A n.. a l y.. s i s.. a n.. d.. G.. e.. o.. m.. e.. t r y.. i.. n.. M.. e t r i c.. S.. p a.. c e.. s \ centerline
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- Patience Webb
- 6 years ago
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1 G O I : 4 8 G 3 v- G G O I : 4 8 / 3 v O I : G / v G J K w * w - J K v w w : - - v - - w wv w w - F I6 FI F 6 Uv w F Uv ˆ v v ˆ wv wk W w - - : W w - v - w v w H w - v w v 8 4 O w v 8 4O w - w ˆ - v - œv - v w v H H w - ˆ - W ˆ w v - w - w w v ˆ wk ˆ H - w z W FI 6 ˆ U v v w w Kw z - Kw - H H W w é w v - 34 H H : w v 3 J K V z : w v N- - 3 Jˆ K V z N v w ˆ ˆ wv v v v N ˆ - I v ˆ H ˆ ˆ w ˆ v N ˆ v w ˆ v O w -- v v I µ w é v µ µ w -k H v w w k é v v σ w v - - z - Kw µ w w v v v v H - - v µ µ w z - -w --w w v v w v v w z w : v 3 4 w v w w k v v 3 J K v V w k é w - v v w w w v w wk F N w * : j k wk k F N : - - v v w - : j k 6 - : v v 6 I ˆ
2 w ˆ 34 H zw k k [ v w k F v v w k k ˆ k w ˆ ˆ [ v v v w z w I w 3 w H I 4 w w ˆ 3 6 ] w v w w v z H k I w v z 6 w v k [ z w v v F w w v [ ] w : k I ˆ w ˆ k k w k w w - - v w v H -- k-k k v w k F v v w -- -k-k w --k - -v w - w w k - v v v w w z w I w w kw w 3 w v H I 4 w w - -w w - - H k 3 6 k w v w w v v z v H k W 5 I w v w w z 6 w H - -v k k -- -z w- - k v w w w w k v F w w v w : k I : v v w w w k- w H - - w k w v v v O W : O w k k O w é é - µ H zwkk v w k é F - v v w é k k kv w [ v v v w W w z w I w w kw w 3 w v - z H I 4 w w w H [ 3 6 ] w v w w é v v z v - H k W 5 I w v w w z 6 w H - w vk [ z w [ v w w w k w k é v v F w µ w µ µ w v µ µ w : k v I µ µ w k k w H w k w µ v γ v v Ω W : Ω [] Ω γ wv γ v j w Ω W µ ε ε z z ε : zε
3 J K v w J K z v - -k - - J K v k w w z z k w k [ [ ε ε - -k k k ] v wv w z v z z z z z - - k- k v ] k wv k z v [ 3 3 F w w F w w z µ; z w w W V v k v W - w V χ v w Ω w 3 Ω V Ω v - F ˆ 3 k w w O w [ O V O Ω χ Ω k w W v w w k [ ; w k ˆ 4 O O W I - - V -k- Ω w-w -- v ˆ w - - V k v v Ω w Ω 4 I V O w w ] 5 O - w O O w - - w w- - O k w - V Ω 5 O V O Ω U Ω w U - -O Ω V O w O U- U 3 O w Ω U - ˆ O Ω µ Ω w [ U w w O O O w O k ww v H U w Ω w U O N H U V Ω O [ U- -O N Ω V z O V O z 8 - V w 8 4 I V O [ O ˆ O w w 5 O V O O O w U O
