˆ Eq: ˆ ˆ v / ˆ b ˆ / R ˆ η w 4 ww - ˆ ˆ / ˆ / ˆ ϑ π ˆ X T k R b w w b v ˆ ˆ
7 44 Dv B F 44 Dv 44 P B Dv 4 B F T C F P O 4 T Ω P bk bk 4 T O ˆ bk bk S ɛɛs θ θ ɛ β σ Λ S T b C Lb S T w w θ - - v --v -- w S - q Lb -w DS -q ɛ Lb-D OS b ˆ k - -S - -T- Lb - S T -O S - - k- 5 L4 W S 5 ɛ S P b 4 L 4 w W v LbD ˆ ˆ S S T P ˆ - O 4 S b k ˆ P W W ˆ S L b P - w - W w W w S ˆ x L L-b W w v ˆ S - q- - S q x P ˆ ˆ Eq: 4 ˆ ˆ S ˆ - ˆ W W x x ɛ : χɛχ χ w ɛ χ ± b : x w O ɛχ b x χ v χ b- x bk O * - bk P 7 ˆ 8 6 ˆ 5 bk W w W ˆ S ɛ χ - - w ˆ - - v Lb w T - x ˆ ɛ - ɛ w w ˆ v k T - - q ɛ x w ˆ q x k - b S W x ˆ w ˆ ˆ ˆ b ˆ x ˆ v b ˆ w T x ˆ w k
8 qv Tw b 45 q v qv Tw Tw b b / b W b W / ˆ b W D R V D R V x H x w H x D R B V x w V x B V H ˆ / T bj v > H - w T Ô bj w v H H > 5 b x k B q 4 w w O-x x x B w H V vb bk 5 bk b b / T b j v H ˆ x k B w O ˆ x : q x 4 H D q w RΓ w H ˆ 5 q x > 5 b F H vb w Eq: 5 w-x : w H x k q q 4 k D-q H : w q χh χ R C H H F H q w v b w b J D F x w H : H v b b H b H b w b J D 5 F x w w x : x x x : : + x J x Dq ˆ J x ˆ x xv -w v x x v v w b F Fb P k w 4 4 bw F b D - J w Fb q- F H q w P k- 4 4 T Ω W b- D- - -J * H : w - H H H ˆ T C O bk Eq: bk b w b J D F x w x : + x J x x v v w b F Fb P k 4 4 W b D J ˆ ˆ w ˆ ˆ T ˆ O ˆ ˆ b ˆ
9 46 Dv B F 46 Dv B F 46 w Dv ˆ B w Rw F b w Rw x x - - -z x - - C b C R C z V F z F q- D - - R- -x & RΓ V k D R w- T - V - - R V q b w D R x D k H V R w T V b b w w B - - z B O - ˆ q D ˆ R ˆ H ˆ V ˆ ˆ b w - D w- B ψ -x -- R b - - B z w O T ˆ V ˆ θẽ x q 6 b - w - q 6-7 F - I Rw Rw - bk bk b q w 7F D T bk bk w - I - - H V θ D - F w b T b w ψσ B T w ˆ D O F b w V β * D w σ V x F ˆ ˆ q b q q- b D bk R C w ˆ 7 F b D RΓ I b ˆ D ˆ T ˆ T w F b D V F b D ˆ R ˆ
10 qv Tw b 47 w q v Tw b 47 w v qv JTw b : 47 F k w J kw v bk J : F bk k Jk J J w w v J k k q- I I J J J F k q J k Ow v b bk bk Ow v J k J J q- b J J q Ow v b 5 J L Iww bk bk q q b 6 w w v Iww 5 W w L Iww 6 J w w v Iww W w ˆ ν ; ν J F b w : ζ / q q / W ζ F b ζ w : z v : ζ / W k q- W q z q Λ : z v q T : 5 L Iww z ṅ L 6 w w v Iww H C w vlb : : bk bk 4 bk bk : bk bk k -q W bk w bk bk ζ W w T bk bk L ζ /ζ H w v - : Rw - Rw : k C C ˆ ; q W Rw Rw z w - - w b - - F b w w z : z / q ˆ / ˆ W L I z w b Rw Rw P w--- 5 w w b z w- v b q ˆ : Λ I- D - Λ R z Γ T w - b ˆ - v W P / 5 -w W RΓ T b - - b C b Lb Lb - D : C Lb R C ˆ T v ˆ q ˆ ˆ W H T 9 - R C T - - Eq: - 9 H T H w v : b : 4 b & & W b ˆ ˆ z ˆ z ˆ / z b ˆ b w b w I w b 5 w b b Lb D Lb R v W R T b 9 H T
