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Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
Np complete problems a minimalist mutatis
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Np complete problems a minimalist mutatis

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  • 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 NP Complete Problems-A Minimalist Mutatis Mutandis Model- Testament Of The Panoply *1 Dr K N Prasanna Kumar, 2Prof B S Kiranagi And 3Prof C S Bagewadi *1 Dr K N Prasanna Kumar, Post doctoral researcher, Dr KNP Kumar has three PhD’s, one each in Mathematics, Economics and Political science and a D.Litt. in Political Science, Department of studies in Mathematics, Kuvempu University, Shimoga, Karnataka, India Correspondence Mail id : drknpkumar@gmail.com 2 Prof B S Kiranagi, UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri, University of Mysore, Karnataka, India 3 Prof C S Bagewadi, Chairman , Department of studies in Mathematics and Computer science, Jnanasahyadri Kuvempu university, Shankarghatta, Shimoga district, Karnataka, IndiaAbstractA concatenation Model for the NP complete problems is given. Stability analysis, Solutional behaviorare conducted. Due to space constraints, we do not go in to specification expatiations and enucleation ofthe diverse subjects and fields that the constituents belong to in the sense of widest commonalty term.IntroductionNP Complete problems in physical reality comprise of(1) Soap Bubble(2) Protein Folding(3) Quantum Computing(4) Quantum Advice(5) Quantum Adiabatic algorithms(6) Quantum Mechanical Nonlinearities(7) Hidden Variables(8) Relativistic Time Dilation(9) Analog Computing(10) Malament-Hogarth Space Times(11) Quantum Gravity(12) Anthropic ComputingWe give a minimalist concatenation model. We refer the reader to rich repository, receptacle, andreliquirium of literature available on the subject: Please note that the classification is done based on thephysical parameters attributed and ascribed to the system or constituent in question with acomprehension of the concomitance of stratification in the other category. Any little intrusion intocomplex subjects would be egregiously presumptuous, an anathema and misnomer and will never dojustice to the thematic and discursive form. Any attempt to give introductory remarks, essentialpredications, suspensional neutralities, rational representations, interfacial interference and syncopatedjustifications would only make the paper not less than 500 pages. We shall say that the P-NP problemitself is not solved let alone all the NP complete problems. We have taken a small step in this direction.More erudite scholars, we hope would take the insinuation made in the paper for further developmentand proliferation of the thesis propounded.NotationSoap Bubble And Protein Folding System: Variables Glossary : Category One Of Soap Bubbles 176
  • 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 : Category Two Of Soap Bubbles : Category Three Of Soap Bubbles : Category One Of Protein Folding : Category Two Of Protein Folding : Category Three Of Protein FoldingQuantum Computing And Quantum Advice : Category One Quantum Computing : Category Two Of Quantum Computing : Category Three Of Quantum Computing : Category One Of Quantum Advice : Category Two Of Quantum Advice : Category Three Of Quantum AdviceQuantum Adiabatic Algorithms And Quantum Mechanical Nonlinearities : Category One Of Quantum Adiabatic Algorithms : Category Two Of Quantum Adiabatic Algorithms : Category Three Of Quantum Adiabatic Algorithms : Category One Of Quantum Mechanical Nonlinearities : Category Two Of Quantum Mechanical Nonlinearities : Category Threeof Quantum Mechanical NonlinearitiesHidden Variables And Relativistic Time Dilation : Category One Of Hidden Variables : Category Two Of Hidden Varaibles : Category Three Of Hidden Variables : Category One Of Relativistic Time Dilation : Category Two Of Relativistic Time Dilation : Category Three Of Relativistic Time DilationAnalog Computing And Malament Hogarth Space Times :Category One Of Analog Computing : Category Two Of Analog Computing : Category Three Of Analog Computing : Category One Of Malament Hogarth Space Times : Category Two Of Malament Hogarth Space Times : Category Three Of Malament Hogarth Space TimesQuantum Gravity Anthropic Computing : Category One Of Quantum Gravity(Total Gravity Exists) : Category Two Of Quantum Gravity : Category Three Of Quantum Gravity : Category One Of Anthropic Computing : Category Two Of Anthropic Computing : Category Three Of Anthropic Computing 177
  • 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) : ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Accentuation coefficients( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) , ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Dissipation coefficientsGoverning Equations: System: Soap Bubble And Protein FoldingThe differential system of this model is now ( )( ) )( ) ( )( ) ( )] 1 [( ( )( ) )( ) ( )( ) ( )] 2 [( ( )( ) )( ) ( )( ) ( )] 3 [( ( )( ) )( ) ( )( ) ( )] 4 [( ( )( ) )( ) ( )( ) ( )] 5 [( ( )( ) )( ) ( )( ) ( )] 6 [( ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factorGoverning Equations: System: Quantum Computing And Quantum AdviceThe differential system of this model is now ( )( ) )( ) ( )( ) ( )] 7 [( ( )( ) )( ) ( )( ) ( )] 8 [( ( )( ) )( ) ( )( ) ( )] 9 [( ( )( ) )( ) ( )( ) (( ) )] 10 [( ( )( ) )( ) ( )( ) (( ) )] 11 [( ( )( ) )( ) ( )( ) (( ) )] 12 [( ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factorGoverning Equations: System: Quantum Adiabatic Algorithms And Quantum MechanicalNonlinearities:The differential system of this model is now ( )( ) )( ) ( )( ) ( )] 13 [( ( )( ) )( ) ( )( ) ( )] 14 [( 178
  • 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) )( ) ( )( ) ( )] 15 [( ( )( ) )( ) ( )( ) ( )] 16 [( ( )( ) )( ) ( )( ) ( )] 17 [( ( )( ) )( ) ( )( ) ( )] 18 [( ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factorGoverning Equations: System: Hidden Variables And Relativistic Time DilationThe differential system of this model is now ( )( ) )( ) ( )( ) ( )] 19 [( ( )( ) )( ) ( )( ) ( )] 20 [( ( )( ) )( ) ( )( ) ( )] 21 [( ( )( ) )( ) ( )( ) (( ) )] 22 [( ( )( ) )( ) ( )( ) (( ) )] 23 [( ( )( ) )( ) ( )( ) (( ) )] 24 [( ( )( ) ( ) First augmentation factor ( ) ( ) (( ) ) First detritions factorGoverning Equations: System: Analog Computing And Malament Hogarth Space TimesThe differential system of this model is now ( )( ) )( ) ( )( ) ( )] 25 [( ( )( ) )( ) ( )( ) ( )] 26 [( ( )( ) )( ) ( )( ) ( )] 27 [( ( )( ) )( ) ( )( ) (( ) )] 28 [( ( )( ) )( ) ( )( ) (( ) )] 29 [( ( )( ) )( ) ( )( ) (( ) )] 30 [( ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factorGoverning Equations: System: Quantum Gravity And Anthropic ComputingThe differential system of this model is now ( )( ) )( ) ( )( ) ( )] 31 [( ( )( ) )( ) ( )( ) ( )] 32 [( ( )( ) )( ) ( )( ) ( )] 33 [( 179
