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Localization of multiple nodes based on correlated measurements and shrinkage estimation

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This is a repository copy of

Localization of multiple nodes based on correlated

measurements and shrinkage estimation

.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/84022/

Version: Accepted Version

Proceedings Paper:

Salman, N, Mihaylova, L and Kemp, AH (2014) Localization of multiple nodes based on

correlated measurements and shrinkage estimation. In: Proceedings of the 2014

Workshop on Sensor Data Fusion: Trends, Solutions, Applications, SDF 2014. 2014

Workshop on Sensor Data Fusion: Trends, Solutions, Applications, SDF 2014, 08-10 Oct

2014, Bonn. IEEE , 1 - 6. ISBN 9781479973873

https://doi.org/10.1109/SDF.2014.6954712

eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse

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⋅″ ←÷•↑ °↵°≈→ ♥≈ ±°←•≥•ƒ≈ ←÷≈ °≈→≡±→≥↵″≈ ±≡ ←♥± ≥↔×⌠ ←•°×≈ ⊂ ∈⊆⊆ ×±↵←•±″ ↵×≠±→•←÷≥↑∫ ±↵←•±″ ≈↑←•≥↵←•±″ ±≡ ⊂↑ •↑ ↵ ″±″⌠ו″≈↵→ ≈↑←•≥↵←•±″ °→±×≈≥∫ ⇐↔≈ ←± ←÷≈ ″±″⌠ ו″≈↵→ ″↵←↔→≈ ←÷≈ ×±↵←•±″ ≈↑←•≥↵←•±″ •↑ ≠≈″≈→↵××♣ °≈→≡±→≥≈… ↔↑•″≠ ↵″ •←≈→↵←•♠≈ ↵×≠±→•←÷≥ ♥•←÷ ↵ ×±↑≈ •″•←•↵× ≈↑←•≥↵←≈ ↵″… ≈♥←±″ ↑←≈° ↔°…↵←≈∫ ¬±♥≈♠≈→ ×±↵←•±″ ±±→…•″↵←≈↑ ↵″ ↵×↑± ≈ ≈↑←•≥↵←≈… ↔↑•″≠ ↵ ו″≈↵→ ×≈↵↑← ↑↓↔↵→≈↑ ⊆ ↵°°→±↵÷∫ ⋅″ ←÷•↑

∧⊆ ←≈÷″•↓↔≈ ≡±→ ≥↔×←•°×≈ ⊂ ×±↵וƒ↵←•±″∫ ⊆≈±″… ♥≈ ↵×↑± ≡±→≥↔×↵←≈ ↵ ♥≈•≠÷←≈… ⊆ ⊃⊆ ↑±×↔←•±″ ←± ←÷≈ ≥↔×←•°×≈ ⊂ ×±↵←•±″ °→±×≈≥∫ ⇓ ×±↑≈… ≡±→≥ ≈♦°→≈↑↑•±″ •↑ …≈→•♠≈… ≡±→ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ±≡ ≥↔×←•°×≈ ⊂ →≈↵…•″≠↑ ↵← ←÷≈ ⊆↑∫ ⊂÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ≈♦°→≈↑↑•±″ …≈°≈″…↑ ±″ ←÷≈ ♠↵→•↵″≈ ↵″… ±♠↵→•↵″≈ ♠↵×↔≈↑ ±≡ ←÷≈↑≈ →≈↵…•″≠↑∫ ⋅″ °→↵←•↵× ↑≈″↵→•±↑ ↵″… •″ ×↵→≠≈ ″≈←♥±→∝↑ ←÷≈↑≈ ♠↵×↔≈↑ ↵→≈ ↔″∝″±♥″ ↵″… ←÷≈ ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ •↑ ″±← ×↵→≠≈ ≈″±↔≠÷ ←± °→±♠•…≈ ↵″ ↵↔→↵←≈ ↑↵≥°×≈ ≈↑←•≥↵←≈ …↔≈ ←± ו≥•←≈… →≈↑±↔→≈↑ ↵″… °↵∝≈← ×±↑≈∫ ⊂± ±↔″←≈→ ←÷•↑ °→±×≈≥ ♥≈ ↵°°×♣ ←÷≈ ↑÷→•″∝↵≠≈ ←≈÷″•↓↔≈ ←± ≈↑←•≥↵←≈ ←÷≈ ♠↵→•↵″≈ ↵″… ±♠↵→•↵″≈ ≈×≈≥≈″←↑ →≈↓↔•→≈… •″ ←÷≈ ≈♦°→≈↑↑•±″ ≡±→ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ •″ ←÷≈ ⊃⊆ ≥≈←÷±…∫ ⊂÷≈ ↑÷→•″∝↵≠≈ ←≈÷″•↓↔≈ •↑ ↵×↑± ↵°°×•≈… ←± ≈↑←•≥↵←≈ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ≡±→ ←÷≈ ∧⊆ ←≈÷″•↓↔≈∫ ⇐↔≈ ←± ←÷≈ ×↵∝ ±≡ ↑←→↔←↔→≈ ↵″… ≡±→ ↵ ו≥•←≈… ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦ •↑ ″±″ •″♠≈→←•×≈ →≈″…≈→•″≠ •← •≥°→↵←•↵× ←± ↔↑≈ •″ ←÷≈ ∧⊆ ≥≈←÷±…∫ ⊂÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≈↑←•≥↵←±→ ±″ ←÷≈ ±←÷≈→ ÷↵″… •″←→±…↔≈↑ ↵ ↑←→↔←↔→≈ ←± ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ↵″… ÷≈″≈ •← •↑ •″♠≈→←•×≈ ≈♠≈″ ≡±→ ו≥•←≈… ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑∫ ⋅← •↑ ↑÷±♥″ ♠•↵ ↑•≥↔×↵←•±″ ←÷↵← ←÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦ ±≥°↵→≈… ←± ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦ ÷↵↑ ↵ ↑≥↵××≈→ ≈↑←•≥↵←•±″ ≈→→±→∫ ⇑±″↑≈↓↔≈″←×♣ ←÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦ ♥÷≈″ ↔↑≈… •″ ←÷≈ ∧⊆ ↵″… ⊃⊆ ↵×≠±→•←÷≥ →≈↑↔×←↑ •″ ↑↔°≈→•±→ °≈→≡±→⌠ ≥↵″≈∫ ⇔↑°≈•↵××♣ •″ ←÷≈ ↵↑≈ ±≡ ∧⊆ ↵↑≈ ♥÷≈→≈ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≈↑←•≥↵←±→ •↑ ″±″ •″♠≈→←•×≈ ≡±→ ↵ ו≥•←≈… ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ←÷•↑ °→±×≈≥ •↑ →≈↑±×♠≈… ♥•←÷ ←÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦∫

