• No results found

Finite-difference calculation of the electronic structure of artificial graphene, the 2D hexagonal AlwGa1-wAs/GaAs structure with tunable interactions

N/A
N/A
Protected

Academic year: 2019

Share "Finite-difference calculation of the electronic structure of artificial graphene, the 2D hexagonal AlwGa1-wAs/GaAs structure with tunable interactions"

Copied!
22
0
0

Loading.... (view fulltext now)

Full text

(1)

This is a repository copy of Finite-difference calculation of the electronic structure of

artificial graphene, the 2D hexagonal AlwGa1-wAs/GaAs structure with tunable

interactions.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/88011/

Version: Accepted Version

Article:

Thirayatorn, R, Moontragoon, P, Amornkitbamrung, V et al. (2 more authors) (2015)

Finite-difference calculation of the electronic structure of artificial graphene, the 2D

hexagonal AlwGa1-wAs/GaAs structure with tunable interactions. Computer Physics

Communications, 191. 106 - 118. ISSN 0010-4655

https://doi.org/10.1016/j.cpc.2015.02.005

(c) 2015, Elsevier. Licensed under the Creative Commons

Attribution-NonCommercial-NoDerivatives 4.0 International

http://creativecommons.org/licenses/by-nc-nd/4.0/

[email protected] https://eprints.whiterose.ac.uk/

Reuse

Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website.

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

(2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

≠→↵°÷≈″≈ ←÷≈ ⇐ ÷≈♦↵≠±″↵× ⇓× ∧↵

⇓↑〉∧↵⇓↑ ↑←→↔←↔→≈ ♥•←÷

←↔″↵×≈ •″←≈→↵←•±″↑

∈∫ ⊂÷•→↵♣↵←±→″↵ ∠∫ ±±″←→↵≠±±″↵  ⊇∫ ⇓≥±→″∝•←↵≥→↔″≠↵ ⊆∫ ≈↵″↑•→• ∅∫ ⋅∝±″•

⇐≈°↵→←≥≈″← ±≡ ∠÷♣↑•↑ ⊆•≈″≈ ∏÷±″ ∏↵≈″ ⊄″•♠≈→↑•←♣ ∏÷±″ ∏↵≈″ ððð ⊂÷↵•×↵″…∫

⋅″←≈≠→↵←≈… ↵″±←≈÷″±×±≠♣ ∈≈↑≈↵→÷ ⇑≈″←≈→ ⋅∈⇑ ⇐≈°↵→←≥≈″← ±≡ ∠÷♣↑•↑ ⊆•≈″≈ ∏÷±″ ∏↵≈″ ⊄″•♠≈→↑•←♣

∏÷±″ ∏↵≈″ ððð ⊂÷↵•×↵″…∫

⊆÷±±× ±≡ ∠÷♣↑•↑ ⊆↔→↵″↵→≈≈ ⊄″•♠≈→↑•←♣ ±≡ ⊂≈÷″±×±≠♣ ↵∝÷±″ ∈↵←÷↵↑•≥↵ ðððð ⊂÷↵•×↵″…∫

⊆÷±±× ±≡ ⇔×≈←→±″• ↵″… ⇔×≈←→•↵× ⇔″≠•″≈≈→•″≠ ⊄″•♠≈→↑•←♣ ±≡ ≈≈…↑ ≈≈…↑ ⊄∏∫

⇓↑←→↵←

⊂÷≈ ≈″≈→≠♣ …•↑°≈→↑•±″ →≈×↵←•±″ ±≡ ←♥± …•≥≈″↑•±″↵× ÷≈♦↵≠±″↵× ×↵←←•≈ ±≡ ∧↵⇓↑ ↓↔↵″←↔≥ ♥•→≈↑ ≈≥⌠ ≈……≈… •″ ⇓× ∧↵

÷≈∝≈… ♣ ±≥°↵→•″≠ ←÷≈ ±↔″… ↑←↵←≈ ≈″≈→≠•≈↑ ±≡ ♠↵→•±↔↑ ←♥± …•≥≈″↑•±″↵× ↑♣↑←≈≥↑ ♥•←÷ ↵°°→±⌠ °→•↵←≈ ±↔″…↵→♣ ±″…•←•±″↑ ♥•←÷ ↵″↵×♣←• ↑±×↔←•±″↑ ±→ ←÷≈ →≈↑↔×←↑ ±←↵•″≈… ♣ ⇑∇⊆∇ ↑±≡←♥↵→≈

→≈↵× ≠→↵°÷≈″≈ ↵″… ↵→≈ ←↵•×±→↵×≈ ♣ ↵°°→±°→•↵←≈ ↑←→↔←↔→≈ ≈″≠•″≈≈→•″≠∫

∏≈♣♥±→…↑

↵″… ↑←→↔←↔→≈ ≈″≠•″≈≈→•″≠

ðð ⊆⇑ ⇐ ⇐⇐ð

∫ ⋅″←→±…↔←•±″

∧→↵°÷≈″≈ ↵ ←♥±⌠…•≥≈″↑•±″↵× ⇐ ÷≈♦↵≠±″↵× →♣↑←↵×ו″≈ ↵××±←→±°≈ ♥•←÷ ↵ →≈≠↔×↵→ ↑°⌠±″…•″≠ ±≡ ↵→±″ ÷↵↑ ≈≈″ ↵←←→↵←•″≠ ←÷≈ →≈↑≈↵→÷ ↵←←≈″←•±″ …↔≈ ←± ↵ ♠↵→•≈←♣ ±≡ •←↑ ↔″•↓↔≈ °→±°≈→←•≈↑ ≈•″≠ ↵ ↑←→±″≠ ו≠÷← ″≈↵→×♣ ←→↵″↑°↵→≈″← ≥↵←≈→•↵× ↵″… ↵″ ≈♦≈××≈″← ±″…↔←±→ ±≡ ÷≈↵← ↵″… ≈×≈←→•⌠ •←♣ ⊕ ℘∫ ⋅″ °↵→←•↔×↵→ •← ↑÷±♥↑ ↵″ ↔″↔↑↔↵× ≡≈↵←↔→≈ •″ •←↑ ⇐ ↵″… ↑←→↔←↔→≈ ♥•←÷ ≥↵↑↑×≈↑↑ ⇐•→↵

≡≈→≥•±″↑ ⇐•→↵ ±″≈↑ ↑÷±♥•″≠ •″←→•″↑•↵××♣ ↔×←→↵÷•≠÷ ↵→→•≈→ ≥±•ו←•≈↑ ↑•″≈ ←÷≈ ↓↔↵↑•°↵→←•×≈↑ •″ •← ≈÷↵♠≈ ו∝≈ →≈×↵←•♠•↑←• ≈×≈≥≈″←↵→♣ °↵→←•×≈↑ ♥•←÷ ƒ≈→± →≈↑← ≥↵↑↑ ⊕ ℘∫ ⇐↔≈ ←± ←÷≈↑≈ ±↔←↑←↵″…⌠ •″≠ ÷↵→↵←≈→•↑←•↑ ≠→↵°÷≈″≈ ↑÷±♥↑ ≈♦←→↵±→…•″↵→♣ °±←≈″←•↵× ≡±→ …≈♠•≈ ↵°°×•↵←•±″↑ ו∝≈ ≠→↵°÷≈″≈