  • 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) )( ) ( )( ) (( ) )] 34 [( ( )( ) )( ) ( )( ) (( ) )] 35 [( ( )( ) )( ) ( )( ) (( ) )] 36 [( ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factorSystem: Soap Bubble-Protein Folding –Quantum Computing-Quantum Advice-Quantum AdiabaticAlgorithms –Quantum Mechanical Nonlinearities-Hidden Variables-Relativistic Time Dilation-AnalogComputing-Malament Hogarth Space Times-Quantum Gravity-Anthropic Computing 37 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 38 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 39 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients forcategory 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficientfor category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficientfor category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fourth augmentationcoefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentationcoefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentationcoefficient for category 1, 2 and 3 40 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 41 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 42 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 180
  • 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients forcategory 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detrition coefficients forcategory 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients forcategory 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritioncoefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detrition coefficientsfor category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detrition coefficientsfor category 1, 2 and 3 43 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 44 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 45 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficientsfor category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficientfor category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficientfor category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentationcoefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are fifth augmentationcoefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentationcoefficient for category 1, 2 and 3 46 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 181
  • 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 47 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 48 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are first detrition coefficients forcategory 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients forcategory 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficientsfor category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritioncoefficients for category 1,2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritioncoefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are sixth detritioncoefficients for category 1,2 and 3 ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 49 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 50 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 51 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are first augmentation coefficients forcategory 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second augmentationcoefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentationcoefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ) are fourthaugmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentationcoefficients for category 1, 2 and 3 182
  • 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentationcoefficients for category 1, 2 and 3 52 ( )( ) ( )( ) ( ) –( )( )( ) –( )( )( ) ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) 53 ( )( ) ( )( ) ( ) –( )( )( ) –( )( )( ) ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( ) –( )( )( ) –( )( )( ) 54 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients forcategory 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second detrition coefficientsfor category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are third detrition coefficients forcategory 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritioncoefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detritioncoefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth detritioncoefficients for category 1, 2 and 3 55 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 56 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 57 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 183
  • 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentationcoefficients for category 1, 2,and 3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth augmentationcoefficients for category 1, 2,and 3 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are sixth augmentationcoefficients for category 1, 2,and 3 58 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 59 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 60 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( )–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 61 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 62 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 63 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 184
  • 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentationcoefficients for category 1,2, and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentationcoefficients for category 1,2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentationcoefficients for category 1,2, 3 64 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 65 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 66 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficientsfor category 1,2, and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detritioncoefficients for category 1,2, and 3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detritioncoefficients for category 1,2, and 3 ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 67 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) 185
  • 11. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 68 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) 69 ( )( ) [ ] ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - are fourthaugmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - fifth augmentationcoefficients ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) sixth augmentationcoefficients 70 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 71 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 72 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritioncoefficients for category 1, 2, and 3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth detritioncoefficients for category 1, 2, and 3 186