⊂÷≈ →≈↑← ±≡ ←÷≈ °↵°≈→ •↑ ±→≠↵″•ƒ≈… ↵↑ ≡±××±♥↑∫ ⊆≈←•±″ ⋅⋅ •″←→±…↔≈↑ ←÷≈ ↑•≠″↵× ≥±…≈× ↵″… ←÷≈ ≥↔×←•°×≈ ⊂ ×±↵←•±″ °→±×≈≥∫ ⊆≈←•±″ ⋅⋅⋅ …≈♠≈×±°↑ ←÷≈ ∧⊆ ↵″… ←÷≈ ⊃⊆ ≈↑⌠ ↵″… ±♠↵→•↵″≈ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←±→∫ ⊆≈←•±″ ⊇ °→≈↑≈″←↑ ←÷≈ ↑•≥↔×↵←•±″ →≈↑↔×←↑ ♥÷•÷ ↵→≈ ≡±××±♥≈… ♣ ←÷≈ ±″×↔↑•±″↑∫ ⊂÷≈ …≈→•♠↵←•±″ ±≡ ←÷≈ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←±→ •↑ °→±♠•…≈… •″ ←÷≈ ⇓°°≈″…•♦∫ ⋅⋅∫ ⊆∩⊆⊂⇔ ∇⇐⇔ ∈″ •↑ ←÷≈ ↑≈← ±≡ …•≥≈″↑•±″↵× →≈↵× ″↔≥≈→↑ •↑ ←÷≈ ←→↵″↑°±↑≈ ±°≈→↵←±→ ⇔ →≈≡≈→↑ ←± ←÷≈ ≈♦°≈←↵←•±″ ±°≈→↵←±→ ⊕•∂ →≈≡≈→↑ ←± ←÷≈ ≈×≈≥≈″← ↵← ←÷≈←÷→±♥ ↵″… ←÷ ±×↔≥″ ±≡ ≥↵←→•♦  →≈°→≈↑≈″←↑ ←÷≈ …•≥≈″↑•±″↵× •…≈″←•←♣ ≥↵←→•♦∫  →≈°→≈↑≈″←↑ ←÷≈ ″±→≥↵× …•↑←→•↔←•±″ ♥•←÷ ≥≈↵″ ↵″… ♠↵→•↵″≈ ∫∝∝ →≈°→≈↑≈″←↑ ←÷≈ ∨→±≈″•↔↑ ″±→≥∫ ⇓ ←♥± …•≥≈″↑•±″↵× ⌠⇐ ″≈←♥±→∝ •↑ ±″↑•…≈→≈… ±″↑•↑←•″≠ ±≡  ⊆↑ ♥•←÷ ∝″±♥″ ×±↵←•±″↑ •∑∂  ⊕♦•∑∂∑ ♣•∑∂℘⊂ •∑∂  ∈ ≡±→ •∑ ∂  ∑ ∑  ⊂÷≈ ″≈←♥±→∝ ↵×↑± ±″↑•↑←↑ ±≡  ⊂↑ ♥÷•÷ ÷↵♠≈ ↔″∝″±♥″ ±±→…•″↵←≈↑ ∝∑×  ⊕♦∝×∑ ♣∝∑×℘⊂ ∝∑× ∈ ≡±→ ∝∑ ×  ∑    ∑  ←÷↵← ↵→≈ ←± ≈ ≈↑←•≥↵←≈…∫ ⊂÷≈ →≈≈•♠≈… °±♥≈→ ↵← ←÷≈ ⊆↑ …↔≈ ←± →↵″…±≥ ↑÷↵…±♥•″≠ •↑ ×±≠⌠″±→≥↵××♣ …•↑←→•↔←≈…∫ ⊂÷•↑ ≥±…≈× •↑ ↵↑≈… ±″ ≈≥°•→•↵× →≈↑↔×←↑ ±←↵•″≈… •″ ⊕℘ ⊕℘∫ ⋅← •↑ ↵↑↑↔≥≈… ←÷↵← ←÷≈ ⊂↑ ←→↵″↑≥•← …↔→•″≠ °→≈↵↑↑•≠″≈… ≈°±÷↑ ←± ↵♠±•… •″←≈→≡≈→≈″≈ ≈←♥≈≈″ …•≡≡≈→≈″← ⊂ ←→↵″↑≥•↑↑•±″↑∫ ⊂÷↔↑ ←÷≈ …•↑←↵″≈…•∝ ≈←♥≈≈″ ←÷≈∝←÷ ⊂ ↵″… ←÷≈•←÷ ⇓ •↑ →≈×↵←≈… ←± ←÷≈ °↵←÷⌠×±↑↑ ↵← ←÷≈←÷ ⇓ •∝ ↵″… ←÷≈ ∠⇔  ↵↑ ⊕℘

•∝ ð⌡ ð ×±≠

ð…•∝⌡♥•∝∑ 

♥÷≈→≈ ð•↑ ←÷≈ °↵←÷⌠×±↑↑ ↵← ←÷≈ →≈≡≈→≈″≈ …•↑←↵″≈ …ð …ð

…•∝ ↵″… •↑ ″±→≥↵××♣ ←↵∝≈″ ↵↑  ≥ ↵″… ♥•∝ •↑ ↵ ƒ≈→±⌠

≥≈↵″ ∧↵↔↑↑•↵″ →↵″…±≥ ♠↵→•↵×≈ →≈°→≈↑≈″←•″≠ ←÷≈ ≥↔×←•°↵←÷ ×±≠⌠″±→≥↵× ↑÷↵…±♥•″≠ ≈≡≡≈← •∫≈∫ ♥•∝  ð •∝ ∫ ⋅←

•↑ ↵↑↑↔≥≈… ←÷↵← •↑ ∝″±♥″ ♠•↵ °→•±→ ÷↵″″≈× ≥±…≈ו″≠ ±→ ↵↔→↵←≈ ≈↑←•≥↵←•±″∫ ⊂÷≈ °↵←÷⌠×±↑↑ •↑ ↵×↔×↵←≈… ↵↑

•∝ ð ×±≠ð ð ×±≠ð•∝  ♥÷≈→≈ ∠∝ •↑ ←÷≈ ←→↵″↑≥•← °±♥≈→ ↵← ←÷≈∝←÷ ⊂ ↵″… ∠•∝ •↑ ←÷≈

→≈≈•♠≈… °±♥≈→ ↵← ←÷≈•←÷ ⇓∫ ⊂÷≈ …•↑←↵″≈

•∝ •↑ ≠•♠≈″ ♣

…•∝ ♦∝ ♦•⌡ ♣∝ ♣• 

⊂÷≈ ±↑≈→♠≈… °↵←÷⌠×±↑↑ •″ …⇒ ≡→±≥…ð←±…•∝ƒ•∝ •∝ð

↵″ ≈ ≈♦°→≈↑↑≈… ↵↑

ƒ•∝≡• ∝ ⌡♥•∝∑ 

♥÷≈→≈≡• ∝  ×″…•∝ ↵″…  ×″ ðð ∫  ↵″ ≈ ♥→•←←≈″ •″

↵ ≥↵←→•♦ ≡±→≥ ↵↑ ∅  ⌡  ♥÷≈→≈ ←÷≈ ∝←÷ ±×↔≥″ ±≡ ↵″… →≈°→≈↑≈″←↑ ←÷≈ →≈↵…•″≠↑ ↵″… ↵↑↑±•↵←≈… ″±•↑≈ ≈×≈≥≈″←↑ ↵← ←÷≈ ⊆↑ ≡→±≥ ←÷≈ ∝←÷ ⊂ →≈↑°≈←•♠≈×♣ ↵″…⊕∨ ℘ •∝ ≡• ∝∫ ∨±→ ±″♠≈″•≈″≈  ↵″

↵×↑± ≈ ♥→•←←≈″ •″ ↵ ℵ→±×× ±↔← ≡±→≥ •∫≈∫ ƒ ♠≈  ←÷≈″ ≡±→ ←÷≈∝←÷ ←↵→≠≈← ♥≈ ÷↵♠≈

ƒ∝

ƒ∝

∫∫ ∫ ƒ ∝

 ≡ ∝ ≡ ∝ ∫∫ ∫ ≡ ∝ ⌡ ♥∝ ♥∝ ∫∫ ∫ ♥ ∝ ±→

ƒ  ⊂÷≈″ ♥≈ ÷↵♠≈ ♥ ð∝∝ ♥÷≈→≈∝∝

•∝⋅ ↵″…

⇔ ♥×∝×∫ ¬≈→≈∝× •↑ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ≈←♥≈≈″ ∈⊆⊆ →≈↵…•″≠↑ ↵← ←÷≈ ⊆↑ ≡→±≥ ←÷≈∝←÷↵″…×←÷

(4)

…≈°≈″…↑ ±″ ←÷≈ →≈×↵←•♠≈ …•↑←↵″≈ ≈←♥≈≈″ ←÷≈ ⊂↑ ↵″… ↵″ ≈ ≥±…≈×≈… ↵±→…•″≠ ←± ∧↔…≥↔″…↑±″ ⊕℘ ↵↑ ≡±××±♥↑ ∝× ≈♦° …∝× … ∑ 

♥÷≈→≈ … •↑ ←÷≈ ℵ…≈±→→≈×↵←•±″ …•↑←↵″≈∫ ∨•≈×… ≥≈↵↑↔→≈≥≈″←↑

•″ ⊕℘ ↑↔≠≠≈↑← ♠↵×↔≈↑ ≡±→ … ≡±→ …•≡≡≈→≈″← ≈″♠•→±″≥≈″←↑∫ ⇔↓∫

 ♥÷≈″ ♥→•←←≈″ ≡±→ ↵×× ⊂↑ •↑ ≠•♠≈″ ♣

ƒ ♠≈  ƒƒ∑ ∑ƒ



±→

ƒ  ⌡ 

♥÷≈→≈ ♥ ð ♥÷≈→≈≈″±≥°↵↑↑≈↑ ±→→≈×↵←•±″↑ ≈←♥≈≈″ ↵×× ⊂↑∑ ⇑ ∫∫ ∫ ∫∫∫ ∫ ∫∫ ∫∫∫ ⇑   ⊂♥± ×±↵←•±″ ↵×≠±→•←÷≥↑ ←± ↑±×♠≈  ↵→≈ …•↑↔↑↑≈… •″ ←÷≈ ″≈♦← ↑≈←•±″∫ ⋅⋅⋅∫ ∇⇑⇓⋅∅⇓⊂⋅∇ ⇓∧∇∈⋅⊂¬⊆ ∨∇∈ ⊄⊂⋅∠⇔⊂⊆ ⇓∫ ∧≈″≈→↵וƒ≈… ×≈↵↑← ↑↓↔↵→≈↑ ↵×≠±→•←÷≥