11 5 Dv B F 5 Dv B F 5 z Dv B k z F F- ζ z k z ˆ F ˆ kζ Fk k z k ζ k k S ξ z ζ L 6 ˆ bw S x b b z L 6 bw Eq: S x 5 z k x z L k b b b k b v b k χ ζ kξ Fk φ χ 5 k φ k µ k χ z χ k k χ x ˆ χ F ˆ k χ ζ χ 5 k - χ z k T-- k ˆ k ˆ T v v z - -q - - q q - - -w w q b T k/χ bχ k χ k -b k v ˆ C q b k C - k Rw 5 w q χ k - ˆ B - k w χ B φ w 5 q - q L b bv 5 bv χq b q bv µ bv bb L k / b b k ˆ -b b bxbx b- b bx bxb b b b χ χ b b χ z χ z b χζ χ χ ζ & χ k χ w k χ ˆ χ w k k & w w χ x x kq q v v b 5 χ χ 5 χ w χ χ w B b w w ˆ ˆ w q ˆ q ˆ b b bx bx b bx bx b ˆ ˆ b ˆ b ˆ ˆ ˆ ˆ z k ˆ w k w x q v 5 w ˆ
12 5 Dv B F 6 5 Dv 5 B Dv T B F 6 F B T bv B z bv - 6 z bk T S bk B bv x v C z Lb - x v w Λ PR wvx-r Eq: 8 wvx- S - - x v Lb S b b bb T z b k θ P R wvx k z R C X k F w PR wvxr wvx 8 ˆ F v b ˆ ˆ + q S b + b ν b b T ˆ k ˆ bk k z L z b bk b b / ˆ b Fb k F F X -- - ˆ x F - - ˆ - ˆ - -w ˆ - - q -v - - T k ˆ k ζ kf ζ / -Ξ - - ξ - + F q- ν ν w-w - W - L - w L-q Fb -b - F -q B-x - -z - - X- L Fb F ν w vv- - - v bk bk L q b ww bv bv bv bv W bv w bv bv bv L q ˆ bq bv bv Bb x b w bv bv bv bv L - b x v v w b F v ˆ q - ˆ ww F - b ˆ bxbx w bxbxbxbx bxbxbxbx L q b bx F bxb Bx x z ˆ X ˆ T ξ ξ z b F v w - F b F + F b- F F ξ F T x ζ/ ζf + b b bx bx bx bx bx bx bx I I bx bx > χd bx b χ x ẽχ x D bx v bx -q bx 8 bx x χ j ˆ -w q x x 4 qv ˆ x q χ ˆ v + ˆ 8 F ˆ j ˆ q F 4 ˆ ˆ qv q ˆ ˆ F ˆ x T z / ˆ z + ˆ ˆ F ˆ ˆ ˆ x ˆ q ˆ I > D x ˆ w x v 8 j q 4 q v q
13 qv Tw b 5 -x w- q v T- Tw -- b qv -w- --Tw b z- I x w-z- w T - - -x w x w -- ˆ T ˆ w z -x - ˆ b I -q wz - x x- k- ˆ β I q x v- x k L 6 -b b- v L 6 b z ℶ χ w k w v x χ ζ χ b I x w k k k χ b ν w v v b z v b L 6 b b kν : z χ ζ : ζ χ bk χ :ζ χ χ - bk : χ z bk T T 4 q q q 7 q w z 7 w -x - w- -- w T B -- - L x w w q- k k- wt z- q-q z- B L k- w v - ---I v-q -- v- L x- -- F x- - ˆ k F Eq: ˆ ˆ ˆ -b z ˆ ˆ F ˆ : ˆ z b ˆ ˆ x ˆ b : z v b - F b F b F b- χ β z B χ T x q ˆ q ˆ 7 w ˆ ν F + ν F b χζ χ χ - F b b F B x w w T B L q qq z ν F + > ν k F zq ˆ ˆ vq v L ˆ x x F ˆ b ˆ b ˆ b ˆ F ˆ + ˆ b z B ˆ ˆ ˆ F ˆ + ˆ F
14 6 Dv B F R 6 Dv 6 B Dv B F F bk bk D B T R D L j B D B R T D L j B b - S b é -E S S 5 É S D 67 B D B L B T T é D Iww L j R é B C I I bk bk 5 D : / / xv B / b / L/ 596 b B L S T - B w q Iww B S 5 x : x D W B D B 64 C v 67 I I 5 z : xv F b 596 B bk bk -L wz - -B --w é - B q x D B L B T 5S x B L ST w ν b 5 9 v I : D-W T - B- -- kf - - q -- v D- C I I 5 : / / - P B- 8 -E- x B x- - - v x v / b / / 596 B-z L B w k -- q - F B- - - B - -- D - B - - E -w -z - DW B - D v b - H bk E 5 bk x qv v S- z ˆ x B H - L 7 S T F B b B D B 5 9 L -v -I : L q ˆ w z 8 D T B - F kf O q- v - P 8 -x v T B v 99 J D B k F Eq v T B