  • 12. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detritioncoefficients for category 1, 2, and 3Where we suppose(A) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 73(B) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(C) ( )( ) ( ) ( )( ) 74 ( )( ) ( ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 75With the Lipschitz condition, we place a restriction on the behavior of functions( )( ) ( ) and( )( ) ( ) ( ) and ( ) are points belonging to the interval[( ̂ ) ( ) ( ̂ ) ( ) ] . It is to be noted that ( ) ( ) ( ) is uniformly continuous. In theeventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the firstaugmentation coefficient Would be absolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 76(D) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) :(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) (̂ )( ) ( ̂ )( ) and ( ̂ )( ) and the constants( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )satisfy the inequalities 77 ( ) ( ) ( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose(F) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )(G) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 187
  • 13. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Definition of ( )( ) ( )( ) : 78 ( )( )( ) ( ) ( )( ) ( ̂ )( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(H) ( )( ) ( ) ( )( ) ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : 79Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . Itis to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ ) ( ) then the function ( )( ) ( ) , the SECOND augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 80(I) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 81There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose(J) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 82The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : 83 188
  • 14. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 84 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( ) ( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . Itis to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ )( ) then the function ( )( ) ( ) , the THIRD augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 85(K) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants 86( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose(L) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 87(M) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : 88 ( ) ( ) ( ) ( ) ( ) ( ̂ ) ( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( )(N) ( )( ) ( ) ( )( ) 89 ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) :Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 90 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . Itis to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if 189
  • 15. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( ̂ )( ) then the function ( )( ) ( ) , the FOURTH augmentation coefficient, would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 91(O) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( )( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 92(P) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose(Q) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 93(R) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( )(S) ( )( ) ( ) ( )( ) 94 ( )( ) ( ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) :Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 95 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( ) ( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ ) ( ) ( ̂ ( ) ) ] . Itis to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ )( ) then the function ( )( ) ( ) , the FIFTH augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 96(T) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 97 190
  • 16. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012(U) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 98(V) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( )(W) ( )( ) ( ) ( )( ) 99 ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 100 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ ) ( ) ( ̂ ) ( ) ] . Itis to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ )( ) then the function ( )( ) ( ) , the first augmentation coefficient attributable toterrestrial organisms, would be absolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 101( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 102There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( ) 191
  • 17. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the 103conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the 104conditionsDefinition of ( ) ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Theorem 1: if the conditions IN THE FOREGOING,NAMELY FIRST FIVE CONDITIONS are fulfilled,there exists a solution satisfying the conditions ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )if the conditions SECOND FIVE CONDITIONS above are fulfilled, there exists a solution satisfying 105the conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )if the conditions THIRD MODULE OF FIVE CONDITIONS above are fulfilled, there exists a solution 106satisfying the conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )if the conditions FOURTH MODULE OF FIVE CONDITUIONS CONCOMITANT TO A-E above are 107fulfilled, there exists a solution satisfying the conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ( )Proof: Consider operator defined on the space of sextuples of continuous functions which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )( 192
  • 18. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012By 108 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 109 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 110 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 111 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 112 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () 113 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 114Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By 115 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 116 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 117 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 118 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 119 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 120 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 121Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By 122 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 123 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 124 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 125 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 193