⋅← •↑ ×≈↵→ ←÷↵←  °→≈↑≈″←↑ ↵ ″±″⌠ו″≈↵→ ≈↑←•≥↵←•±″ °→±×≈≥ ↵″… ↵ ×±↑≈ ≡±→≥ ≈♦°→≈↑↑•±″ ←± ↑±×♠≈  •↑ ″±← →≈↵…•×♣ ↵♠↵•×↵×≈∫ ⇓ ↑±×↔←•±″ ←±  •↑ ±←↵•″≈… ♣ ≥•″•≥•ƒ•″≠ ←÷≈ ≡±××±♥•″≠ ±↑← ≡↔″←•±″ ⊕℘

   ƒ ≡ ⊂⇑ ƒ ≡  ð ♣ ←÷≈•∫≈∫   ⌡   ♥÷≈→≈ ∨  …•↵≠ ⊕ ↵ ∑∨ ↵ ∑ ∑∨ ↵ ℘ ↵″… ∨ •↑ ←÷≈ √↵±•↵″ ≥↵←→•♦ ±≡ ←÷≈ ←÷ ⊂ ↵″… •←↑ ≈×≈≥≈″←↑ ↵→≈ ≠•♠≈″ ♣ ∨  ◊≡ ∝ ◊♦∝ ∝ ↵∝ ◊≡ ∝ ◊♣∝ ∝ ↵∝ ◊≡ ∝ ◊♦∝ ∝ ↵∝ ◊≡ ∝ ◊♣∝ ∝ ↵∝ ∫∫ ∫ ∫∫∫ ◊≡ ∝ ◊♦∝ ∝ ↵∝ ◊≡ ∝ ◊♣∝ ∝ ↵∝ ≡±→ ◊≡• ∝ ◊♦∝ ∝ ↵∝  ♦↵ ∝ ♦• •∝ ↵″… ◊≡• ∝ ◊♣∝ ∝ ↵∝  ♣↵ ∝ ♣• •∝∫

⊄↑•″≠  •″ ð ↵″… ≥•″•≥•ƒ•″≠ ♥•←÷ →≈↑°≈← ←± ∑↑± ←÷↵← ←÷≈ ″≈♦← ≈↑←•≥↵←•±″ •↑ ≠•♠≈″ ♣

↵⌡

 ↵⌡↵ 

♥÷≈→≈

↵ ∨⊂⇑ ∨ ∨⊂⇑ ƒ ≡  ⊂÷↔↑ ≠•♠≈″ ↵″ •″•←•↵× ≈↑←•≥↵←≈  ←÷≈ ≠≈″≈→↵וƒ≈… ×≈↵↑← ↑↓↔↵→≈↑ ∧⊆ ±″♠≈→≠≈↑ ←± ←÷≈ ≥•″•≥↵∫ ⊂÷≈ ↵×≠±→•←÷≥ •↑ ↵⌡ ↵  ≈±≥≈↑ ↑≥↵××≈→ ←÷↵″ ↵ ≈→←↵•″ ←÷→≈↑÷±×…∫ ⊂± ↵♠±•… ←÷≈ ±≥°×≈♦•←♣ ±≡ ←÷≈ •←≈→↵←•♠≈ °→±≈…↔→≈ ↵″… ↵ ×±↑≈ •″•←•↵× ≈↑←•≥↵←≈ ±″…•←•±″ ↵← ←÷≈ ≈♦°↵″↑≈ ±≡ ↵↔→↵♣ ↵ ⊃⊆ ↑±×↔←•±″ •↑ °→≈↑≈″←≈… •″ ←÷≈ ″≈♦← ↑↔↑≈←•±″∫ ⇒∫ ⊃≈•≠÷←≈… ו″≈↵→ ×≈↵↑← ↑↓↔↵→≈↑ ↵×≠±→•←÷≥ ⇓ ↑ו≠÷← ≥↵″•°↔×↵←•±″ ±≡  →≈″…≈→↑ •← ←± ≈ ו″≈↵→∫ ⊂÷•↑

↵″… ↵″↵×♣ƒ≈… •″ ⊕℘∫ ∨±→ ↵ ↑•″≠×≈ ←↵→≠≈← ∈⊆⊆ ≥≈↵↑↔→≈≥≈″←↑ ←÷≈ ו″≈↵→•ƒ≈… ≥±…≈× •↑ …•↑↔↑↑≈… •″ ⊕℘∫ ¬≈→≈ ♥≈ ≈♦←≈″… ←÷•↑ ≥≈←÷±… ≡±→ ←÷≈ ≥↔×←•°×≈ ⊂ ↵↑≈ ↵↑ ≡±××±♥↑∫ ∨→±≥  •← ↵″ ≈ →≈↵…•×♣ ↑÷±♥″ ←÷↵←

⇔ 

•∝

≈♦° ƒ•∝  ℜ…•∝∑ 

♥÷≈→≈ •∝ ≈♦°   •∝   ∫ ⊆•≥•×↵→×♣ ÷±±↑•″≠ ↵ →≈≡≈→≈″≈ ⇓ •← ↵″ ≈ ↑÷±♥″ ⇔  →∝

≈♦° ƒ→∝  ℜ…

→∝∑ 

♥÷≈→≈ →∝  ≈♦° 

 →∝

   ∨±→ ו″≈↵→•ƒ↵←•±″ ←÷≈ ↑↓↔↵→≈ ±≡

≈↵÷ …•↑←↵″≈ ≈↓↔↵←•±″ •↑ ↑↔←→↵←≈… ≡→±≥ ←÷≈ ↑↓↔↵→≈ ±≡ ↵ →≈≡≈→≈″≈ …•↑←↵″≈ ≈↓↔↵←•±″∫ ⊂÷•↑ →≈↑↔×←↑ •″ ↵ ו″≈↵→ ↑♣↑←≈≥ ♥÷•÷ •↑ →≈°→≈↑≈″←≈… •″ ≥↵←→•♦ ≡±→≥ ↵↑



♥÷≈→≈ ⊕∑ ∑•↑ ←÷≈ ±↑≈→♠↵←•±″ ♠≈←±→ ↵← ←÷≈ ⊆↑ ≡→±≥ ←÷≈ ⊂↑∫ ⊂÷≈ ו″≈↵→•ƒ≈… ±↑≈→♠↵←•±″ ≡→±≥ ←÷≈∝←÷ ⊂ ↵←

←÷≈ ⇓↑ •↑ ←÷↔↑ ≠•♠≈″ ↵↑  →∝ ∝ →∝⌡  →∝ ∝ →∝⌡  ∫∫ ∫ →∝  ∝ →∝⌡  

≡±→ →∝ →∝≈♦° ƒ→∝ ↵″… •∝ •∝≈♦° ƒ•∝ ∫ ⊃÷•×≈

(5)

⇓∝   ♦ ♦→∝ ♣ ♣→∝ ♦ ♦→∝ ♣ ♣→∝ ∫∫ ∫ ∫∫∫ ♦  ♦→∝ ♣  ♣→∝ 

•↑ ←÷≈ ″±•↑≈ ♠≈←±→ ♥÷•÷ ÷↵↑ ƒ≈→± ≥≈↵″ ↵″… ↵   ±♠↵→•↵″≈ ≥↵←→•♦ ⇑ ≠•♠≈″ ♣  ⇑♠ ⇑♠ ∫∫ ∫ ∫∫∫ ∫ ∫∫ ∫∫∫ ⇑    ⇔♦°→≈↑↑•±″↑ ≡±→ ←÷≈ •″…•♠•…↔↵× ≈×≈≥≈″←↑ ±≡ ⇑ ↵→≈ ≠•♠≈″ ↵↑ ≡±××±♥↑∫ ⊂÷≈ …•↵≠±″↵× ≈×≈≥≈″←↑ ±≡⇑∝∝↵→≈ ≠•♠≈″ ♣ ⊕⇑∝∝ ••⇔ →∝ •∝ …→∝⌡…•∝  … •∝≈♦°   •∝   …  •∝⌡…→∝≈♦°   →∝   …  →∝  ↵″… ←÷≈ ±≡≡⌠…•↵≠±″↵× ≈×≈≥≈″←↑ ±≡ ⇑∝∝↵→≈ ≠•♠≈″ ♣ ⊕⇑∝∝ •∂ ⇔ →∝ •∝ …→∝⌡…•∝ →∝ ∂∝ …→∝⌡…∂∝  …→∝≈♦°   →∝   …  →∝   ⊂÷≈ …•↵≠±″↵× ≈×≈≥≈″←↑ ±≡⇑∝× ↵→≈ ≠•♠≈″ ↵↑ ⊕⇑∝× •• ⇔ →∝ •∝ …→∝⌡…•∝ →× •× …→×⌡…•×