w b D B w E v b - v H D D B - 5 S B L ˆ F T x- b vh w S T w 7 - v 7 II b ˆ 5 9 J B D D B B L - - O EqL v - T w I : b C - j T T kf v I v 5 59 bk q v q P8 bk C D - B T C F- qq - 8 x B O - - v - L - T q - v I F - B J b ˆ k T C b E q 9855 bk 9 bk D B F Eq -v T w b w - B 4 R C D D v Bv v - I v D E D P B D - V F v T L é é P S P b - v w 7 b 6 P D - L -v é 6 - II H C J S 7 bk F bk T qv D B x T w b H j O Eq v- v C T - w b C J j - T - F B 4 P R v - I v B j 5 9 B D : B L v bk P bk P S C P - 5 vi L I v S T C --q L q - q q I 8 F D - B ˆ - - b- - F O q - v 7 T T C b v E q J x-v bk 9 D 4 B bk R F C Eq v D- v v T - w I- b v w v D q bk D 5 bk B P D - V F - L - T - q b v S w P ˆ 7 v I I J bk 6 4 bk 7 5 P D L C D B O Eq v j - T 4 6 ˆ v S - I v 5 bk 7 bk F T qv T w b - j P - C T C q bk ˆ 8 bk J- F - B 4 P - R j - B - L ˆ - : v - - I P F - S ˆ P 5 -v I- b L I v S 7 T C b E q ˆ x v 4 R C D v v I v 6 5 P D V L
15 qv Tw b 6 bk 9 bk q v Tw F - b 6qv b Tw b b q 6 9 FIv ö b 7 97 b q 4 Iv 66 7 bk 97 4bk T -T- F b b w Fw-F- z z-z k v z-k -v I vj j z 7-97 C 4 66 z O C- C - T wf z zk v - b - v - I- wqw O- - -C - j - - Pb- v- S-I P w-q b -w -- b j- j z J -P S - - w -S - H H P- b- -V b C O b -S -- - W 59 S B 6 4 b v I wq w w D H H q v j P S P b v bk - v bk H J L H - 4 VJ - H J S - V P S9 v 4b T T - - C b j W q 59 S 48 5 S w Hb H H - - B B j V ˆ 9 j bk b Iw w bk D H - H D k - J q -v - - v W v S ˆ B Hb -J J W L D H C P H ˆ B - bk 4 bk J H- V P S q v v 4 T C 7 b j - L v 9 I ww H - W L v J L q 4 v B J P I I : - S bk H V P S ˆ v 4 T E 5 bk B Hb L B99 - j -6 S q w -j Iw w D S 5 Hb Yk 99 D k J 6 4 B 4 j Hb J W - ˆ C P - Iw - w j - H - W B Iw L w v D 9 F D T j v D k q b v I P bk 7 bk - L Rw Rw 6 4 ww H 5 D v W - H - W L Ub H S z E L - v B P - I I : - E- 4 b B - Hb J W L C 55 - P 99 - S w B L Yk 99 T V b J w B P - z - -R Iw w - H W L v L ˆ T xw Iw w q bk 9 bk F b I v J 9 I D ww ˆ k v T j- -v H - W - b v I- w x P q é q - L 5 D v v q B P I I : B H P W R T 4 v L 5 V P - S I B U -b E q H vt w S - I v z T Ω E - / - b E B U q J 6 L C L 55 bk bk 99 L- - T W 6 W Rw C I L S Rw S V b w ˆ - J C 6 F T x 8 S - - w w Y -B k Yk 98 P z- R T- - w w- - v- z- q- Iw w C L q - - b I v bk bk -J- - D - vb k - v S- -z - Iw - -w x H W Rw Rw L - - B P - - R v T -- bk 9 F 4 bk D - T v - - j 5 v bk b 5 v bk I V P P S I B - q E q vt w T C O - U q - 5 J D v bk 6 bk L Rw Rw W W C - - I L S H W - C L 6 F - - U b - H T x S 8 S - w Y k 98 b C L D v B L T V b ˆ J w B P z R T ˆ x w v z q q ˆ b I v J k v S z w vc w x B P R T
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