  • 19. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012̅ ( ) 126 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () 127 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 128Consider operator defined on the space of sextuples of continuous functions which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By 129 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 130 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 131 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 132 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 133 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () 134 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 135Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By 136 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) `137 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 138 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 139 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 140 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () 141 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 142Consider operator defined on the space of sextuples of continuous functionswhich satisfy 194
  • 20. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By 143 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 144 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 145 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 146 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) 147 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () 148 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( ) 149(a) The operator maps the space of functions satisfying the system into itself .Indeed it isobvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 150 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(( ) is as defined in the statement of theoremAnalogous inequalities hold also for ( ) 151(b) The operator maps the space of functions into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( ( )( ) )( )( ) ( ̂ )( ) )( ) ( (̂ ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for ( ) 152(a) The operator maps the space of functions into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 195
  • 21. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( )Analogous inequalities hold also for ( ) 153(b) The operator maps the space of functions into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 154 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theorem ( ) 155(c) The operator maps the space of functions into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( (̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 156 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) (̂ )( ) ] ( ̂ )(( ) is as defined in the statement of theorem ( ) 157(d) The operator maps the space of functions satisfying the system into itself .Indeed it isobvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 158 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theoremAnalogous inequalities hold also for ( )( ) ( )( ) 159It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 160 ( ) ( )( ) [( ̂ ( ) ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 161 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ) ( ) ] ( ̂ ) ( )( ̂ )( ) 196
  • 22. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )In order that the operator transforms the space of sextuples of functions into itself ( ) 162The operator is a contraction with respect to the metric ( ) ( ) ( ) ( ) (( )( )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 163Definition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) (̂ )( )∫ ( )( ) | | ( ) ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( )| | (( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ) (( ( ) ( ) ( ) ( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compactthen it suffices to consider that ( )( ) ( )( ) depend only on andrespectively on ( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 164From GOVERNING EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ ) ( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 165Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( ) 197
  • 23. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) then 166Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) ( ( ) ) ( ) ( )( ) 167Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) ( ( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions of equations ( )( ) ( )( ) 168It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( ̂ )( ) large to have (̂ )( ) 169 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 170 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ) ( ) ] ( ̂ ) ( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( ) 171The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows )( ) ( )( ) | (̂ )( )|( 198
  • 24. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 (( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( (̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact ( ) ( )then it suffices to consider that ( ) ( ) depend only on andrespectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where () () 172From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] () ( () ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 173Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) () ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions ofSOLUTIONAL EQUATIONS OF THE HOLISTIC GOVERNING EQUATIONS ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( ) 199
  • 25. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( ̂ )( ) (̂ )( ) large to have (̂ )( ) 174 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( ) 175The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of ̃ ̃ :( (̃) ( ) ) ̃ ( ) (( )( ))It results 176 ( ) ̃ ( )| ( ) ( ) (̂ )( ) (̂ )( )|̃ ∫( )( ) | | ( ) ( ) ( ) ( ) ( ) (̂ )( ) (̂ )( )∫ ( )( ) | | ( ) ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( )| | (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compactthen it suffices to consider that ( )( ) ( )( ) depend only on andrespectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( )From THE SOLUTIONS TO THE GOVERNING EQUATIONS AND THE CONCATENATED EQUATIONSit results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( 200
  • 26. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( ) 177Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions OF THECONCATENATAED GOVENING EQUATIONS ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 178 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ ) ( )( ̂ )( ) (̂ )( ) 179 ( ) ( ) ( ) [(( ̂ ) ( ) ) ( ̂ ) ( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( )The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 201