⇑±→→≈↑°±″…•″≠ ↵↔←÷±→

⇔≥↵•× ↵……→≈↑↑ ≥°↵•→±◊∝∝↔∫↵∫←÷∠∫ ±±″←→↵≠±±″

(3)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

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

ð

→≈°×↵•″≠ ≥≈←↵×↑ •″ ←÷≈ ≥↵″↔≡↵←↔→≈ ±≡ ↵•→→↵≡← ↵″… ↵→↑ ♣ ≠→↵°÷≈″≈⌠°×↵↑←• ±≥°±↑•←≈↑∫

↵ ←↔″↵×≈ °×↵←≡±→≥ ≡±→ •″♠≈↑←•≠↵←•″≠ ≥↵↑↑×≈↑↑ ⇐•→↵ ↓↔↵↑•°↵→←•×≈↑ ↑↔÷ ↵↑ ←÷≈•→ ≈÷↵♠•±→ ↔″…≈→     ℘∫ ⊂÷≈↑≈ ↑←→↔←↔→≈↑ ↵″ ≈ ≡↵→•↵←≈… ♣ ↵…♠↵″≈… ″↵″±↑↵×≈ ≡↵→•↵←•±″ ←≈÷″±×±≠♣ °→±≈↑↑≈↑ ♥÷•÷ ≈″↵×≈ ÷•≠÷ ↓↔↵×⌠



≠→↵°÷≈″≈ ↑←→↔←↔→≈↑ ♥÷•÷ ♥≈→≈ …≈↑•≠″≈… ↵″… ≡↵→•↵←≈… ↵→≈ ↑♣″←÷≈←• ″↵″±°↵←←≈→″≈… ∧↵⇓↑ ÷≈←⌠ ≈→±↑←→↔←↔→≈↑ ♥•←÷ ↔×←→↵÷•≠÷⌠≥±•ו←♣ ⇐ ≈×≈←→±″ ≠↵↑ ⊕ð      ℘∫ ⊂÷≈↑≈ ♥≈→≈

±≡ •∝≈× …•↑∝↑ ♥•←÷ ←÷≈ ÷±″≈♣±≥ ≠≈±≥≈←→♣ ↵″… ←÷≈″ ♣ ≈←÷•″≠ ↵♥↵♣ ←÷≈ ≥↵←≈→•↵× ±↔←↑•…≈ ←÷≈

ð

…•↑∝↑ ♣ •″…↔←•♠≈⌠±↔°×≈… →≈↵←•♠≈ •±″ ≈←÷•″≠∫ ⊆↵″″•″≠ ←↔″″≈ו″≠ ≥•→±↑±°≈ ⊆⊂ ♥↵↑ ↔↑≈… ←± °↵←←≈→″ ←÷≈ ∇♦♣≠≈″ ↵″… ⇑↵→±″ ↵←±≥↑ ±≡ ⇑∇ ≥±×≈↔×≈↑ •″ ↵ ÷±″≈♣±≥ °↵←←≈→″∫ ⇓″±←÷≈→ ≈♦↵≥°×≈

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



∧↵ ⇓↑〉∧↵⇓↑〉⇓× ∧↵ ⇓↑

↑♣↑←≈≥∫

∫ ⇑±≥°↔←↵←•±″↵× ≥≈←÷±…

↑←→↔←↔→≈ ±≡ ⇓× ∧↵ ⇓↑〉∧↵⇓↑〉⇓× ∧↵ ⇓↑ •↑ ±″↑•…≈→≈… ♥•←÷ ←÷≈ ⇓× ±″←≈″←   ≈↓↔↵× ←± ð∫ð∫ ð

±″…↔←•±″ ↵″… …•↑±″←•″↔•←♣ ≈←♥≈≈″ ⇓× ∧↵ ⇓↑ ↵″… ∧↵⇓↑ •↑ ←↵∝≈″ ↵↑ ð∫ ⊕≈⊇℘ ≡±→ ð 

⊕  

≈♦•↑←↑ ⊕ð   ♥↵↑ ÷≈→≈



↵°°→±♦•≥↵←≈… ♣ ↵ ↑•≥°×≈→ ≈♦°→≈↑↑•±″

   °

 ð⌡ ð 

⌡  ⌡

  •↑ …≈↑→•≈… ♣

   ° ð 

 ð⌡ ð  ⌡ 



♥÷≈→≈ ð ð •↑ ←÷≈ ≈″←≈→ ±≡ ←÷≈ ∧↵⇓↑ ♥•→≈

↵″… ←÷≈ ♥•→≈ →↵…•↔↑ ↵↑ ↑÷±♥″ •″ ∨•≠∫  ↵∫

(4)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

↵

♦ ″≥

″

 ð  ð  ð

⌠ ⌠ ⌠ ⌠ ð    

⌠ð∫ ⌠ð∫ ⌠ð∫ ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð



∨•≠↔→≈  ð∧↵ð⇓↑〉 ∧↵⇓↑〉 ⇓×ð∧↵ð⇓↑ ↵″…  ♥•←÷•″ ↵ →÷±≥±•…↵× ↔″•← ≈××∫

(5)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⇓±→…•″≠ ←± ←÷≈ °≈→•±…••←♣ ±≡ ←÷•↑ ↑←→↔←↔→≈ ♥≈ ÷±±↑≈ ↵ ⇐ ↔″•← ≈×× ↵↑ ↵ →÷±≥±•… ♥•←÷

ð

↵↑•↑ ♠≈←±→↑↵ 

°

 ℜ ⌡ ℜ ↵″…↵

°

 ℜ ℜ∫ ⊂÷≈ ×↵←←•≈ ±″↑←↵″← •↑ 

°

 ð ♥÷≈→≈ ð

•↑ ←÷≈ …•↑←↵″≈ ≈←♥≈≈″ ←♥± ″≈•≠÷±↔→•″≠ ∧↵⇓↑ ♥•→≈↑ ↵↑ ↑÷±♥″ •″ ∨•≠∫ 

≥↵↑↑ ≥≈←÷±… ←÷≈ ⇐ ≈×≈←→±″ ≈″≈→≠•≈↑ ↵″… ≈″♠≈×±°≈ ♥↵♠≈≡↔″←•±″↑ ↵→≈ ≡±↔″… ≡→±≥

′

→ ⌡  

♥÷≈→≈ •↑ ←÷≈ ≡→≈≈ ≈×≈←→±″ ≥↵↑↑

←÷≈ ⇓× ±″←≈″←   ð ð ⌡ ð ð ≡±→ ð∫ ⊕  



↵↑ ←÷≈ °±←≈″←•↵×

   ð ð ⌡ ð ð ð ð

°

 ð⌡ ð 

⌡ 



⇓±→…•″≠ ←± ←÷≈ ÷±↑≈″ ↔″•← ≈×× ″≈♥ ±±→…•″↵←≈↑   ♥≈→≈ •″←→±…↔≈… ←± ↑•≥°×•≡♣ ←÷≈ •≥°×≈⌠ ≥≈″←↵←•±″ ±≡ °≈→•±…• ±↔″…↵→♣ ±″…•←•±″↑ ↵″… ↵→≈ ←÷≈ ±±→…•″↵←≈↑ ↵×±″≠↵↵″…↵…•→≈←•±″↑