4 3 z - - H - W z z H µ w k W - v - - w µ w W k v - - O w Ω w O O - W w Ω Ω W v v O w k v O w Ω w k v -- O O w Ω H w Ω µ O O w k v W Ω w v z H w Ω -O z- w O H H Hk w w O µ : : w k v v w w v k - v z v - w H O w w z w v k k - z w H - - k w k v w k w - - w w [ v w --- H w -z-w- H -k-k- é O k - k-- v z - -w -- z w H -- - w - k k j N w w H w v - v w- - H w k v z H µ H k ˆ ˆ j N w w ˆ w v 3 v : w µ 3 k v v v w 3 v w w v w v v z µ w H 33 v w I w v zw é k k w H wv v k w 3 ˆ 3 4 k w [ w w v é v w w k w v v w w z H j : w N w w w v v 3 3 é I w v - H9 wv v ˆ w v w w : w v w - v ˆ w w v ˆ 34 ˆ 34
5 J K v J 3 K 5 v J K v 3 5 w v 3 5 v w v v Hk w w µ w v v Hk w - - µ [ I - v w Hk H k w ˆ 34 I ˆ w ] Hk Hk w w w - -w - w - w [ - - I w k I- I w v w v Hk w H k ˆ 34 Hk µ ˆ 34 H k w w v w w ˆ w v w v [ I W v w W w v [ 34 ] v - v v N w w Nw w w w w w v 3 v w v w v 3 3 [ H k ˆ 34 w v 3 w w µ ] µ µ v v v w w 3 v [ 34 ] v µ w µ : 3 6 v w v 36 v w v v v W w v w v v N w w µ v v w µ w v v 3 w v w v 3 ˆ ˆ v w v : 3 [ ˆ v µ v w µ µ v v w 3 w v v w w v w v Hk w w w v ˆ v
6 µ µ v w w v w w v w 8
7 3 8 F K- w v 3 8 K 3 8 Hk F K w v F K K Hk w v K - ε K K K w H k K Hk K v µ v Hk K + ε w - - v- - - K ˆ w v K w [ H k K + v ˆ K w w v v Hk K [ ] w w - k 3 9 v w H µ 4 V H K ˆ 34 Hk K + ε O O v V V O O [ w ε w k ] v V k 3 9 w v w H 4 VO - - z- H -k- k Ω 4 Ω v V - ˆ 34 V Ω O O Ω w k v V V w O O : O [ Ω z O k w [ 4 -v ˆ - w- - - v- -- V Ω Ω H k - K - V - + Ω v - - w Ω - - : - Ω --k ] -w - - k- Ω w Ω k v w v χ V O w k v w w k O w v k O O χ k 3 χ 9 Ω+ k χ Ω O H χ Ω w v Ω k χ v Ω Ω H O w H O χ χ Ω+ χ Ω χ Ω 8 4 V 8 H O O V O O w k v V w O z k [ 4 O O V O : O O w O v w v V
8 J K v v J K w v v J K v V O v v w v O w v O V Ω χ Ω Ω k v w k vv w V O O V O Ω Ω : O Ω O : O v V Ω ε V v V O k v V w Ω V Ω V + ε Ω O w O : O V O v O V O -w - - v - Ω Ω - - V O w O Ω Ω V Ω + ε O O V O O V + v O O w w v Ω H Ω : Ω [ Ω w v O V Ω + ˆ ε w ε O ] NH w v Ω w O : O Ω Ω O V O w µ w - Ω N w v O O w µ O V O + O O µ O [ 34 ] Ω µ v µ w O v w v µ Ω v µ O H w w O : O µω k v O O µ + v Ω w O v w µ N k w v w O v w µ v O k Ω O k µ k w k [ W O k v O w k ] 8 v W w k 8 [ O ] v w [ w O O k [ ˆ
9 4 W O w 4 W - O v w v W O w : 4 3 Ω O v w w v v O -I µ k- - O v w v Ω I O -w- - w I k é Hk O O O w I - - Ω w w O k v I w k Hk Ω Ω O w 4 4 w k Ω k w v I O v w w k w v w µ v Ω [ H k O µk Ω O O µ v O µ v - k w w - j v w Ω v Ω v w w w j v w O k O w-- -- v - - w Hk O [ O v ˆ w 6 v O v w ˆ k ˆ O Ω w O v Hk Ω O µ w v O v O 6 µ j v 6 Ω : w w w 6 - Ω 4 5 w Ω U K w v Ω Hk K w O w K K z U K w v z H k Hk O K ˆ K w 83 ˆ ε K ˆ 6 ˆ K + ε O zµz ˆ 83 O O ˆ O w ˆ ˆ 6 4 5