  • 27. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Definition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) (̂ )( )∫ ( )( ) | | ( ) ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalit follows )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis The result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compactthen it suffices to consider that ( )( ) ( )( ) depend only on andrespectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 180it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below. ( )Remark 5: If is bounded from below and (( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for 202
  • 28. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions of THECONCATENATED EQUATIONSAnalogous inequalities hold also for ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) ( )( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )(̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions satisfying Intoitself ( ) 181The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) (̂ )( )∫ ( )( ) | | ( ) ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows )( ) ( )( ) | (̂ )( ) 182|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the THE CONCATENATED 203
  • 29. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012EQUATIONS HYPOTHESISED the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compactthen it suffices to consider that ( )( ) ( )( ) depend only on andrespectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( )From 19 to 28 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ ) ( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 183Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions ofConcatenated Governing Equations Of The Totalistic System Analogous inequalities hold also for ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 204
  • 30. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 (̂ )( ) 184 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ ) ( )( ̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( ) 185The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the Hypothesized Governing Equations Of The Totalistic System it follows )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered asnot conformal with the reality, however we have put this hypothesis ,in order that we canpostulate condition necessary to prove the uniqueness of the solution bounded by ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact ( ) ( )then it suffices to consider that ( ) ( ) depend only on andrespectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( )From 69 to 32 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 186 205
  • 31. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 187Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions of equationsOF THE GLOBAL AND UNIVERSALISTIC SYSTEMBehavior Of The Solutions Of Equation Representative And Constitutive Of The Totalistic And Global 188System:Theorem 2: If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )(b) Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : ( ) ( )(c) By ( ) ( ) and respectively ( )( ) ( )( ) the roots of theequations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :- 189(d) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 206
  • 32. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are definedThen the solution of the GOVERNING EQUATIONS OF THE GLOBAL SYSTEM satisfies the 190inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 191Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior Of The Solutions Of Equation Hypothesizing The Globality Of The System AndConsequential Concatenated Form:Theorem 2: If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(e) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 207
  • 33. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots(f) of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( ) ( ) ( ) ( )Definition of ( ̅ ) ( ̅ ) (̅ ) (̅ ) :By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) theroots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-(g) If we define ( )( ) ( )( ) ( )( ) ( )( ) by( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution of the system satisfies the inequalities (( )( ) ( )( ) ) ( ) ( )( )( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 208
  • 34. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions of equation 37 to 42Theorem 2: If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of theequations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 192(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution of The Global Concatenated Equations satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )( )( ) is defined ABOVE (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) 209
  • 35. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsTheorem 2: If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(e) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of theequations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) 210
  • 36. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )Then the solution of GLOBAL EQUATIONS satisfies the inequalities (( )( ) ( )( ) ) ( ) ( )( )where ( )( ) is defined ABOVE (( )( ) ( )( ) ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) ( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions of GLOBAL EQUATIONS: 193If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(g) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(h) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of theequations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( ) ( ) ( ) ( )Definition of ( ̅ ) ( ̅ ) (̅ ) (̅ ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(i) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 211
  • 37. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( ) are defined ABOVEThen the solution of THE CONCATENATED GLOBAL EQUATIONS satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined ABOVE (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior Of The Solutions Of Equation Constitutive Of Global Status Of The System (Module Six): If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(j) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : ( ) ( )(k) By ( ) ( ) and respectively ( )( ) ( )( ) the roots of theequations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 212
  • 38. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(l) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( ) are defined ABOVEThen the solution of THE GLOBAL CONCATENATED EQUATIONS satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined IN THE FOREGOING (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )MODULE ONEProof : From THE GLOBAL EQUATIONS CONCATENATED FOR MODULE ONE we obtain 213
  • 39. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in FIRST MODULE OF THE CONCATENATED SYSTEM OFGLOBAL SYSTEM we get easily the result stated in the theorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition 214