♥÷≈→≈ ↵″… ↵→≈ ←÷≈ ⇐ ×↵←←•≈ ↵↑•↑ ♠≈←±→↑ 

°

 ℜ

 ⌡ ℜ ↵″… 

°

 ℜ

 ℜ∫ ⊂÷≈→≈≡±→≈

↵″… ←↵∝≈ ♠↵×↔≈↑ ≈←♥≈≈″ ð ↵″…  ↵″… ↵→≈ →≈×↵←≈… ←± →≈←↵″≠↔×↵→ ±±→…•″↵←≈↑ ↵″… ↵↑

ð

°

 

°

 



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

°

 

°

   

 



⊂÷≈ ⊆÷→ϒ±…•″≠≈→ ≈↓↔↵←•±″ ↵″ ″±♥ ≈ ♥→•←←≈″ •″ ←÷≈ ″≈♥ ±±→…•″↵←≈ ↑♣↑←≈≥ ≈∫≠∫ •″ ←÷≈ ±″↑←↵″←

 

 

 ⌡

 

 

 ↵″… 

(6)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

 

⌡  ⌡ 

 

 

⌡  ⌡ 



  ⌡ ⌡ ⌡ 

 ⌡⌡  

 ±±→…•″↵←≈↑ •↑



′

• ∑∂  •∑∂⌡ •⌡∑∂

 ⌡

•∑∂   •∑∂⌡ •∑∂ 



•⌡∑∂⌡ •⌡∑∂  • ∑∂⌡⌡ • ∑∂  ⌡ 

  ⌡  ⌡ 

⌡  ⌡  ⌡

⌡ ⌡ ⌡ ⌡ 

 ⌡⌡  

⌡ 

 

⌡  ⌡  ⌡ 

⌡ ⌡

⌡ ⌡ ⌡ ⌡ 

 ⌡⌡  

♥÷≈→≈  ′

  

′

  

′

  ↵″…   ∫

∫∫ ⊇↵ו…•←♣ ±≡ ←÷≈ ≥≈←÷±…

↵×↔×↵←≈… ←÷≈ ±↔″… ↑←↵←≈ ≈″≈→≠•≈↑ ±≡ ←÷→≈≈ ↑♣↑←≈≥↑  ↑•″≠×≈ ↓↔↵″←↔≥ ♥•→≈  …±↔×≈ ↓↔↵″←↔≥ ♥•→≈ ♥•←÷ →≈←↵″≠↔×↵→ ±♦ ±↔″…↵→♣ ±″…•←•±″↑ ↵″…  …±↔×≈ ↓↔↵″←↔≥ ♥•→≈ ♥•←÷ →÷±≥±•…↵× ±♦

ð

±↔″…↵→♣ ±″…•←•±″↑∫

∫∫∫ ⊆•″≠×≈ ↓↔↵″←↔≥ ♥•→≈

   ″≥ ↵″… ±≡ ð  ≈⊇ •″ ↵ ↑↓↔↵→≈ ±♦ ♥≈→≈  ð ð ⌡ð ð

 

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

(7)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

↑←→↔←↔→≈ ↑←↵←≈↑ ↑←↵←≈ ≈″≈→≠♣ ≈⊇

∨⇐ ∨⇔ ↵″↵×♣←• ↑±×↔←•±″

 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 ⌠ð∫ð

 ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð

⊂↵×≈ 

℘ ↵″… ≡→±≥ ←÷≈ ↵″↵×♣←• ↑±×↔←•±″ ⊕℘∫

∫∫∫ ⇐±↔×≈ ↓↔↵″←↔≥ ♥•→≈↑

 ð ð ⌡ ð ð

  ♥≈ ÷↵♠≈ ↵×↑± ↵×↔×↵←≈… ←÷≈ ±↔″… ↑←↵←≈ ð

≈″≈→≠•≈↑ ±≡ ∧↵⇓↑ …±↔×≈ ♣ו″…→•↵× ↓↔↵″←↔≥ ♥•→≈↑ ♥•←÷ ↵ →↵…•↔↑ ±≡   ↵″…  ″≥ ♥÷•÷ ♥≈→≈ ↑≈°↵→↵←≈… ♣ ð

°

 ″≥ ≈≥≈……≈… •″ ⇓× ∧↵ ⇓↑ ≥↵←→•♦ ±≡ →≈←↵″≠↔×↵→ ↑÷↵°≈ ±♦ ±≡ ←÷≈

↑•ƒ≈  ð° ″≥ ↵″…  ð ″≥ ↵↑ ↑÷±♥″ •″ ∨•≠∫ ↵ ↵″… ±≡ →÷±≥±•…↵× ↑÷↵°≈ ♥•←÷  ð   ↵↑ ↑÷±♥″ •″ ∨•≠∫ ∫

⋅″ ←÷≈ ↵↑≈ ±≡ →÷±≥±•…↵× ±♦ ←÷≈ ≈″♠≈×±°≈ ♥↵♠≈ ≡↔″←•±″↑ ♥≈→≈ ≠→•……≈… ↵×±″≠ ←÷≈ ↵″…



…•→≈←•±″↑∫ ⊂÷≈ ⇐•→•÷×≈← ±↔″…↵→♣ ±″…•←•±″↑ ↵→≈ ð   ð   ð     ↵″…

ð   ð   ð    ∫ ⊂÷≈→≈≡±→≈ ←÷≈ ♥↵♠≈≡↔″←•±″       ↵″

≈ ↵×↔×↵←≈… ≡→±≥  ♥÷≈″ •↑ ↵ ↑♣≥≥≈←→• ≥↵←→•♦ ±≡ ↑•ƒ≈    ↵″… ←÷≈

±″←≈″← ±≡ ↔°°≈→ ←→•↵″≠×≈ ±≡ ←÷•↑ ≥↵←→•♦ •↑

 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 



(8)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

♦ ″≥

″

 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  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∫ð ⌠ð∫ð ⌠ð∫ð