↵ ⌡↵ ↵ ↵ ð

♥÷≈→≈ ↵   →∝  →× …→∝…→×≈♦°   →∝⌡ →× ⌡  ∝× →∝ →×   ↵   •∝  •× … •∝…•×≈♦°   •∝⌡ •×⌡  ∝× •∝ •×   ↵ …→∝…→× ↵″… ↵ …•∝…•× ∨•″↵××♣ ←÷≈ ±≡≡ …•↵≠±″↵× ≈×≈≥≈″←↑ ±≡ ⇑∝× ↵→≈ ≠•♠≈″ ♣ ⊕⇑♠∝×℘•∂  ⇔ →∝ •∝ …→∝⌡…•∝ →× ∂× …→×⌡…∂×    ¬≈→≈ ←÷≈ ±→→≈×↵←•±″ ≈←♥≈≈″ ←÷≈ ≥≈↵↑↔→≈≥≈″←↑ ±≡ ←÷≈ ↑↵≥≈ ⊂ ↵← …•≡≡≈→≈″← ⇓↑ •↑ ″±← ±″↑•…≈→≈…∫ ⊂÷•↑ ↵↑↑↔≥°←•±″ •↑ ↵↑≈… ↔°±″ ←♥± ↵→≠↔≥≈″←↑∫ ∨•→↑← ↵°↵→← ÷≈″≈ →≈←↵•″•″≠ ו←←×≈ ±→→≈×↵←•±″∫ ⊆≈±″… ≈♠≈″ •≡ ←÷≈ ⊆↑ ↵→≈ °×↵≈… ×±↑≈ ←± ≈↵÷ ±←÷≈→ …•↑←↵″≈ ↵″… ÷≈″≈ ±→→≈×↵←•±″ ≈←♥≈≈″ ←÷≈≥ •↑ ∝″±♥″ ↵″… ↵″ ≈↵↑•×♣ ≈ •″±→°±→↵←≈… •″ ∫ ♥÷≈→≈    →∝  →× …→∝…→×≈♦°   →∝⌡ →× ⌡  ∝× →∝ →×   ↵″…  …→∝…→×

⊂÷≈ ⊃⊆ ↑±×↔←•±″ •↑ ±←↵•″≈… ♣ ≥•″•≥•ƒ•″≠ ←÷≈ ±↑← ≡↔″⌠ ←•±″ ⊃ ⊆    ⇓  ⊂ ⇑♠   ⊂÷≈ ⊃⊆ ≈↑←•≥↵←≈ •↑ ±←↵•″≈… ↵↑ ≡±××±♥↑

⊃ ⊆⇓ƒƒ 

♥÷≈→≈ ⇓ƒ

↵″… ƒ

♠ ∫ ⊆•″≈ ←÷≈ ↵←↔↵× …•↑←↵″≈↑ ↵→≈ ″±← ↵♠↵•×↵×≈ ←÷≈•→ ≈↑←•≥↵←≈… ♠↵×↔≈↑ ↵→≈ ↔↑≈… •″ ∫ ⊂÷≈ ♠↵→•↵″≈ ↵″… ±→→≈×↵←•±″ ≈×≈≥≈″←↑ ±≡⇑↵″ ≈ ≈↑←•≥↵←≈… ↵↔→↵←≈×♣ ♣ ←÷≈ ↑↵≥°×≈ ≈↑←•≥↵←•±″ ±≡ ⇑♠≡±→ ↵ ×↵→≠≈ ″↔≥≈→ ±≡ →≈↵…•″≠↑ ±→ ↑″↵°↑÷±←↑∫ ∨±→ ↵ ו≥•←≈… ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ↵ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←±→ •↑ °→±°±↑≈… •″ ↑≈←•±″ ⋅⊇∫ ⊂÷≈ ⊃⊆ ←≈÷″•↓↔≈ °→≈↑≈″←≈… •↑ →≈×↵←•♠≈ ×±♥ •″ ±≥°×≈♦•←♣ ↵″… …±≈↑ ″±← →≈↓↔•→≈ ↵″ •″•←•↵× ≈↑←•≥↵←≈∑ ÷±♥≈♠≈→ •←↑ °≈→≡±→≥↵″≈ •↑ ↑↔⌠±°←•≥↵×∫ ⊂÷•↑ •↑ ≈↵↔↑≈ •″≡±→≥↵←•±″ ≡→±≥ ↵×× ←÷≈ ⊆↑ •↑ ″±← ↔←•×•ƒ≈… ↵↑ ←÷≈ ≥≈↵↑↔→≈≥≈″← ≡→±≥ →≈≡≈→≈″≈ ⊆ •↑ ↔↑≈… ←± ו″≈↵→•ƒ≈ ←÷≈ ↑♣↑←≈≥∫

(6)

    ×≈← ⇑ℜ ≈ ←÷≈ ↑↵≥°×≈ ≈↑←•≥↵←±→ ±≡ ←÷≈″ •←↑ ≈×≈≥≈″←↑ ↵→≈ ≠•♠≈″ ♣ ℜ ⇑ •∂  ∠  ∠ °

ƒ•° ƒ• ƒ∂° ƒ∂∑ 

♥÷≈→≈ •∑ ∂  ∑     ↵″… ƒ• ↵″… ƒ∂ →≈°→≈↑≈″← ←÷≈ ↑↵≥°×≈

≥≈↵″↑∫ ⊂÷≈ ↑↵≥°×≈ ≈↑←•≥↵←±→ •″  •↑ ↔″•↵↑≈… ↵″… ≈↵↑♣ ←± ≠≈″≈→↵←≈∫ ⇐≈↑°•←≈ ←÷≈↑≈ ↵…♠↵″←↵≠≈↑ ≡±→ ↵ ↑≥↵×× ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑∠⇑ℜ ♥•×× ≈♦÷••← ↵ ×↵→≠≈ ≈→→±→ ♠↵→•↵″≈ …↔≈ ←± ←÷≈ ×↵→≠≈ ″↔≥≈→ ±≡ ↔″∝″±♥″ °↵→↵≥≈←≈→↑ ←± ≈ ≈↑←•≥↵←≈…∫ ⊃≈ ↵″ ÷±♥≈♠≈→ •″←→±…↔≈ ↑±≥≈ ↑←→↔←↔→≈ ←± ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≠≈″≈→↵××♣ ♣ →≈…↔•″≠ ←÷≈ ″↔≥≈→ ±≡ ≡→≈≈ ♠↵→•↵×≈↑∫ ⊆↔÷ ↑←→↔←↔→≈… ≈↑←•≥↵←≈↑ ±≡≡≈→ ↵ →≈×↵←•♠≈×♣ ↑≥↵×× ≈→→±→ ♠↵→•↵″≈ ⇓″ ±°←•≥↵× ↑←↵←•↑←•↵× ↑±×↔←•±″ •↑ ←± →≈↵÷ ↵″ ±°←•≥↵× ←→↵…≈±≡≡ ≈←♥≈≈″ ←÷≈ ←÷≈ ↔″•↵↑≈… ↔← ÷•≠÷ ♠↵→•↵″≈ ≈↑←•≥↵←≈ ↵″… ←÷≈ ×±♥ ♠↵→•↵″≈ ↔← •↵↑≈… ≈↑←•≥↵←≈∫ ⊂÷↔↑ ←÷≈ ↔″•↵↑≈… ↑↵≥°×≈ ±♠↵→•↵″≈ •↑ ↑÷→•″∝≈… ←±♥↵→…↑ ↵ •↵↑≈… ↑←→↔←↔→≈… ±♠↵→•↵″≈ ≈↑←•≥↵←≈∫

⊂± ≡±→≥↵וƒ≈ ←÷≈ …•↑↔↑↑•±″ ↵″… …≈♠≈×±° ↵″ ≈♦°→≈↑↑•±″ ≡±→ ⇑ð ←± ≈ ←÷≈ ←→↔≈ ±♠↵→•↵″≈ ≥↵←→•♦ ±≡ ←÷≈ °↵←÷⌠×±↑↑ →≈↵…•″≠↑ ƒ∫ ≈← ⇑ℜ ≈ ↵″ ↔″•↵↑≈… ≈↑←•≥↵←≈ ±≡ ⇑ð∫ ≈← ↵×↑± ←÷≈ ↑÷→•″∝↵≠≈ ←↵→≠≈←⊂ ≈ ↵″±←÷≈→ •↵↑≈… ↑←→↔←↔→≈… ≈↑←•≥↵←≈ ±≡ ⇑ð ♥•←÷ →≈×↵←•♠≈×♣ ↑≥↵×× ″↔≥≈→ ±≡ °↵→↵≥≈←≈→↑ …↔≈ ←± ♥÷•÷ ⊂♥•×× ±″↑•↑← ±≡ ↵ →≈×↵←•♠≈×♣ ↑≥↵×× ≈→→±→ ♠↵→•↵″≈∫ ⊂÷↔↑ •″↑←≈↵… ±≡ ÷±±↑•″≠ ≈←♥≈≈″ ←÷≈ ←♥± ≈♦←→≈≥≈↑ ←÷≈ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←±→⊆±≥•″≈↑ ±←÷ ≈↑←•≥↵←≈↑ •″ ↵ ♥≈•≠÷←≈… ≡↵↑÷•±″ •∫≈∫