  • 40. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also defines( )( ) for the special caseAnalogously if ( )( ) ( )( ) ( )( ) ( )( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( )MODULE NUMBERED TWOProof : From GLOBAL EQUATIONSCONCATENATED SYSTEM we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( ) ((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( )From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( )( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(f) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- 215
  • 41. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in the system equations we get easily the result stated in thetheorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) ( ) ( ) ( ) ( )Analogously if ( ) ( ) ( ) ( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important ( ) ( )consequence of the relation between ( ) and ( ̅ )MODULE BEARING NUMBER THREE ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtains(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( )Definition of ( ̅ )( ) :-From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( )( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ](̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( )( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- 216
  • 42. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in the system equations we get easily the result stated in thetheorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( )Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( )( ) and ( ̅ )( )MODULE BEARING NUMBER FOUR IN THE CONCATENATED GLOBAL SYSTEM ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( , ( )( ) [ ( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) ( ̅ )( ) [ ( ( )( ) (̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(f) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 217
  • 43. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in THE CONCATENATED SYSTEM OF THE GLOBAL ORDERwe get easily the result stated in the theorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )( ) for the special case .Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important ( ) ( )consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )MODULE BEARING NUMBER FIVE IN THE GLOBAL EQUATIONS WHICH ARE CONCATENATED THEFOLLOWING NATURALLY HOLDS AND IS PROVED. ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(g) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( ) 218
  • 44. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012(h) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(i) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in thetheorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( )( ) ( ) then ( ) ( ) and as a consequence ( ) ( ) ( ) this also defines( )( ) for the special case .Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( )MODULE NUMBERED SIX IN THE CONCATENATED GLOBAL EQUATIONS OBTAINEDCONSEQUENTIAL TO THE CONCATENATION PROCESS ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :-It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) )From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(j) For ( )( ) ( )( ) ( ̅ )( ) 219
  • 45. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(k) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ̅ )( ) ( ̅ )( )(l) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in thetheorem.Particular case :FOR THE ENTIRE GLOBAL SYTEM WITH RESPECT TO MODULE SIXIf ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also defines( )( ) for the special case .Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important ( ) ( )consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )We can prove the following FOR THE CONCATENATED SYSTEM OF EQUATIONS FOR THE GLOBALORDER(FIRST MODULE TO SIXTH MODULE)Theorem If ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 220
  • 46. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined ABOVE are satisfied , then the systemSECOND MODULE OF QUANTUM COMOPUTING AND QUANTUM ADVICE IN THE CONCATENATEDEQUATIONS HAS TO SATISFY IN THE HOLISTIC EQUATIONAL ORDER:( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined ABOVE are satisfied , then the systemTheorem If ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfied , then the systemWe can prove the followingIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined by equation are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 194( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined ABOVE are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) , ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined ABOVE are satisfied , then the system( )( ) [( )( ) ( )( ) ( )] 195 221
  • 47. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) [( )( ) ( )( ) ( )] 196( )( ) [( )( ) ( )( ) ( )] 197( )( ) ( )( ) ( )( ) ( ) 198( )( ) ( )( ) ( )( ) ( ) 199( )( ) ( )( ) ( )( ) ( ) 200has a unique positive solution , which is an equilibrium solution for the system (GLOBAL SYSTEM)( )( ) [( )( ) ( )( ) ( )] 201( )( ) [( )( ) ( )( ) ( )] 202( )( ) [( )( ) ( )( ) ( )] 203( )( ) ( )( ) ( )( ) ( ) 204( )( ) ( )( ) ( )( ) ( ) 205( )( ) ( )( ) ( )( ) ( ) 206has a unique positive solution , which is an equilibrium solution( )( ) [( )( ) ( )( ) ( )] 207( )( ) [( )( ) ( )( ) ( )] 208( )( ) [( )( ) ( )( ) ( )] 209( )( ) ( )( ) ( )( ) ( ) 210( )( ) ( )( ) ( )( ) ( ) 211( )( ) ( )( ) ( )( ) ( ) 212has a unique positive solution , which is an equilibrium solution for THE GLOBAL EQUATIONS( )( ) [( )( ) ( )( ) ( )] 213( )( ) [( )( ) ( )( ) ( )] 214( )( ) [( )( ) ( )( ) ( )] 215( )( ) ( )( ) ( )( ) (( )) 216( )( ) ( )( ) ( )( ) (( )) 217( )( ) ( )( ) ( )( ) (( )) 218has a unique positive solution , which is an equilibrium solution for the system WHICH IS HOLISTICDEFINED BY THE CONCATENATED SYSTEM OF EQUATIONS WHICH ARE CONSEQUENTIAL TO THEMODULE EQUATIONS( )( ) [( )( ) ( )( ) ( )] 219( )( ) [( )( ) ( )( ) ( )] 220( )( ) [( )( ) ( )( ) ( )] 221( )( ) ( )( ) ( )( ) ( ) 222( )( ) ( )( ) ( )( ) ( ) 223 222
  • 48. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012( )( ) ( )( ) ( )( ) ( ) 224has a unique positive solution , which is an equilibrium solution for the system( )( ) [( )( ) ( )( ) ( )] 225( )( ) [( )( ) ( )( ) ( )] 226( )( ) [( )( ) ( )( ) ( )] 227( )( ) ( )( ) ( )( ) ( ) 228( )( ) ( )( ) ( )( ) ( ) 229( )( ) ( )( ) ( )( ) ( ) 230has a unique positive solution , which is an equilibrium solution for the systemIndeed the first two equations have a nontrivial solution if FOR SOAP BUBBLE ANDPROTEIN FOLDING ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )Indeed the first two equations have a nontrivial solution if FOR QUANTUM COMPUTINGAND QUANTUM ADVICE ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if FOR THE QUANTUMADIABATIC ALGORITHMS AND QUANTUM MECHANICAL NONLINEARITIES ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if HIDEN VARIABLES ANDRELATIVISTIC TIME DILATION ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if FOR ANALOG COMPUTINGAND MALAMENT HOGARTH SPACE TIMES ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if FOR QUANTUM GRAVITYAND ANTHROPIC COMPUTING ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations 223