∨•≠↔→≈  ð∧↵ð⇓↑〉 ∧↵⇓↑〉 ⇓×ð∧↵ð⇓↑ ↓↔↵″←↔≥ ♥•→≈↑ ♥•←÷  →≈←↵″≠↔×↵→ ±↔″…↵→♣ ±″…•←•±″↑ ↵″…  ♥•←÷ →÷±≥±•…↵× ±↔″…↵→♣ ±″…•←•±″↑∫

(9)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

♥÷≈→≈ ←÷≈ ≥↵←→•♦ ↵″… ↵→≈

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 



↵″…

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



♥÷≈→≈ ←÷≈ ↑•ƒ≈↑ ±≡ ≥↵←→•≈↑ ↵″… ↵→≈     ↵″…   ⌡  ⌡ ∫ ⊂÷≈ ≈•≠≈″♠↵×↔≈↑ ↵″… ≈•≠≈″♠≈←±→↑ ±≡ ↵→≈ ←÷≈ ↑←↵←≈ ≈″≈→≠•≈↑ ↵″… ←÷≈ ≈″♠≈×±°≈ ♥↵♠≈ ≡↔″←•±″↑ ←÷≈ ×↵←←≈→ ≠•♠≈″ •″ ←÷≈   ±±→…•″↵←≈ ↑♣↑←≈≥ •∫≈∫ ≡              

         ≠∫



⋅″ ↵↑≈ ±≡ →≈←↵″≠↔×↵→ ±♦ ←÷≈ ≈″♠≈×±°≈ ♥↵♠≈ ≡↔″←•±″↑ ♥≈→≈ ≠→•…≈… ↵×±″≠ ↵″… …•→≈←•±″↑ ↵″… ←÷≈ °↵→↵≥≈←≈→↑ ♥≈→≈ ←→↵″↑≡±→≥≈… ↵↑   ′

   

′

    ð ↵″…

  ∫ ⊂÷≈ →≈↑↔×←↑ ♥≈→≈ ±≥°↵→≈… ↵≠↵•″↑← ←÷≈ ↑←↵←≈ ≈″≈→≠•≈↑ ↵″… ♥↵♠≈≡↔″←•±″↑ ±←↵•″≈…

≡→±≥ ←÷≈ ⇑∇⊆∇ ↑±≡←♥↵→≈ ⊕  ↵  ↵″… ⊂↵×≈

∫ ⊂÷≈ ←♥± ≥≈←÷±…↑ ↵→≈ •″ ≠±±… ↵≠→≈≈≥≈″← °↵→←•↔×↵→×♣ ≡±→ ×±♥⌠≈″≈→≠♣ ↑←↵←≈↑ ↵″… ←÷≈ …≈↑→•≈…

ð

∫ ∈≈↑↔×←↑

∫∫ ⇐ ≈″≈→≠♣ ↵″… ↑←→↔←↔→≈

⊂÷≈ °≈→•±…• °±←≈″←•↵× → ♥•←÷ °≈→•±… ±≡ •″ ±←÷ ↵″… …•→≈←•±″↑ ↑÷±♥″ •″ ∨•≠∫   ÷↵↑ ←÷≈ °→±°≈→←♣



→  →⌡∈ 

♥÷≈→≈ ∈•↑ ↵″♣ ←→↵″↑×↵←•±″ ♠≈←±→ ≡±→ ←÷•↑ ↔″•← ≈××∫ ⊂÷≈ ♥↵♠≈ ≡↔″←•±″↑ ←÷≈″ ≥↔↑← ↑↵←•↑≡♣ ←÷≈

(10)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 ↑← ↑←↵←≈ ♦ ″≥ ♣ ″ ≥ 

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 ð ð ð  ð  ð  ←÷ ↑←↵←≈ ♦ ″≥ ♣ ″ ≥ 

ð ð ð ð ð  ð  ð  ð∫ðð 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∫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 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 ⌠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 ð ð ⌠ð ⌠ ð  ð  ð  ♦ ð⌠

⌠ð∫ð ⌠ð∫ð ð ð∫ð ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð ð ð∫ðð ð∫ð ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð ð ð∫ðð ð∫ð ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð ð ð∫ðð ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð ð ð∫ðð ð∫ð ð∫ð 

∨•≠↔→≈  ⊂÷≈ ≈″♠≈×±°≈ ♥↵♠≈ ≡↔″←•±″↑ ±≡ …±↔×≈ ∧↵⇓↑ ♣ו″…→•↵× ↓↔↵″←↔≥ ♥•→≈↑ ≈≥≈……≈… •″ ⇓×ð∧↵ð⇓↑ ≥↵←→•♦

≥≈←÷±…∫

(11)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

″≥ ↑←↵←≈↑

↑←↵←≈ ≈″≈→≠♣ ≈⊇

→÷±≥±•… →≈←↵″≠↔×↵→

∨⇐ ∨⇔ ∨⇐ ∨⇔

 ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð  ⌠ð∫ð ⌠ð∫ðð ⌠ð∫ð ⌠ð∫ð  ⌠ð∫ðð ð∫ð ⌠ð∫ðð ð∫ðð

 ð∫ðð 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∫ð ⌠ð∫ð  ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð  ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð  ⌠ð∫ð ⌠ð∫ðð ⌠ð∫ð ⌠ð∫ð  ð∫ð ⌠ð∫ðð ⌠ð∫ðð ⌠ð∫ðð  ð∫ð ⌠ð∫ðð ⌠ð∫ðð ⌠ð∫ðð

 ⌠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∫ð ⌠ð∫ð ⌠ð∫ðð  ⌠ð∫ðð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð  ⌠ð∫ððð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð

⊂↵×≈  ⊂÷≈ ↑←↵←≈ ≈″≈→≠•≈↑ ±≡ …±↔×≈ ♣ו″…→•↵× ↓↔↵″←↔≥ ♥•→≈ ♥•←÷ →÷±≥±•…↵× ↵″… →≈←↵″≠↔×↵→ ±♦ ±↔″…↵→♣

⇒×±÷ ±″…•←•±″↑ •∫≈∫ ←÷≈ ♥↵♠≈ ≡↔″←•±″↑ ↵″ ≈ ♥→•←←≈″ ↵↑ ←÷≈ °→±…↔← ±≡ ↵ °≈→•±…• ≡↔″←•±″   ↵″…

     ð

↵″… ♥÷≈→≈ ←÷≈ °≈→•±…••←♣ ±≡   •↑ ←÷≈ ↑↵≥≈ ↵↑ ←÷↵← ±≡ ←÷≈ °±←≈″←•↵×∫ ⊂÷≈→≈≡±→≈ ←÷≈ °↵→←•×≈  •∫≈∫