⌡  ⇑ℜ 

♥÷≈→≈ •↑ ←÷≈ ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ ↵″… •←↑ ♠↵×↔≈ •↑ ≈←♥≈≈″ð ↵″… ∫ ⋅← •↑ ″±←≈… ←÷↵← •≡   ←÷≈″ ←÷≈ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←≈ •↑ ←÷≈ ↑÷→•″∝↵≠≈ ←↵→≠≈← ↵″… ←÷≈ ↔″•↵↑≈… ±♠↵→•↵″≈ •↑ ≠•♠≈″ ″± ♥≈•≠÷←∫ ∇″ ←÷≈ ±←÷≈→ ÷↵″… •≡  ð∑←÷≈″ ″± ↑÷→•″∝↵≠≈ ←↵∝≈↑ °×↵≈ ↵″… ←÷≈ ↔″•↵↑≈… ≈↑←•≥↵←±→ •↑ ÷±↑≈″ ↵↑ ←÷≈ ↑÷→•″∝↵≠≈ ↑≈×≈← ←÷≈ ↑÷→•″∝↵≠≈ ←↵→≠≈←⊂↵″… ↑≈±″…×♣ ♥÷↵← ♠↵×↔≈ ↑÷±↔×… ≈ ≠•♠≈″ ←± ←÷≈ ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ ∫ ⇓ ″↔≥≈→ ±≡ ↑÷→•″∝↵≠≈ ←↵→≠≈←↑ ↵″ ≈ ↔↑≈… •″  ≈∫≠∫ ⊕℘ ו↑←↑ ↑•♦ ←↵→≠≈← ↑÷→•″∝↵≠≈ •″←≈″↑•←•≈↑∫ ⋅″ ≠≈″≈→↵× ↑≈×≈←•±″ ±≡ ⊂ ↑÷±↔×… ≈ …→•♠≈″ ♣ ←÷≈ ×±♥≈→ …•≥≈″↑•±″ ↑←→↔←↔→≈ •″ ←÷≈ …↵←↵∫ ⊂÷↔↑ •″ ←÷≈ ↵↑≈ ±≡ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ⇑ð ≡±→ ≥↔×←•°×≈ ⊂↑ ←÷≈ ″↵←↔→↵× ↑÷→•″∝↵≠≈ ←↵→≠≈← •↑ ←÷≈ ±″↑←↵″← ±→→≈×↵←•±″ ↑÷→•″∝↵≠≈ ←↵→≠≈←∫ ∨±→ ←÷≈ ±″↑←↵″← ±→→≈×↵←•±″ ↑÷→•″∝↵≠≈ ←↵→≠≈←⊕⊂ •• ℜ ⇑ ••↵″… ⊕⊂

•∂  ℜ ⇑ℜ ••

ℜ ⇑

∂∂ ♥÷≈→≈ ℜ•↑ ←÷≈ ↵♠≈→↵≠≈ ±→→≈×↵←•±″

±≡ ↵×× ←÷≈ ±→→≈×↵←•±″↑ ≈←♥≈≈″ ←÷≈ ″≈←♥±→∝ ″±…≈↑ •∫≈∫

ℜ  

      •   ∂• ℜ ⇑ •∂ ℜ ⇑ •• ℜ ⇑ ∂∂    ∇°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″↑•←♣

↑÷→•″∝↵≠≈ •″←≈″↑•←♣ ℜ ←÷≈ ∨→±≈″•↔↑ ″±→≥ ±≡ ←÷≈ …•≡≡≈→≈″≈ ≈←♥≈≈″ ←÷≈ ↑÷→•″∝↵≠≈ ≈↑←•≥↵←≈ ↵″… ←÷≈ ←→↔≈ ±♠↵→•↵″≈ ≥↵←→•♦ •↑ ≥•″•≥•ƒ≈… ♥•←÷ →≈↑°≈← ←± ∫ ⊂÷•↑ •↑ ↵÷•≈♠≈… ♣ ≥•″•≥•ƒ•″≠

←÷≈ ≡±××±♥•″≠ ±↑← ≡↔″←•±″

  ⇔ ⊂⌡  ⇑ℜ ⇑ð

∨  ±→ ≈↓↔•♠↵×≈″←×♣   ⇔   •   ∂ ⊕⊂ •∂⌡   ℜ ⇑ •∂ ⊕

ð

•∂ 

 ⇔♦°↵″…•″≠  ×≈↵…↑ ←±  ≠•♠≈″ ↵← ←÷≈ ←±° ±≡ ←÷≈ ″≈♦← °↵≠≈∫

∨•″↵××♣ ↔↑•″≠⇔ ⇑ℜ

•∂  ⊕

ð

•∂ •″ ð ←÷≈ ♠↵×↔≈ ±≡ℜ

•↑ ±←↵•″≈… ≡→±≥ ←÷≈ ≡±××±♥•″≠ ≈♦°→≈↑↑•±″

ℜ 

  •

 

∂ ⊇↵→ ⇑ℜ •∂ ⇑±♠ ⊕⊂℘•∂∑ ⇑ℜ •∂   •   ∂ ⇔ ⊕⊂℘•∂ ℜ ⇑ •∂    ⊆•″≈ ←÷≈ ♠↵→•↵″≈ ↵″… ±♠↵→•↵″≈↑ •″  ↵→≈ ↵×↑± ≈↑←•⌠ ≥↵←≈… ←÷≈♣ ↵→≈ →≈°×↵≈… ♥•←÷ ←÷≈•→ ↑↵≥°×≈ ≈↑←•≥↵←≈↑ ←± ±←↵•″ ←÷≈ ±°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ •∫≈∫ ℜ    •   ∂ ⊇↵→ ℜ ⇑ •∂ ⇑±♠

•∂ ⇑ℜ

•∂  

•  

∂ ⇔ ⊕⊂℘•∂ ⇑ℜ •∂

 



⇔♦°→≈↑↑•±″↑ ←± ±←↵•″ ⊇↵→ ⇑ℜ

•∂ ↵″… ⇑±♠ ⊕⊂ •∂∑ ℜ ⇑ •∂ ↵→≈ …≈→•♠≈… •″ ←÷≈ ⇓°°≈″…•♦∫

⊂÷≈ ♠↵×↔≈ ±≡ℜ⊕ð ℘•″ ≥±↑← ↵↑≈↑∫ ⋅″ ←÷≈ ↔″ו∝≈×♣ ≈♠≈″← •≡ ℜ ≈♦≈≈…↑ ←÷≈↑≈ ו≥•←↑ ←÷≈ ≡±××±♥•″≠ ±↔″… •↑ •≥°±↑≈…∫

ℜ  ≥↵♦ ð∑≥•″ ∑ℜ 

⊂÷↔↑ ←÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦⊆±←↵•″≈… ≡→±≥  •↑ ↔↑≈… •″ ←÷≈ ≈♥←±″ ↑←≈°  ←± ↔°…↵←≈ ←÷≈ ∧⊆ ↵×≠±→•←÷≥ ∫ ⊆•≥•×↵→×♣ ←÷≈ ♠↵→•↵″≈ ↵″… ±♠↵→•↵″≈ ♠↵×↔≈↑ ±←↵•″≈… ≡→±≥ ⊆ ↵→≈ ↔↑≈… •″ ←÷≈ ≠≈″≈→↵←•±″ ±≡ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ ≡±→ ←÷≈ ⊃⊆ ↵×≠±→•←÷≥∫ ⊇∫ ⊆⋅⊄⇓⊂⋅∇ ∈⇔⊆⊄⊂⊆

(7)