  • 49. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](e) By the same argument, the equations FOR THE GLOBAL SYSTEM admit solutions if ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that ( )(f) By the same argument, the equations FOR THE GLOBAL SYSTEM admit solutions if ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 224
  • 50. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ))(g) By the same argument, the equations FOR THE GLOBAL SYSTEM admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ))(h) By the same argument, the equations FOR GLOBAL SYSTEM admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ))(i) By the same argument, the equations FOR GLOBAL SYSTEM admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ))(j) By the same argument, the equations FOR GLOBAL SYSTEM admit solutions if ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that ( )Finally we obtain the unique solution of the CONSEQUENTIAL CONCATENATED EQUATIONS OFTHE HOLISTIC TOTALISTIC SYSTEM ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solution of GLOBAL EQUATIONSFinally we obtain the unique solution of GLOBAL EQUATION OF THE SYSTEM: 225
  • 51. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solution of GLOBAL SYSTEMFinally we obtain the unique solution of the GLOBAL GOVERNING EQUATIONS (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solution of GOVERNING GLOBAL EQUATIONSFinally we obtain the unique solution of GLOBAL EQUATIONS ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solution of GOVERNING GLOBAL EQUATIONSFinally we obtain the unique solution of GOVERNING GLOBAL EQUATIONS (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solution of GOVERNING GLOBAL EQUATIONS,Finally we obtain the unique solution of CONCATENATED SYSTEM OF GLOBAL EQUATIONS (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solution of GLOBAL EQUATIONSASYMPTOTIC STABILITY ANALYSISTheorem 4: If the conditions of the previous theorem are satisfied and if the functions( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stable.Proof: DenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , ( ) 226
  • 52. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Then taking into account equations PERTAINING TO THE GLOBAL SYSTEM IN QUESTION andneglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) ( )Belong to ( ) then the above equilibrium point is asymptotically stable(FOR QUANTUMCOMOPUTING AND QUANTUM ADVICE)Definition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Taking into account equations PERTAINING TO THE GLOBAL SYSTEM and neglecting the terms ofpower 2, we obtain FOR THE GLOBAL SYSTEM, AS THE CONTRIBUTION FROM THE MODULE OFQUANTUM COMPUTING AND QUANTUM ADVICE. IT IS NECESSARY THAT THE MODULES MUST BEBORNE IN MINS AND WE SHALL NOT REPEAT THIS EXPRESSIVELY IN THE WORK. 231 (( )( ) ( )( ) ) ( )( ) ( )( ) 232 (( )( ) ( )( ) ) ( )( ) ( )( ) 233 (( )( ) ( )( ) ) ( )( ) ( )( ) 234 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 235 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 236 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) ( )Belong to ( ) then the above equilibrium point is asymptotically stable.(THIRD MODULECONTRIBUTION)Definition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain 237 (( )( ) ( )( ) ) ( )( ) ( )( ) 227
  • 53. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 238 (( )( ) ( )( ) ) ( )( ) ( )( ) 239 (( )( ) ( )( ) ) ( )( ) ( )( ) 240 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 241 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 242 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )FOURTH MODULE CONTRIBUTION TO THE GLOBAL EQUATIONS If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) ( )Belong to ( ) then the above equilibrium point is asymptotically stable.DenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations GLOBAL EQUATIONS and neglecting the terms of power 2, weobtain 243 (( )( ) ( )( ) ) ( )( ) ( )( ) 244 (( )( ) ( )( ) ) ( )( ) ( )( ) 245 (( )( ) ( )( ) ) ( )( ) ( )( ) 246 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 247 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 248 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )A CONTRIBUTION TO THE HOLISTIC SYSTEMAL EQUATIONS FROM THE FIFTH MODULE NAMELYANALOG COMPUTING AND MALAMENT HOGHWART SPACETIMESTheorem 5: If the conditions of the previous theorem are satisfied and if the functions( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stable.Proof: DenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations PERTAINING BTO THE GLOBAL FIELD and neglecting the termsof power 2, we obtain 249 (( )( ) ( )( ) ) ( )( ) ( )( ) 250 (( )( ) ( )( ) ) ( )( ) ( )( ) 228
  • 54. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 251 (( )( ) ( )( ) ) ( )( ) ( )( ) 252 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 253 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 254 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )A SIXTH MODULE CONTRIBUTION(QUANTUM GRAVITY AND ANTHROPIC COMPUTING)If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) ( )Belong to ( ) then the above equilibrium point is asymptotically stable.Proof: DenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 255 (( )( ) ( )( ) ) ( )( ) ( )( ) 256 (( )( ) ( )( ) ) ( )( ) ( )( ) 257 (( )( ) ( )( ) ) ( )( ) ( )( ) 258 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 259 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 260 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )The characteristic equation of this system is(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) 261((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ) 229
  • 55. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) 230
  • 56. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012 (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )And as one sees, all the coefficients are positive. It follows that all the roots have negative real part,and this proves the theorem.AcknowledgmentsThe introduction is a collection of information from various articles, Books, News Paper reports,Home Pages Of authors, Journal Reviews, the internet including Wikipedia. We acknowledge allauthors who have contributed to the same. In the eventuality of the fact that there has been any actof omission on the part of the authors, We regret with great deal of compunction, contrition, andremorse. As Newton said, it is only because erudite and eminent people allowed one to piggy ride 231