ðð

    ⌡  

♥÷≈→≈ ∝ •↑ ←÷≈ ♥↵♠≈ ♠≈←±→ →♣↑←↵× ≥±≥≈″←↔≥∫ ⊂÷≈ ¬↵≥•×←±″•↵″ ≥↵←→•♦ •↑ ↵ ÷≈→≥•←•↵″ ≥↵←→•♦ ±≡ ↑•ƒ≈ ±≡   ♥÷≈→≈ •↑ ←÷≈ ″↔≥≈→ ±≡ ≠→•… °±•″←↑ ↵×±″≠ ↵″… …•→≈←•±″↑ ↵″… •←↑

(12)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

≥↵←→•♦ ≈×≈≥≈″←↑ ↵→≈

 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 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 ð ð ð

ð ð ð ð

∫∫

∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫ ∫∫ ∫∫∫ ∫∫∫ ∫∫∫

ð ð ð ð ð

ð ð ð ð ð

ð ð ð ð ð



♥÷≈→≈    ↔  ↵″… ←÷≈ ↑•ƒ≈↑ ±≡ ≥↵←→•≈↑ ↵″… ↵→≈

⊂÷≈ ≈•≠≈″♠↵×↔≈↑ ↵″… ≈•≠≈″♠≈←±→↑ ±≡ ←÷≈ ¬↵≥•×←±″•↵″ ≥↵←→•♦ ↵→≈ ←÷≈ ↑←↵←≈ ≈″≈→≠•≈↑ ↵″… ♥↵♠≈⌠ ≡↔″←•±″↑ ±≡ ↵″… ↵″… ♥↵♠≈ ♠≈←±→     

≠

≠  ↵↑ ↑÷±♥″ •″ ∨•≠∫

 ↵″… ∫ ⋅″ ±→…≈→ ←± ÷≈∝ ←÷≈ ♠↵ו…•←♣ ±≡ ←÷•↑ ≥≈←÷±… ≡±→ ←÷≈ °≈→•±…• ±↔″…↵→♣ ±″…•←•±″↑ ←÷≈ →≈↑↔×←↑ ♥≈→≈ ±≥°↵→≈… ♥•←÷ ←÷±↑≈ ±←↵•″≈… ♥•←÷ ←÷≈ ≥±→≈ ±″♠≈″←•±″↵× ≡±→ °≈→•±…• ↑←→↔←↔→≈↑

ð

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

   °  ⌡  

(13)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

♥÷≈→≈ ð ♥•×× ≈

ð

′

  ⌡ 

ð⌡  ð 

♥÷≈→≈ ↵→≈ ←÷≈ →≈•°→±↵× ×↵←←•≈ ♠≈←±→↑ ↵″…  ð •↑ ←÷≈ ∨±↔→•≈→ ←→↵″↑≡±→≥ ±≡ ←÷≈ °±←≈″←•↵×

  ↵×↔×↵←≈… ↵↑ ←÷≈ ⇐ ∨±↔→•≈→ ←→↵″↑≡±→≥ ±≡ ←÷≈ ÷±″≈♣±≥ °±←≈″←•↵× ↑÷±♥″ •″ ∨•≠∫  ↵″…



♥•←÷ ⇔↓∫ 

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

↑÷±♥″ •″ ∨•≠∫ ∫ ⊂÷≈ ↵≠→≈≈≥≈″← ≈←♥≈≈″ ←÷≈ ←♥± •↑ ≠≈″≈→↵××♣ ♠≈→♣ ≠±±…∫ ⋅″ ↵↑≈ ±≡ ↑÷↵→°

ð



♥↵♠≈ ≥≈←÷±… ≡±→ ↵″♣ →≈↵ו↑←• ↵↑≈ ♥≈ ÷↵♠≈ ←≈↑←≈… ÷↵♠•″≠    ↵″…  ≠→•… °±•″←↑ °≈→ …•→≈←•±″ ±→ ↵ ←±←↵× ±≡   ↵″… 

ð

←± →≈←↵″≠↔×↵→ ♥•←÷ ↵↑•↑ ♠≈←±→↑  ℜ   ðℜ ↵″…  ℜ 

°

 ðℜ ↵↑ ↑÷±♥″ •″ ∨•≠∫ ∫

⊂÷≈ ⇐ ⊆÷→ϒ±…•″≠≈→ ≈↓↔↵←•±″

′

  ⌡

  ⌡   

℘

⌡ • ∑∂

•  ∑∂

⌡ •∑∂ •  ∑∂

⌡ •∑∂ •⌡  ∑∂

•⌡∑∂

•⌡  ∑∂

⌡ •∑∂ 

•∑∂  ⌡ •∑∂

•∑∂  ⌡ •∑∂

•∑∂⌡  

•∑∂⌡

•∑∂⌡  

⌡ 

±→

♣ •∑∂ 

•∑∂ 

♦ • ∑∂

• ∑∂ ⌡ ⌡ ♦

• ∑∂ ⌡ ♦

•⌡  ∑∂

⌡ ♣

•∑∂  ⌡ ♣

•∑∂⌡  

⌡ ♦ •⌡∑∂

•⌡  ∑∂

♣ •∑∂⌡

•∑∂⌡  

(14)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

↑← ↵″…

∝♦ 〉″≥

∝♣



〉″

⌠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∫ðð 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∫ðð ⌠ð∫ðð

↵

⌠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⇓↑ ↑←→↔←↔→≈∫

(15)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

∧  ∏ ∧

⌠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⇓↑ ↵×±″≠ ↑°≈•↵× ÷•≠÷⌠↑♣≥≥≈←→♣ …•→≈←•±″↑ •″ ←÷≈ ←♥±⌠…•≥≈″↑•±″↵× ÷≈♦↵≠±″↵× ⇒→•××±↔•″ ƒ±″≈ ∧ðð 〉°

ð 〉↵ ∏ð〉

 ðð∫ ⊂÷≈ ×↔≈ ×↵∝ →≈… ↵″… ≠→≈≈″ ו″≈↑ ±→→≈↑°±″… ←± ←÷≈ ″↔≥≈→ ±≡ ≠→•… °±•″←↑ ±≡    ↵″…  →≈↑°≈←•♠≈×♣ •″ ≈↵÷ …•→≈←•±″ ±→ ←± ←÷≈ ″↔≥≈→ ±≡ °×↵″≈ ♥↵♠≈↑ ±≡ 

   

↵″… 

∫ ⊂÷≈ ≈″≈→≠♣ •↑ ≥≈↵↑↔→≈… ≡→±≥ ←÷≈ ±″…↔←•±″ ↵″… ≈…≠≈ ±≡ ←÷≈ ≥↵←→•♦ ≥↵←≈→•↵× ≈↵↔↑≈ ←÷≈ °±←≈″←•↵× ⇔↓∫ 