     •   ∂ ⊇↵→ ⊕ •∂ ⌡  

⊇↵→ ⇑ℜ

•∂ ⌡  ⇑±♠ ⊕ ⊂ •∂∑ ℜ ⇑ •∂ ⌡ ⊕ ⊂ •∂ ℜ ⇑ •∂ ⌡⇔ ℜ ⇑ •∂ ⊕

ð

•∂ 

 ⊂↵∝•″≠ …≈→•♠↵←•♠≈ ♥•←÷ →≈↑°≈← ←± ↵″… ≈↓↔↵←•″≠ ←± ð ♣•≈×…↑

  •   ∂  ♠↵→ ⊕⊂ •∂    ♠↵→ ℜ ⇑ •∂ ⌡     ⇑±♠ ⊕ ⊂ •∂∑ ℜ ⇑ •∂ ⌡  ⇔ ⊕⊂

•∂ ⇑ℜ

•∂ ⊕

•∂ ⇑ℜ

•∂ ⌡⇔

ℜ ⇑

•∂ ⊕

ð

•∂  ð ð

⌠ð ⌠ð ⌠ð ⌠ð ð ð ð ð ð

⌠ð ⌠ð ⌠ð ⌠ð ð ð ð ð ð ♦⌠↵♦•↑ ≥ ⊂ ⊆ ∨•≠↔→≈ ∫ ≈←♥±→∝ …≈°×±♣≥≈″←

ð  ð  ð  ð  ð

ð  ð  ð  ð ⊆″↵°↑÷±←↑ ∠

≈↑↵≥°×≈β•∂∝×⊕ ℘

≈↑÷→•″∝↵≠≈β•∂∝×⊕ ℘

≈↑↵≥°×≈β•∂∝×⊕ ℘

≈↑÷→•″∝↵≠≈β•∂∝×⊕ ℘

≈↑↵≥°×≈β•∂∝×⊕ ℘

≈↑÷→•″∝↵≠≈β•∂∝×⊕ ℘

∨•≠↔→≈ ∫ ⇔→→±→ ±≥°↵→•↑±″ ±≡ ←÷≈ ≈↑←•≥↵←≈… ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦⇑ℜ ↵″… ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦⊆ ∈  ð≥  ∑      … ð ≥  ðð

↵″… ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦ •∫≈∫≈↑÷→•″∝↵≠≈∝⊆ ⇑ð∝

↵″… ↵×↑± ≈←♥≈≈″ ←÷≈ ←→↔≈ ↵″… ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦ •∫≈

↑↵≥°×≈  ⇑ℜ ⇑ð

 ∨∫ ⊂÷≈ ±≥°↵→•↑±″ •↑ …±″≈ ≡±→ →↵″…±≥ ↑÷↵…±♥•″≠ ♠↵→•↵″≈ ≡±→ ≈↵÷ ו″∝ ≈←♥≈≈″ ←÷→≈≈ …•≡≡≈→≈″← ±↔″…↑ •∫≈∫ ≈↵÷ ו″∝ •↑ ≠•♠≈″ ↵ →↵″…±≥ •∂∝× ♠↵×↔≈ ≈←♥≈≈″ ⊕ ℘ ⊕ ℘ ↵″… ⊕ ℘∫ ⋅← •↑ ≈♠•…≈″← ≡→±≥ ∨•≠∫  ←÷↵← ≈↑÷→•″∝↵≠≈ •↑ ×±♥≈→ ←÷↵″ ←÷↵″ ≈↑↵≥°×≈ •″ ↵×× ←÷→≈≈ ↵↑≈↑∫ ⊂÷≈ …•≡≡≈→≈″≈ •″ °≈→≡±→≥↵″≈ •↑ ≥±→≈ °→±≡±↔″… ≡±→ ↑≥↵××≈→ ″↔≥≈→ ±≡ ∠ ⇓×↑± ≡±→ ×↵→≠≈→ •∂∝× ±↔″…↑ ×↵→≠≈→ ≈→→±→ •↑ ↑÷±♥″ ♣ ±←÷ ≈↑←•≥↵←±→↑∫

ð  ð  ð  ð  ð

ð∫ ð∫ ð∫ ð∫ ð∫ ð∫ ð∫ ð∫ ⊆″↵°↑÷±←↑ ∠ β •∂∝×⊕ ℘ β •∂∝×⊕ ℘ β •∂∝×⊕ ℘

∨•≠↔→≈ ∫ ⇓♠≈→↵≠≈ ♠↵×↔≈ ≡±→ ±°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ ℜℜ ±♠≈→  ðð

•″…≈°≈″…≈″← →↔″↑∫ ∑  ∑ ∈ ð≥    ð∑ … ð ≥∑

∨•≠∫  ±≥°↵→≈↑ ←÷≈ ♠↵×↔≈↑ ±≡ ←÷≈ ±°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″⌠ ↑•←♣ ≡±→ …•≡≡≈→≈″← ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ∠ ↵″… …•≡≡≈→≈″← ↑≈←↑ ±≡ ↑÷↵…±♥•″≠ ♠↵→•↵″≈ 

•∂∝×∫ ∨→±≥ ∨•≠∫  ♥≈ …≈…↔≈ ←÷≈

≡±××±♥•″≠∑ • ←÷≈ ±°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ •↑ •″…≈≈… ≈←♥≈≈″ ←÷≈ ←♥± ≈♦←→≈≥≈↑ •∫≈∫ ℜℜ⊕ð ℘∫ •• ℜℜ …≈→≈↵↑≈↑ ↵↑ ←÷≈ ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ •″→≈↵↑≈↑ ←÷•↑ •↑ ≈♦°≈←≈… ↵↑ ♥•←÷ ≥±→≈ ↑″↵°↑÷±←↑ ←÷≈ ↑↵≥°×≈ ≈↑←•≥↵←±→⇑ℜ ≈±≥≈↑ ↵ ≥±→≈ ↵↔→↵←≈ ≈↑←•≥↵←±→ ±≡ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦⇑ð ↵″… ÷≈″≈ ↵ ×±♥≈→ ♥≈•≠÷← •↑ ≠•♠≈″ ←± ←÷≈ ←↵→≠≈← ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ⊂↵±→…•″≠ ←± ∫ •••ℜℜ •↑ ×↵→≠≈→ ♥÷≈″ ←÷≈ ±↔″… ±≡ ↑÷↵…±♥•″≠ ♠↵→•↵″≈ 

•∂∝ו↑ ×↵→≠≈

(8)

ð  ð  ð  ð  ð ð     ⊆″↵°↑÷±←↑ ∠

⊆β

•∂∝×⊕ ð℘

⊃⊆↑↵≥°×≈β•∂∝×⊕ ð℘

⊃⊆↑÷→•″∝↵≠≈β•∂∝× ⊕ ð℘

⊆β

•∂∝×⊕ ℘

⊃⊆↑↵≥°×≈β•∂∝×⊕ ℘

⊃⊆↑÷→•″∝↵≠≈β•∂∝× ⊕ ℘

∨•≠↔→≈ ∫ ⇓♠≈→↵≠≈ ∈⊆⇔ ±≥°↵→•↑±″ ≈←♥≈≈″ ⊃⊆↑÷→•″∝↵≠≈ ↵″…

⊃⊆↑↵≥°×≈  ∑  ∑ ∈ ð≥    ð∑ … ð ≥

≈↑←•≥↵←•±″ ±≡ ←÷≈ ±♠↵→•↵″≈ ≥↵←→•♦ °≈→≡±→≥↑ ≈←←≈→ ←÷↵″ ⊃⊆↑↵≥°×≈

↑≥↵××≈→ ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑∫ ¬±♥≈♠≈→ ≈♠≈″ ≡±→ ×↵→≠≈→ ♠↵×↔≈↑ ±≡∠⊃⊆↑÷→•″∝↵≠≈↑←•×× °≈→≡±→≥↑ ↑↔°≈→•±→ ←±⊃⊆↑↵≥°×≈∫

ð

ð ð  ð  β •∂∝× ⊆∠  ⊃⊆↑↵≥°×≈∠  ⊃⊆↑÷→•″∝↵≠≈∠ 

⊆∠ ð ⊃⊆↑↵≥°×≈∠ ð

⊃⊆↑÷→•″∝↵≠≈∠ ð

∨•≠↔→≈ ∫ ⇓♠≈→↵≠≈ ∈⊆⇔ ±≥°↵→•↑±″ ≈←♥≈≈″ ⊃⊆↑÷→•″∝↵≠≈ ↵″…

⊃⊆↑↵≥°×≈  ∑  ∑ ∈ ð≥    ð∑ … ð ≥

⋅″ ∨•≠∫  ♥≈ ±≥°↵→≈ ←÷≈ ↵♠≈→↵≠≈ ∈⊆⇔ ±≡ ←÷≈ ×±↵←•±″ ≈↑←•≥↵←≈ ±≡ ⊃⊆↑↵≥°×≈∑ ⊃⊆↑÷→•″∝↵≠≈ ↵″… ⊆ ±→→≈⌠