10 J K v J V K v J K v V V Ω K w µω w v O K w K Hk Ω Ω K + ε O - - -w v - w ε Hk K Hk O O k 4 6 O O v w w - K v z 4 w w - I H k K H k K O k 4 6 w v - - O v w z K Hk K K 4 K + v I K w v w w K v Hk w k 4 6 v K w w w w w v v w w- - - v Iw - - w V 5 w z v v I w V v 4 5 I I 5 K w v z Hk z zµz I w K H k K z K Hk z z z z : z w w w v - ˆ ˆ - z : z v - v w v v v w k w w v v v- -w- - v -k - -- w v : 5 Hk z Hk I z w z Hk z w w Hk w 5 j w w w w j j j j w 5 j w j j w j w w j w w j w I j w j Z w j j j z z j w H k w z j z j j z j Z z w j z w z w j v j 84 w j w z w w z j w j w : 84 z w v v [ 34 ] v v w k w v
11 w v j j j I w j j j j j µ j j jµ j j j µ 53 v v H w w z Hk z j µ µ v 5 4 I j j z µz z 55 V µ ε w + ε χ χ V χ w v U U w χ U w U χ χ µ w v v w + ε µ χ µ µ F w ν w v 5 6 I
12 ν V w v j - j [ 34 ] 5 w v I w - j w v j j z j νz z w I w w j w w j j w 85 j k w j [ 34 j ] w j w j k w j 5 3 v-v-h --w w- j z Hk z j w w j k w j ˆ 34 ˆ j v - ˆ j 5 j 4 ˆ j ˆ j I j ˆ ˆ j ˆ j : z z v v H w z w z V w z H k z j ˆ ˆ ˆ j v ˆ j v w 5 4 V w I V z z w-- v U z U z 5 w 5 U w U w w V ˆ 34 w w v -v --v - -w V w w [ 34 ] v w v w v w v V ˆ 34 - F w w v + v I [ 34 ]
13 J K v J G K v v J K G v v G v δ w K K ˆ K Hk w K Hk µ K + δ [ H k ˆ K + ˆ v- v v- w v w K v v w ν K ν [ ] ν V w V + K w v / z w v V w w v z w V ˆ 34 w / V w k V µ w µ ε w ν K V w Hk K z : z + v w v / z w v v w v H w w V w K z νz ν w K z : z Hk z z zµz w w v / I ν ν µ / -v - w- - w - - -v H ˆ V ˆ z z w w I ˆ w Hk µ w ˆ V 59 z z v w z w K z H k ˆ z µz K z zµz w -v - - w : - z I 6 H - w [ I v-- 34 w - ] w w H - w I v w v H H - w v V w v µ w k v z 5 z ˆ ˆ ˆ H k z H k z z z z z z z w w - - v I - H - w I w w / v w H - w - w w - H w I v
14 w - - v w w [ v v Q v ] - v Q - v + w z - w -- k[ - -k k : / / N w v Q v w : + w v k w w v [ 6 N z ˆ w k [ w v w v - v- --k w v k 6 w ˆ - - ˆ w W w ˆ / w v / w w N w ˆ w ˆ W - w - v k+ χ k k k+ w k+ χ k w k w k w k k k w w k 6 k w w k 3 k k k k+ k Z k w k w k k w Z k v k ˆ w v k w k µ ˆ ˆ k Hk Hk k k ˆ k ˆ v k k Z j w v w k k Z j w v w w W k+ χ k k+ χ w k w k k w w w k k [ k k k ˆ k + 3 ˆ k k ˆ k+ k + χ k ˆ k+ χ k ˆ k w + w k k k k k k ] w v k 4 k k v k w χk k ˆ k ˆ k + k µ Z k k µ 8 k k ˆ v k 8 [ H k k ˆ k k k Z j [ ] ˆ ˆ ˆ k + k ˆ ˆ