(16)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

♦ ″≥

″

 ð  ð  ð

    ð   

⌠ð∫ ⌠ð∫ ⌠ð∫ ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð

∨•≠↔→≈  ð∧↵ð⇓↑〉 ∧↵⇓↑〉

⇓×ð∧↵ð⇓↑∫

       ′

 ≈  ↵″…  ′

 ≈  ♥÷•×≈ ↵″… ↵→≈ ←÷≈ ″↔≥≈→ ±≡ ≠→•… °±•″←↑ •″ ↵″… …•→≈←•±″↑ →≈↑°≈←•♠≈×♣∫ ⊂÷≈ ¬↵≥•×←±″•↵″ ≥↵←→•♦  •↑ ↵ ↑↓↔↵→≈ ¬≈→≥•←•↵″ ≥↵←→•♦ ±≡ ↑•ƒ≈  ↵″… ←÷≈ ↔°°≈→ ←→•↵″≠×≈ ≥↵←→•♦ ≈×≈≥≈″←↑ ↵→≈



  ð ð ð ð

  ð ð ð ð

  ð ð ð

∫ ∫∫

♣  ♣  ð ð

♣  ♣  ð

♣  ♣ 



(17)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

♥÷≈→≈

  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 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

♥•←÷

 ⌡

 

⌡ 

⌡  

⌡ 

 ♣

•∑∂ 

 ♦

• ∑∂

 ♦

•⌡  ∑∂

 ♣

•∑∂⌡  

↵″…  ♣ ♣ ♦ ♦∫ ⊂÷≈ ↑•ƒ≈↑ ±≡ ≥↵←→•≈↑  ↵″…↵→≈ ∫ ⊂÷≈ ≈•≠≈″♠↵×↔≈↑ ↵″… ≈•≠≈″♠≈←±→↑ ±≡ ←÷≈ ¬↵≥•×←±″•↵″ ≥↵←→•♦ ↵→≈ ←÷≈ ↑←↵←≈ ≈″≈→≠•≈↑ ↵″… ♥↵♠≈≡↔″←•±″ ±≡ ↵″… ≡±→ ♥↵♠≈ ♠≈←±→    

♦ ♦≠ ♣ ♣≠ ∫

⇓″ ≈♦↵≥°×≈ ↵×↔×↵←•±″ ±≡ ←÷•↑ ←♣°≈ ↑÷±♥″ •″ ∨•≠∫ 

ð

 ∧↵ ⇓↑〉

(18)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⇓× ∧↵ ⇓↑〉 ⇓× ∧↵

≡≈←•♠≈ ≥↵↑↑ ↵↑≈ ↵″… ←÷≈↑≈ ♥•×× …≈°≈″… ±″ ←÷•↑ ±″↑←↵″← ≥↵↑↑ ♠↵×↔≈ ←↵∝≈″ ↔← ←÷≈ ↵″… ↑←→↔←↔→≈ …±≈↑ ↑÷±♥ ⇐•→↵ °±•″←↑ •″ ←÷≈ °±↑•←•±″⌠…≈°≈″…≈″← ≥↵↑↑ ↵↑ ♥≈××∫

 ↑÷±♥↑ ←÷≈ ↵″…



↵″… ≈×× ↑•ƒ≈↑ ð …•↑←↵″≈ ≈←♥≈≈″ ←♥±

•↑ ♣ ↵″… ≠↵° ±°≈″•″≠ ↵← ←÷≈ ⇐•→↵ °±•″←↑∫ ⊂÷•↑ •↑ ↵÷•≈♠↵×≈ ♣ →≈↵∝•″≠ ←÷≈ ↑♣≥≥≈←→♣ •∫≈∫ ←÷≈ ≈↓↔•♠↵×≈″≈ ±≡ ↑•←≈↑ ⇓ ↵″… ⇒ •″ ←÷≈ ÷≈♦↵≠±″↵× ×↵←←•≈ ♠•↵ ÷↵″≠•″≠ ←÷≈ ⇓× ±″←≈″← ±″×♣ •″ ←÷≈ ♥•→≈↑ ↵← ↑•←≈ ⇓ ↑↔×↵←←•≈ ±≡ ←÷≈ ÷±″≈♣±≥ ×↵←←•≈ ↑←→↔←↔→≈ ↵↑ •××↔↑←→↵←≈… •″ ∨•≠∫ …∫

ð

∫ ⇑±″×↔↑•±″↑

±≡ ⇓× ∧↵ ⇓↑〉 ∧↵⇓↑〉 ⇓× ∧↵

∧↵ ⇓↑ ≥↵←→•♦ ♥↵↑ ←↵∝≈″



±↔″… ↑←↵←≈ ≈″≈→≠•≈↑ ♥•←÷ ↵″↵×♣←• ↑±×↔←•±″↑ •″ ←÷≈ ↵↑≈ ±≡ ↑•″≠×≈ ↓↔↵″←↔≥ ♥•→≈ ±→ •″ ↵↑≈↑ ±≡

ð

∫ ⇓∝″±♥×≈…≠≥≈″←

∨↔″…•″≠ ≡±→ ←÷•↑ ♥±→∝ •↑ °→±♠•…≈… ♣ ⊂÷↵•×↵″… ∈≈↑≈↵→÷ ∨↔″… ⊂∈∨ ⋅″←≈≠→↵←≈… ↵″±←≈÷″±×⌠ ±≠♣ ∈≈↑≈↵→÷ ⇑≈″←≈→ ⋅∈⇑ ∏÷±″ ∏↵≈″ ⊄″•♠≈→↑•←♣ ∏∏⊄ ↵″… ←÷≈ ⋅″↑←•←↔←≈ ±≡ ←÷≈ ∠→±≥±←•±″ ±≡ ⊂≈↵÷•″≠ ⊆•≈″≈ ↵″… ⊂≈÷″±×±≠♣ ⋅∠⊆⊂ ←÷≈ ↵″±←≈÷″±×±≠♣ ⇑≈″←≈→ ⇓∇⊂⇔⇑ ⊆⊂⇐⇓ •″•↑←→♣ ±≡ ⊆•≈″≈ ↵″… ⊂≈÷″±×±≠♣ ⊂÷↵•×↵″… ←÷→±↔≠÷ •←↑ °→±≠→↵≥ ±≡ ⇑≈″←≈→ ±≡ ⇔♦≈××≈″≈ ≈←⌠



♥±→∝∫

∈≈≡≈→≈″≈↑

⊕℘ ∏∫ ⊆∫ ±♠±↑≈×±♠ ⊇∫ ⋅∫ ∨↵×∝± ∫ ⇑±×±≥± ∠∫ ∈∫ ∧≈××≈→← ∫ ∧∫ ⊆÷♥↵ ↵″… ∏∫ ∏•≥ ⇓ →±↵…≥↵° ≡±→ ≠→↵°÷≈″≈ ↵←↔→≈ ð ⌠ðð ð∫