↑°±″…•″≠ ←± •″→≈↵↑•″≠ ♠↵×↔≈ ±≡ ↑÷↵…±♥•″≠ ♠↵→•↵″≈ •∂∝× 

⋅″ ←÷•↑ ↵↑≈ •∂∝× •↑ ↑↵≥≈ ≡±→ ↵×× ⊆⌠⊂ ו″∝↑ •∫≈∫ •∂∝×  •∑ ∂∑ ∝∑ ×

∠  ↵″…∠  ð⋅← •↑ ↵≠↵•″ ″±←≈… ←÷↵← ←÷≈⊃⊆↑÷→•″∝↵≠≈

°≈→≡±→≥↑ ↑↔°≈→•±→ ←± ±←÷ ⊃⊆↑↵≥°×≈ ↵″… ⊆∫ ⋅← •↑ ↵×↑± •″←≈→≈↑←•″≠ ←± ″±←≈ ←÷↵← ←÷≈ ≡±→∠  ∑⊃⊆↑↵≥°×≈ °≈→≡±→≥↑ ≈♠≈″ ♥±→↑≈ ←÷↵″ ←÷≈ ⊆ ←÷•↑ •↑ ↵×↑± ≈♠•…≈″← ≡→±≥ ∨•≠∫  ≡±→ ∠  ⊂÷≈ →≈↵↑±″ ≈•″≠ ←÷≈ ×↵→≠≈ ≈→→±→ •″ ←÷≈ ↑↵≥°×≈ ≈↑←•≥↵←≈ ℜ ⇑♥•←÷ ↵ ↑≥↵×× ″↔≥≈→ ±≡ ∨•≠∫  ±≥°↵→≈↑ ←÷≈ °≈→≡±→≥↵″≈ ±≡ ←÷≈ ∧⊆ ↵×≠±→•←÷≥ ♥•←÷ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦∧⊆↑↵≥°×≈∑∧⊆ ♥•←÷ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦ ∧⊆↑÷→•″∝↵≠≈↵″… ←÷≈ ∧⊆ ≈↑←•≥↵←±→ ♥•←÷ •…≈″←•←♣ ±♠↵→•↵″≈ ≥↵←→•♦ ∧⊆•…≈″←•←♣∫ ⊂÷≈ ↵♠≈→↵≠≈ ∈⊆⇔ ±≡ ↵×× ←÷≈ ⊂↑ •↑ ±≥°↵→≈… ♥•←÷ ←÷≈ ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ∠ ⊂÷≈ ≈″←≈→ ±≡ ←÷≈ ″≈←♥±→∝ •↑ ↑≈×≈←≈… ↵↑ ←÷≈ •″•←•↵× ≈↑←•≥↵←≈ ≡±→

ð  ð  ð  ð  ð ð    ⊆″↵°↑÷±←↑ ∠

∧⊆↑↵≥°×≈β•∂∝×⊕ ð℘

∧⊆•…≈″←•←♣β•∂∝× ⊕ ð℘

∧⊆↑÷→•″∝↵≠≈β•∂∝×⊕ ð℘

∨•≠↔→≈ ∫ ⇓♠≈→↵≠≈ ∈⊆⇔ ±≥°↵→•↑±″ ≈←♥≈≈″ ∧⊆↑÷→•″∝↵≠≈ ↵″…

∧⊆↑↵≥°×≈  ∑  ∑ ∈ ð≥      ð∑ …

ð ≥∑ 

∝ ⊕ð∑ð℘ ⊂

↵×× ⊂↑ •∫≈∫ ∝  ⊕ð∑ð℘ ⊂

∝ ⊂÷≈ ∧⊆ ↵×≠±→•←÷≥ •↑ •←≈→↵←≈…   ←•≥≈↑∫ ⇐↔≈ ←± ←÷≈ ×↵∝ ±≡ ↑←→↔←↔→≈ •″ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦ ⇑ℜ •← •↑ ″±″⌠•″♠≈→←•×≈ ≡±→ ∠  ð ↵″… ÷≈″≈ ∧⊆↑↵≥°×≈ °→±…↔≈↑ ″± →≈↑↔×← ≡±→ ←÷≈↑≈ ♠↵×↔≈↑ ±≡ ∠

∠  ð•↑ ♣ ″± ≥≈↵″↑ ↵ ≥•″•≥↔≥ ♠↵×↔≈ ≡±→ ←÷≈ •″♠≈→←••×•←♣ ±≡⇑ℜ•″ ≡↵← ←÷≈ ≥•″•≥↔≥ ″↔≥≈→ ±≡ ↑″↵°↑÷±←↑ ♥•×× •″→≈↵↑≈ ♥•←÷ ←÷≈ •″→≈↵↑≈ •″ ←÷≈ ″↔≥≈→ ±≡ ⊂↑∫ ∨±→ ∠  ð ←÷≈ ∧⊆↑↵≥°×≈↑÷±♥↑ ↔″↵≈°←↵×≈ ≈→→±→ ↵″… °≈→≡±→≥↑ ♥±→↑≈ ←÷↵″ ∧⊆•…≈″←•←♣ ∇″ ←÷≈ ±←÷≈→ ÷↵″… ←÷≈ ↑÷→•″∝↵≠≈ ±♠↵→•↵″≈ ≥↵←→•♦ ⊆ •↑ ±←÷ •″♠≈→←•×≈ ↵″… ±″↑•↑←↑ ±≡ ↵ ↑≥↵×× ≈→→±→ ♠↵→•↵″≈ ±″↑≈↓↔≈″←×♣ ∧⊆↑÷→•″∝↵≠≈ °≈→≡±→≥↑ ≈←←≈→ ←÷↵″ ∧⊆↑↵≥°×≈ ↵″…∧⊆•…≈″←•←♣ ⊇⋅∫ ⇑∇⇑⊄⊆⋅∇⊆

(9)

⇓⇑∏∇⊃⇔⇐∧⇔⊂ ⊃≈ ↵∝″±♥×≈…≠≈ ←÷≈ ↑↔°°±→← ≡→±≥ ←÷≈ ⊄∏ ⇔″≠•″≈≈→•″≠ ↵″… ∠÷♣↑•↵× ⊆•≈″≈↑ ∈≈↑≈↵→÷ ⇑±↔″•× ⇔∠⊆∈⇑ ≡±→ ←÷≈ ↑↔°°±→← ♠•↵ ←÷≈ ⇒↵♣≈↑•↵″ ⊂→↵∝•″≠ ↵″… ∈≈↵↑±″•″≠ ±♠≈→ ⊂•≥≈ ⇒⊂↵∈±⊂ ≠→↵″← ⇔∠〉∏ð〉∫ ⇓∠∠⇔⇐⋅∪ ∨±××±♥•″≠ ⊆÷↵≡≈→ ↵″… ⊆←→•≥≥≈→ ⊕℘ ↵″… ∏♥↵″ ⊕℘ ♥÷±↑≈ →≈↑°≈←•♠≈×♣ ♥≈ …≈→•♠≈ ±°←•≥↵× ↑÷→•″∝↵≠≈ •″←≈″↑•←♣ •″ ←÷≈

∨±→ ∠ ↑″↵°↑÷±←↑ ±≡ °↵←÷⌠×±↑↑ ≥≈↵↑↔→≈≥≈″←↑ ƒ• ←÷≈ ↑↵≥°×≈

≥≈↵″ •↑ ≠•♠≈″ ♣ƒ•∠  ∠°ƒ•°∫ ≈←

♠•∂° ƒ•° ƒ• ƒ∂° ƒ∂ 

≡±→ •∑ ∂  ∑ ∑   ↵″… °  ∑ ∑ ∠ ⇓×↑± ×≈← ♠•∂  ∠  ∠ °♠•∂°⊂÷≈″ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ •↑ ≠•♠≈″ ♣ ℜ ⇑ •∂ ∠ ∠ ♠•∂  ∠  ∠ °

ƒ•° ƒ• ƒ∂° ƒ∂ 

∨±→ ≈↵↑≈ ±≡ ↔″…≈→↑←↵″…•″≠ ♠•∂° ↑÷±↔×… ≈ ♠•≈♥≈… ↵↑ ↵

→↵″…±≥ ♠↵→•↵×≈∫ ⊂÷≈ ↔″•↵↑≈… ♠↵→•↵″≈ ±≡ •″…•♠•…↔↵× ≈×≈≥≈″←↑ ±≡ ⇑ℜ •↑ ≠•♠≈″ ♣

⊇↵→ ⇑ℜ

•∂  ∠ ∠ ⊇↵→ ♠•∂  →↵″…±≥ ♠↵→•↵×≈ ♥≈ ÷↵♠≈ ⊇↵→ ♠•∂   ∠♠↵→ ♠•∂  ±→ ⊇↵→ ♠•∂   ∠  ∠  ∠ ° ♠•∂° ♠•∂   ⊂÷↔↑ ←÷≈ ♠↵→•↵″≈ ±≡ ←÷≈ ↑↵≥°×≈ ±♠↵→•↵″≈ ≥↵←→•♦ •↑ ≠•♠≈″ ♣