15 J K v w J K k v v --v J K v w w k v k v v - - v w v w Hk k v k Hk w k k+ χ k k w Hk ˆ k k w k ˆ H k k+ χ ˆ k k k ˆ k Hk + w Hk k k ˆ w H k k Hk k k+ µ k [ ˆ k k ˆ w ˆ k k F v w ˆ H k v k w ˆ k k + Hk ˆ H k k k F- v- w- - ˆ - k - k Hk - ˆ k + - k k k ˆ k k w k k w v ˆ k ˆ k + w H k k k + k k Hk w k w k w v k F + k k v w w Hk k + w Hk k w k 3 w v k [ 6 ˆ + k ˆ k + w k w w H k Hk k w ˆ + v w - z ˆ v k w ˆ Hk I ˆ k w w w v ˆ k v + 4 v w ˆ 6 w + w v ˆ k w ˆ v v k z ˆ w H k z - w F w v ˆ z Ω+ ˆ k I ˆ ˆ Ω W k w w ˆ v k v H k v v v z Ω ˆ + H k z ˆ F w ˆ 6 µ w k v Ω w ˆ O ˆ + ˆ O F W ˆ w F F O Ω W ˆ v F ] w F F µ w w k k v v Ω w O w Ω F v z I w v w F w F Ω w v O v W v v F v v z F Ω 88 F w w k z v O w O k F F w O O W v O w w w O v w O w ˆ 88 w k v O F w F F O F F ˆ
16 O : O v v : Ω : O v O v v [ 34 ] v + O w - - Ω w + w µ µ w v w [ 34 k ] w O w w w w w µ w w v ˆ 34 w v γ ˆ w µ k w v ˆ v W v γ ˆ z v γ [ 34 ] v k w - kw w v W v - w γ -z - γ γ γ - - -v -k w w γγ w ˆ µ ˆ v ˆ 34 γ Ω ˆ ˆ 34 v Ω w v ˆ ˆ v ˆ Ω w µ k v w [ w w ˆ w ˆ ] v w O w - v W v ˆ z 89 v O v w kw ˆ w v w w [ ˆ O w -v ] w w w w w w ˆ ˆ - 89 ˆ ˆ ˆ 34 O ˆ
17 J K v J v K v J K v -I -k k- - z - v z w - v v w v v w v I kk z v - z -- --v k-k-k H- - k- w - - I k v kv w v z v w - - v z vk ˆ k k H ˆ k w v v v v w v v v w w O v v k k O k w H w v v v w v W Y v w Ω v v µ Ω w Ω w O w w W v Y - v v w G v Y w - O Y Ω w ˆ w O Y Y v v G v Y Ω O w Y Y F I w w F I µ v w v w w Y w w Y Y µ - Y w Ω w * W Y O ˆ w µ O λ : wµ λ * w : w H µ w H G v Y O µλ µ Ω : λ O : Y Y Y w λ w - Y Ω v Y O v Y F Ω µ I µ Y w w O v w w Y * Y w Y w * Y Y w Y Y w k Y k Y Y k k Y Y -w- k-k k k- ˆ Y χ Y w Y µ ˆ O v O v Y µ Y w vk [ w O v v Y [ ˆ w- - z - Λ -v k- Y - : k Y ] z w Λ Y Ω; Λ Y Y Y Y µ / / w w Y w w O H W w Λ Y Λ w Y N Y Y w w Y w [ O : w Λ w ] W w w Y F w ν ω w w Ω Y Ων Y N Z Y w w Ωω w 3 w F w w v w Y O O Y O Y ˆ O Z ˆ O Λ Y ˆ Ω µ/+ Λ Z v Ω 3 - w v Y Y Y ν V Y V - U / + Z ω U Y v V Y w k U k Ω Y 9 O w Z v Y O Y - Y V w w V U O v w v Z U w v k [ v V U O 9 ˆ Y Y ˆ Y O ; ˆ