⊕℘ ∫ ⇒→•←″≈×× ∈∫ ⊇∫ ∧±→↵÷≈♠ ⇓∫ ∏∫ ∧≈•≥ ∫ ⇓∫ ∠±″±≥↵→≈″∝± ⇓∫ •↑÷÷≈″∝± ∫ ⊂∫ ∧→≈≈″⌠

ð

↵♥↵♣ ⊂∫ ∫ ∨→±≥÷±×… ∏∫ ⊆∫ ±♠±↑≈×±♠ ↵″… ∫ ⇔↵♠≈↑ ∈≈↑±″↵″← ←↔″″≈×ו″≠ ↵″… ″≈≠↵←•♠≈ …•≡⌠ ≡≈→≈″←•↵× ±″…↔←↵″≈ •″ ≠→↵°÷≈″≈ ←→↵″↑•↑←±→↑↵←↔→≈ ⇑±≥≥↔″•↵←•±″↑   ð∫

(19)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⌠ð∫

ð

ð∫ ⌠ð∫ ⌠ð∫ ⌠ð∫ð ð ð∫ð ð∫ ð∫ ⌠ð∫ð

⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ðð ð ð∫ðð 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∫

⌠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 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 ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð

⌠ ⌠ ⌠ ð     ♦ ð⌠

ð  ð  ♦ ð⌠



∨•≠↔→≈  ð∧↵ð⇓↑〉 ∧↵⇓↑〉 ⇓×ð∧↵ð⇓↑∫

(20)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

∧ ∪  ∧ ∩ 

⌠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∫ð

∧ ∪  ∧ ∩ 

⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ⌠ð∫ð ð ð∫ð ð∫ð ð∫ð

∫ ″≥

ðð ″≥ …ð ″≥ …ð ″≥

 ″≥

 ″≥

°±↑•←•±″⌠…≈°≈″…≈″← ≥↵↑↑ ±″↑←↵″← ≥↵↑↑

ƒð∫ðð ƒð∫ð ƒð∫ð

∨•≠↔→≈ 

↑♣≥≥≈←→♣ …•→≈←•±″↑ •″ ←÷≈ ←♥±⌠…•≥≈″↑•±″↵× →≈←↵″≠↔×↵→ ⇒→•××±↔•″ ƒ±″≈ ∧ðð ð∫ð∫ 〉↵ ∪ð∫ð 〉↵ ∩ðð∫  ♥∧↵♥⇓↑〉 ⇓× ƒ∧↵ƒ⇓↑〉 ⇓× ♥∧↵♥⇓↑ ↵″… ←÷≈

↵″… ↑←→↔←↔→≈ ←↔″↵•ו←♣ ↵÷•≈♠↵×≈ ♣ ♠↵→♣•″≠  ←÷≈ ♥•→≈ →↵…•↔↑ ∈  ←÷≈ ≈×× ↑•ƒ≈ …ð ↵″… … ←÷≈ ⇓× ±″←≈″← ƒ •″ ♥•→≈↑ ±″×♣ ↵← ↑•←≈ ⇓ ↑↔×↵←←•≈ ±≡ ←÷≈ ÷±″≈♣±≥ ×↵←←•≈ ↑←→↔←↔→≈∫ ⊂÷≈ ≈″≈→≠♣ •↑ ≥≈↵↑↔→≈… ≡→±≥ ←÷≈

±″…↔←•±″ ↵″… ≈…≠≈ ±≡ ←÷≈ ≥↵←→•♦ ≥↵←≈→•↵× ≈↵↔↑≈ ←÷≈ °±←≈″←•↵× ⇔↓∫ 

(21)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⊕℘ ∏∫ ⊆∫ ±♠±↑≈×±♠ ⇓∫ ∏∫ ∧≈•≥ ⊆∫ ⊇∫ ±→±ƒ±♠ ⇐∫ √•↵″≠ ∫ ⋅∫ ∏↵←↑″≈×↑±″ ⋅∫ ⊇∫ ∧→•≠±→•≈♠↵ ⊆∫ ⊇∫ ⇐↔±″±↑ ↵″… ⇓∫ ⇓∫ ∨•→↑±♠ ⊂♥±⌠…•≥≈″↑•±″↵× ≠↵↑ ±≡ ≥↵↑↑×≈↑↑ ⇐•→↵ ≡≈→≥•±″↑ •″ ≠→↵°÷≈″≈

↵←↔→≈  ⌠ðð ðð∫



⊕℘ ∩∫ ∅÷↵″≠∩∫ ⊃∫ ⊂↵″ ¬∫ ∫ ⊆←±→≥≈→ ↵″… ∠∫ ∏•≥ ⇔♦°≈→•≥≈″←↵× ±↑≈→♠↵←•±″ ±≡ ←÷≈ ↓↔↵″←↔≥

↵←↔→≈  ð⌠ð ðð∫

⊕℘

⇑±≥≥↔″∫ ⊂÷≈±→∫

∠÷♣↑∫  ð∫

ð

⊕℘ ∫ ↵…♠±→″•∝ ∫ ∇→ו←↵ ∫ ⇓∫ ∧±″÷↵→↔∝ ∫ ⊆≥→∝↵ ⊇∫ ±♠↵∝ ⊇∫ √↔→∝↵ ∏∫ ¬→↔↑∝↵ ∅∫ ⊇♣±→″♣ ∅∫ ∈∫ ⊃↵↑•×≈♥↑∝• ∫ ∠±←≈≥↑∝• ↵″… ∏∫ ⊇♣±→″♣∫ ∨→±≥ ×↵←≈→↵××♣ ≥±…↔×↵←≈… ←♥±⌠

≈♥ √∫ ∠÷♣↑∫ ððð ð∫

⊕℘ ⊂∫ ⊄≈÷ו″≠≈→ ∧∫ √±←ƒ↔ ∫ ≈↑↑≈→ ⇐∫ ∧→≈•≡ ⊃∫ ¬±≡↑←≈←←≈→ ⊄∫ ⇒•↑↑±→← ↵″… ⊂ ⇔↑↑ו″≠≈→

∠÷♣↑•↵× ∈≈♠•≈♥ ≈←←≈→↑  ð ð∫



⊕℘ ⇔∫ ∈↵↑↵″≈″ ⇑∫ ⇓∫ ∈±ƒƒ• ⊆∫ ∠•←←↵ו↑ ↵″… ∧∫ ⊇•≠″↵×≈ ⇔×≈←→±″⌠⇔×≈←→±″ ⋅″←≈→↵←•±″↑ •″ ⇓→←•⌠