⊇↵→ ⇑ℜ

•∂  ∠ ∠  ∠ ° ♠•∂° ♠•∂  ∇″ ←÷≈ ±←÷≈→ ÷↵″… ←÷≈ ±♠↵→•↵″≈ ≈×≈≥≈″←↑ ↵→≈ ≠•♠≈″ ↵↑

⇑±♠ ⇑ℜ

•∂∑ ℜ ⇑ ∝×  ∠ ∠  ∠ ° ♠•∂° ♠•∂ ♠∝×° ♠∝×  ⇓″ ≈♦°→≈↑↑•±″ ≡±→ ⇑±♠ ⊕⊂ •∂∑ ℜ ⇑ •∂ •↑ ↑ו≠÷←×♣ ≥±→≈

±≥°×•↵←≈… ÷≈→≈ ♥≈ ↔↑≈ ←÷≈ ↑± ↵××≈… ℵ…≈×←↵ ≥≈←÷±…∫ ∨±→ ←÷•↑ °↔→°±↑≈ ♥≈ ±″↑•…≈→ ←÷≈ •″…•♠•…↔↵× ≈×≈≥≈″←↑ ±≡⇑ℜ ↵↑ ♠↵→•↵×≈↑ ↵″… ≈♦°↵″…⊕⊂

•∂  ℜ ⇑ℜ •• ℜ ⇑ ∂∂ ♠•↵ ⊂↵♣×±→ ↑≈→•≈↑ ↵→±↔″… ←÷≈ ±→→≈↑°±″…•″≠ °±•″← ≈↑←•≥↵←≈↑ ⇑ •• ⇑ ∂∂ ↵″… ⇑ •∂ °→±♠•…≈… ♣ ←÷≈ ↑↵≥°×≈↑∫ ⊃≈ ÷↵♠≈ ⊕⊂ •∂ ⇑ •• ⇑ ∂∂⌡  ⇑ ∂∂ ⇑ •• ⊕⇑ •• ⇑ •• ⌡ ⇑ •• ⇑ ∂∂ ⊕⇑ ∂∂ ⇑ ∂∂ ∑ ð ♥÷≈→≈ •↑ ±←↵•″≈… ↔↑•″≠  ≡±→⇑••∂∂ ↵″…•∂ ⇑±♠ ⊕⊂

•∂∑ ⇑ℜ •∂ 

⇔ ⊕⊂

•∂ ⇔ ⊕⊂℘•∂ ⊕⇑℘•∂ ⇔ ⊕⇑℘•∂ 

↔↑•″≠ ð •″  ×≈↵…↑ ←± ⇑±♠ ⊕⊂

•∂∑ ⇑ℜ •∂   ⇑ ∂∂ ⇑ ••

⇑±♠ ⇑ℜ

••∑ ℜ ⇑ •∂ ⌡ ⇑ •• ⇑ ∂∂

⇑±♠ ⇑ℜ

∂∂∑

ℜ ⇑

•∂ 

⇑±♠ ⇑ℜ

••∑ ℜ ⇑ •∂ ↵″…⇑±♠ ℜ ⇑ ∂∂∑ ℜ ⇑ •∂ ↵↑

⇑±♠ ⇑ℜ

••∑ ℜ ⇑ •∂  ∠ ∠  ∠ °

ƒ•° ƒ• ♠•• ⊕ƒ•° ƒ• ƒ∂° ƒ∂ ♠•∂℘

 ↵″… ↑•≥•×↵→×♣

⇑±♠

∂∂∑ ℜ ⇑ •∂  ∠ ∠  ∠ °

ƒ∂° ƒ∂ ♠∂∂ ⊕ƒ•° ƒ• ƒ∂° ƒ∂ ♠•∂℘ ∑

 ♥÷•÷ ±≥°×≈←≈↑ ←÷≈ …≈→•♠↵←•±″∫

∈⇔∨⇔∈⇔⇑⇔⊆

⊕℘ ∇∫ ≈…±•← ↵″… ∫ ⊃±×≡ •¬±″≈♣ ⋅ ⊆÷→↔″∝ ←÷≈ ⊆↵≥°×≈ ⇑±♠↵→•↵″≈ Ó¿¬®·¨Œô√±↔→″↵× ±≡ ∠±→←≡±×•± ↵″↵≠≈≥≈″← ð⌠ ðð∫

⊕℘ ∇∫ ≈…±•← ↵″… ∫ ⊃±×≡ •⋅≥°→±♠≈… ⇔↑←•≥↵←•±″ ±≡ ←÷≈ ⇑±♠↵→•↵″≈ ↵←→•♦ ±º ͬ±½µ 묫®²­ ©·¬¸ ¿² ß°°´·½¿¬·±² ¬± ᮬº±´·± Í»´»½¬·±²Œô√±↔→″↵× ±≡ ⇔≥°•→•↵× ∨•″↵″≈ ð ð⌠ ðð∫

⊕℘ √∫ ⊆÷↵≡≈→ ∏∫ ⊆←→•≥≥≈→ •⇓ ⊆÷→•″∝↵≠≈ ⇓°°→±↵÷ ←± ↵→≠≈⌠⊆↵×≈ ⇑±⌠ ª¿®·¿²½» Ó¿¬®·¨ Û­¬·³¿¬·±² ¿²¼ ׳°´·½¿¬·±²­ º±® Ú«²½¬·±²¿´ Ù»²±³·½­ôŒ ⊆←↵←•↑←•↵× ⇓°°×•↵←•±″↑ •″ ∧≈″≈←•↑ ↵″… ±×≈↔×↵→ ⇒•±×±≠♣ ⊇±×∫  ″±∫  ðð∫

⊕℘ ∫ ∠↵←♥↵→• √∫ ∫ ⇓↑÷ ⊆∫ ∏♣°≈→±↔″←↵↑ ⇓∫ ∇∫ ¬∫ ⋅⋅⋅ ∈∫ ∫ ±↑≈↑ ↵″… ∫ ⊆∫ ⇑±→→≈↵× •±↵←•″≠ ←÷≈ ″±…≈↑ ±±°≈→↵←•♠≈ ×±↵וƒ↵←•±″ •″ ♥•→≈×≈↑↑ ­»²­±® ²»¬©±®µ­ôŒ ⋅⇔⇔⇔ ⊆•≠″↵× ∠→±≈↑↑•″≠ ↵≠∫ ♠±×∫  ″±∫  °°∫ ëìŠêçô Ö«´ò îððëò

⊕℘ ⋅∫ ∧↔♠≈″ ↵″… ⇑∫⌠⇑∫ ⇑÷±″≠ •⇓ ↑↔→♠≈♣ ±″ ⊂∇⇓ ↵↑≈… ♥•→≈×≈↑↑ ×±↵×⌠ ·¦¿¬·±² ¿²¼ ÒÔÑÍ ³·¬·¹¿¬·±² ¬»½¸²·¯«»­ôŒ⋅⇔⇔⇔ ⇑±≥≥↔″∫ ⊆↔→♠≈♣↑ ↵″… ⊂↔←±→•↵×↑ô ª±´ò ïïô ²±ò íô °°ò ïðéŠïîìô ß«¹ò îððçò

⊕℘ ⇒∫⇐∫ ⊇↵″ ⊇≈≈″ ↵″… ∏∫∫ ⇒↔∝×≈♣ •⇒≈↵≥≡±→≥•″≠ ⇓ ♠≈→↑↵←•×≈ ↵°°→±↵÷ ⋅⇔⇔⇔ ⇓⊆⊆∠ ↵≠òô ª±´ò ëô ²±ò îô °°ò ìŠîìô ß°®ò ïçèèò ⊕℘ ⇒∫ ∇←←≈→↑←≈″ ∫ ⊇•≈→≠ ∠∫ ⊆←±•↵ ↵″… ⇓∫ ≈÷±→↵• •⇔♦↵← ↵″… ×↵→≠≈ ↑↵≥°×≈  ←≈÷″•↓↔≈↑ ≡±→ °↵→↵≥≈←≈→ ≈↑←•≥↵←•±″ ↵″… …≈←≈←•±″ •″ ↵→→↵♣ °®±½»­­·²¹ôŒ ·²∈↵…↵→ ⇓→→↵♣ ∠→±≈↑↑•″≠ ⊆∫⊆∫ ¬↵♣∝•″ √∫ •←♠↵ ↵″… ⊂∫ ͸»°¸»®¼ô Û¼­ò Ò»© DZ®µæ Í°®·²¹»®ó Ê»®´¿¹ô ïççíô °°ò ççŠïëïò ⊕℘ ∫ ∠↵←♥↵→• ⇓∫ ∇∫ ¬≈→± ⋅⋅⋅ ∫ ∠≈→∝•″↑ ∫ ⊆∫ ⇑±→→≈↵× ↵″… ∈∫ √∫ ∇⇐≈↵

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