18 F w U O O F w F w V U Ω V Ω U O v V O k O V Ω U V w v V w v Y Y ν V V Y ν w Y ν Y v V Λ O U V Y Ω µ+ w w w Y Λ Z w w Y V ˆ ˆ Λ Y Z Z ω U w Z ω U Y w w k O v ˆ ˆ w w Y w w F ˆ Z w w ˆ v ˆ Y O ˆ + Y ν w + Z ω v ˆ ˆ Z ˆ w w Z w / w w Z w v Z Ω µ/+ Λ Z k 4 w ˆ ˆ Z v ˆ ˆ Z w v k - v w ˆ w Z U -w k w v [ - F - v w : / v k w v [ Y 5 w k v w w w Z Ωµ w v w -v - w w- w Y w w w w v Y Ω µ/+ w w Y v w Ω ˆ w ˆ 34 + V Z w w V U ˆ / + Z U V w U V U Ω v F w w O w V U Ω w Ω Z w V - Ω U ˆ k 4 ˆ ˆ -v- - - Ω - -- µ/+ - ˆ -v w-- / k w ˆ v ˆ k w / Ω µ/+ v / 9 ˆ w Z ˆ w / v- -k -- 5 O -w w ˆ - / + w v w k w 4 w w -w v v O v w k w w V w V ˆ U w v -w ˆ : U ˆ / v k V U V U [ O F w -w - U O 5 O - v V O k U w w O w O w [ ] w w -w w V w w ˆ w ˆ w / + ˆ O ˆ w w
19 J K v J K v J K v w w -w w w- w w [ w ˆ U ˆ w k v ˆ w / / U / ˆ w / V w ˆ U k v ] w V U w -w V V U U / + U w w k v Ω k V O U k V w v U ˆ w v V ˆ w 34 V U ˆ / + w ˆ U w / + w w -w w v k V v-- - U / / w w v w w w /+ w w [ ˆ w ˆ 34 ˆ / w Ω ˆ µ/+ w w ] 6 w Ωµ w v w w -w w v w ˆ O w Ω ˆ µ/ v ˆ / Ω w / - ˆ ˆ ˆ 6 ˆ / O w [ v V ˆ ˆ V U / ˆ 34 v V U Ω / 3 + w wˆ ˆ w O ˆ W w w ˆ w / ] v ν v O w O V ˆ / I + w V U w ˆ v V U O 3 w -w Wν w w 6 v K w v O νk [ I ] 3 Hk K v z v 9 ˆ / v K- w v O K w 3 w [ ] Hk K v z v ˆ V ˆ 34 V U ˆ / v V U O 3 w w W w w
20 8 - w 8 8 w [ v 34 ] v v - v z z z νz z z w w k v - w kw w k w W v kw w k k v w k W w k w v k v w k [ k k w F - K - z z [ ] - H w k F K H z k J 5 8 z 6 - ] J k k I O [ ] - I - 9 O w [ 3 ] k j ö 3 k j ö N j j N - - [ 4 ] kw Y N Y K j k I V k I 4 N H k Y N Y K j k I 4 N H W V I N H- w k k v w [ 5 ] J à 4 z N H - I - k 5 k J v z I v à J v v 6 [ ] J à J v v k 6 k J v J - v [ ] w F I U 6 v K [ 8 ] H k V k G z k J v J - v v - v z w - Jw 5 8 w 6 - I 3 I 4 N Y 9 9 N w I U v [ ] [ 9 ] k G 8 k V H k I G z - - G O k wk v z- w w [ ] H H - - kk 9 K V [ ] H H kk I N 3 I - v - V - - Y N w [ 3 ] k j v 9 k N G 4 j N 5 5 [ ] H k J K G K N k wk µ H v I U v J k 4 [ 3 k ] J K K N H G H H kk K V 3 [ 4 ] Y J k k H 9 H N 9 3 kk Y K 6 N j 5 9 [ 4 ] k IV G z v H V v v N 4 J N F H [ 5 ] 6 V 555 G z k v w - w J k J K K N [ 6 ] F - H v [ 5 ] J v I v U v- z J 5 J I v k 3 k J K 93 K N - G [ 6 ] J v J J w v [ ] J k 4 v k V J G z v ˆ w I v U v J- F [ 8 ] k 5 V G k z V G z z v w w w w J I 3 94 I k 6 N k Y 9 9 F v N w
21 J K v J k K k J O v J V J K v H w [ ] J [ O ] J O J V J V H H - H w k 8 k v [ 8 ] [ v 8 ] v - 8 www 8 - www / / 83 / / / 83 / 94 www
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