∠÷♣↑•↵× ∈≈♠•≈♥ ≈←←≈→↑ ð ð ð∫

⊕℘

≠→↵°÷≈″≈ •″ ↵ ←♥±⌠…•≥≈″↑•±″↵× ≈×≈←→±″ ≠↵↑∠÷♣↑•↵× ∈≈♠•≈♥ ⇒  ð∈ ðð∫

⊕ð℘

ð

±≥ ×↵←←•≈↑ ≡±→ ≈×≈←→±″↑ ↵←±≥↑ ↵″… °÷±←±″↑ ↵←↔→≈ ↵″±←≈÷″±×±≠♣   ð∫

⊕℘ ∫ ⊂↵→→↔≈×× ⇐∫ ∧→≈•≡ ⊂∫ ⊄≈÷ו″≠≈→ ∧∫ √±←ƒ↔ ↵″… ⊂∫ ⇔↑↑ו″≠≈→ ⇑→≈↵←•″≠ ≥±♠•″≠ ↵″… ≥≈→≠•″≠ ⇐•→↵ °±•″←↑ ♥•←÷ ↵ ∨≈→≥• ≠↵↑ •″ ↵ ←↔″↵×≈ ÷±″≈♣±≥ ×↵←←•≈↵←↔→≈  ðð ð∫

⊕℘ ∏∫ ∏∫ ∧±≥≈↑ ⊃↵→→≈″ ↵→ ⊃∫ ∏± ∨∫ ∧↔•″≈↵ ↵″… ¬∫ ⇑∫ ↵″±÷↵→↵″ ⇐≈↑•≠″≈→ ⇐•→↵ ≡≈→≥•±″↑ ↵″… ←±°±×±≠•↵× °÷↵↑≈↑ •″ ≥±×≈↔×↵→ ≠→↵°÷≈″≈ ↵←↔→≈  ðð ð∫



⊕℘ √∫ ⊆•≥±″ ↵″… ∫ ∧→≈•″≈→ ⇑±″…≈″↑≈…⌠≥↵←←≈→ °÷♣↑•↑ ⇓ …↔± ±≡ ≠→↵°÷≈″≈ ≥•≥•↑ ↵←↔→≈

  ð∫

⊕℘ √∫ ∠≈″≠ ↵″… ⊆∫ ⊆∫ • ⇒↵″… ↑←→↔←↔→≈↑ ±≡ ≠→↵°÷≈″≈ ÷≈♦↵≠±″↵× ×↵←←•≈ ↑≈≥•±″…↔←±→ ↓↔↵″←↔≥ …±←↑⇓°°×•≈… ∠÷♣↑•↑ ≈←←≈→↑  ð ðð∫

⊕℘ ∫ ⇓∫ ∠±″±≥↵→≈″∝± ∈∫ ⊇∫ ∧±→↵÷≈♠ ∧∫ ∫ ∩↔ ⇐∫ ⇑∫ ⇔ו↵↑ ∈∫ √↵ו× ⇓∫ ⇓∫ ∠↵←≈× ⇓∫

ðð

•↑÷÷≈″∝± ⇓∫ ⊆∫ ↵♣±→±♠ ⇑∫ ∈∫⊃±±…↑ √∫ ∈∫⊃↵××↵″∝ ∫ ↔÷↵⌠∏→↔ƒ♣″↑∝• ⇒∫ ⇓∫ ∠•±← ∫ ∠±←≈≥↑∝• ⋅∫ ⊇∫ ∧→•≠±→•≈♠↵ ∏∫ ⊆∫ ±♠±↑≈×±♠ ∨∫ ∧↔•″≈↵ ⊇∫ ⋅∫ ∨↵×∝± ↵″… ⇓∫ ∏∫ ∧≈•≥ ⇑×±″•″≠ ±≡ ⇐•→↵ ≡≈→≥•±″↑ •″ ≠→↵°÷≈″≈ ↑↔°≈→×↵←←•≈↑ ↵←↔→≈   ð∫

⊕℘ ↵←↔→≈  ⌠ ð∫

(22)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⊕℘ ⊆∫ ⇑∫ √↵•″ ↵″… ⇐∫ √∫ ∈±↔×↑←±″⊆±×•… ⊆←↵←≈ ⇔×≈←→±″  ″±∫ °°∫⌠ ∫

ð

⊕℘ ⇓∫ ∏∫ ⊆↵♦≈″↵√∫ ∠÷♣↑∫ ⇑∫  ″±∫ °°∫ ⌠ ð∫

⊕℘ ∫

⊕ð℘

±″ ≈×≈←→±″• ↑←↵←≈↑ •″ ±″≈⌠×↵♣≈→ ↓↔↵″←↔≥ →•″≠ ↑↔°≈→×↵←←•≈∠÷♣↑•↵ ⇔  ⌠ ð∫

ð

⊕℘ ⇑∇⊆∇ ⋅″∫ ð∫ ⇑∇⊆∇ ↔×←•°÷♣↑•↑ ⊄↑≈→↑ ∧↔•…≈ ð∫ ⊇≈→↑•±″ ∫ ⇑∇⊆∇ ⇓⇒ ⊆←±∝÷±×≥ ⊆♥≈…≈″∫

⊕℘ ∏•→∝≥↵″ ⊂∫ ð √↵″↔↵→♣ ∫ ∨•″•←≈ ∈±↔″… ⊆↓↔↵→≈ ⊃≈×× ⇒±↔″… ⊆←↵←≈↑∫ ∈≈←→•≈♠≈… ≡→±≥ ÷←←°〉〉♥♥♥∫°÷♣↑•↑∫↑↑∂↔∫≈…↔〉∉〉↑↓↔↵→≈∫ð∫÷←≥×

⊕℘ ∠∫ ¬↵→→•↑±″ ∉↔↵″←↔≥ ⊃≈××↑ ⊃•→≈↑ ↵″… ⇐±←↑ ⊂÷≈±→≈←•↵× ↵″… ⇑±≥°↔←↵←•±″↵× ∠÷♣↑•↑ ±≡



⊆≈≥•±″…↔←±→ ↵″±↑←→↔←↔→≈↑ ⊃•×≈♣ ⊂÷•→… ⇔…•←•±″ ðð∫

References

Related documents

Urban social movements in Buenos Aires create new territories through practices of territorial. organising, which privileges the spaces of the neighbourhood and everyday

The European Round Table of Industrialists (ERT), the Union of Confederations of European Employers (UNICE), the European Foundation for Management Development (EFMD) and

prospects of scientific and technological cooperation with the New Independent States of the former Soviet Union, and in particular on the Community's participation in the

direct forms of state support to individual businesses, through corporate welfare

Clinical practice guidelines for the management of pain, agitation, and delirium in adult patients in the intensive care unit: Executive summary. American Journal of

Welch tests (F) for density (stems ha -1 ), basal area (m 2 ha -1 ), and aboveground carbon (Mg ha -1 ) of economically important nontimber and timber taxa grouped into dominant

Proposal for a Council Decision on the conclusion of the Agreement in the form of an exchange of letters concerning the provisional application of the Protocol establishing, for

group are in employment in the Community than in all countries in Central and Eastern Europe apart from the Czech Republic. This reflects the fact that unemployment