  • 57. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012on their backs; probably an attempt has been made to look slightly further. Once again, it is statedthat the references are only illustrative and not comprehensive REFERENCES1. Garey, Michael R.; Johnson, D. S. (1979). Victor Klee. ed. Computers and Intractability: A Guideto the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. W. H. Freemanand Co.. pp. x+338. ISBN 0-7167-1045-5. MR 519066.2. Agrawal, M.; Allender, E.; Rudich, Steven (1998). "Reductions in Circuit Complexity: AnIsomorphism Theorem and a Gap Theorem". Journal of Computer and System Sciences (Boston,MA: Academic Press) 57 (2): 127–143. DOI:10.1006/jcss.1998.1583. ISSN 1090-27243. Agrawal, M.; Allender, E.; Impagliazzo, R.; Pitassi, T.; Rudich, Steven (2001). "Reducing thecomplexity of reductions". Computational Complexity (Birkhäuser Basel) 10 (2): 117–138.DOI:10.1007/s00037-001-8191-1. ISSN 1016-33284. Don Knuth, Tracy Larrabee, and Paul M. Roberts, Mathematical Writing § 25, MAA Notes No. 14,MAA, 1989 (also Stanford Technical Report, 1987).5. Knuth, D. F. (1974). "A terminological proposal". SIGACT News 6 (1): 12–18. DOI:10.1145/1811129.1811130. Retrieved 2010-08-28.6. http://www.nature.com/news/2000/000113/full/news000113-10.html7. Garey, M.R.; Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 0-7167-1045-5. This book is a classic, developing thetheory, and then cataloguing many NP-Complete problems.8. Cook, S.A. (1971). "The complexity of theorem proving procedures". Proceedings, Third AnnualACM Symposium on the Theory of Computing, ACM, New York . pp. 151–158.DOI:10.1145/800157.805047.9. Dunne, P.E. "An annotated list of selected NP-complete problems". COMP202, Dept. of ComputerScience, University of Liverpool. Retrieved 2008-06-21.10. Crescenzi, P.; Kann, V.; Halldórsson, M.; Karpinski, M.; Woeginger, G. "A compendium of NPoptimization problems". KTH NADA, Stockholm. Retrieved 2008-06-21.11. Dahlke, K. "NP-complete problems". Math Reference Project. Retrieved 2008-06-21.12. Karlsson, R. "Lecture 8: NP-complete problems" (PDF). Dept. of Computer Science, LundUniversity, Sweden. Retrieved 2008-06-21.]13. Sun, H.M. "The theory of NP-completeness" (PPT). Information Security Laboratory, Dept. ofComputer Science, National Tsing Hua University, Hsinchu City, Taiwan. Retrieved 2008-06-21.14. Jiang, J.R. "The theory of NP-completeness" (PPT). Dept. of Computer Science and InformationEngineering, National Central University, Jhongli City, Taiwan. Retrieved 2008-06-21.15. Cormen, T.H.; Leiserson, C.E., Rivest, R.L.; Stein, C. (2001). Introduction to Algorithms (2nd ed.).MIT Press and McGraw-Hill. Chapter 34: NP–Completeness, pp. 966–1021.ISBN 0-262-03293-7.16. Sipser, M. (1997). Introduction to the Theory of Computation. PWS Publishing. Sections 7.4–7.5(NP-completeness, Additional NP-complete Problems), pp. 248–271. ISBN 0-534-94728-X.17. Papadimitriou, C. (1994). Computational Complexity (1st ed.). Addison Wesley. Chapter 9 (NP-complete problems), pp. 181–218. ISBN 0-201-53082-1.18. Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi - Measurement DisturbsExplanation Of Quantum Mechanical States-A Hidden Variable Theory - published at: "InternationalJournal of Scientific and Research Publications, www.ijsrp.org ,Volume 2, Issue 5, May 2012 Edition".19. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi -Classic 2 Flavour ColorSuperconductivity And Ordinary Nuclear Matter-A New Paradigm Statement - Published At:"International Journal Of Scientific And Research Publications, www.ijsrp.org, Volume 2, Issue 5, May2012 Edition".20. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi -Space And Time, Mass And 232
  • 58. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.6, 2012Energy Accentuation Dissipation Models - Published At: "International Journal Of Scientific AndResearch Publications, www.ijsrp.org, Volume 2, Issue 6, June 2012 Edition".21. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi - Dark Energy (DE) AndExpanding Universe (EU) An Augmentation -Detrition Model - Published At: "International JournalOf Scientific And Research Publications, www.ijsrp.org,Volume 2, Issue 6, June 2012 Edition".22. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi -Quantum ChromodynamicsAnd Quark Gluon Plasma Sea-A Abstraction And Attrition Model - Published At: "InternationalJournal Of Scientific And Research Publications, www.ijsrp.org, Volume 2, Issue 6, June 2012 Edition".23. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi - A General Theory Of FoodWeb Cycle - Part One - Published At: "International Journal Of Scientific And Research Publications,www.ijsrp.org, Volume 2, Issue 6, June 2012 Edition".24. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi -Mass And Energy-A BankGeneral Assets And Liabilities Approach –The General Theory Of ‘Mass, Energy ,Space AndTime’-Part 2 Published At: "Mathematical Theory and Modeling , http://www.iiste.org/ Journals/index.php/MTM www.iiste.org, ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012"25. Dr K N Prasanna Kumar, Prof B S Kiranagi And Prof C S Bagewadi -Uncertainty Of Position Of APhoton And Concomitant And Consummating Manifestation Of Wave Effects - Published At:"Mathematical Theory and Modeling , http://www.iiste.org/Journals/index.php/MTM, www.iiste.org,ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012"First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics,Political Science. Thesis was based on Mathematical Modeling. He was recently awarded D.litt., for his work on‘Mathematical Models in Political Science’--- Department of studies in Mathematics, Kuvempu University,Shimoga, Karnataka, India Corresponding Author:drknpkumar@gmail.comSecond Author: 2Prof. B.S Kiranagi is the Former Chairman of the Department of Studies in Mathematics,Manasa Gangotri and present Professor Emeritus of UGC in the Department. Professor Kiranagi has guidedover 25 students and he has received many encomiums and laurels for his contribution to Co homology Groupsand Mathematical Sciences. Known for his prolific writing, and one of the senior most Professors of thecountry, he has over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his creditseveral books on Lie Groups, Co Homology Groups, and other mathematical application topics, and excellentpublication history.-- UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri,University of Mysore, Karnataka, IndiaThird Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Departmentof Studies in Computer Science and has guided over 25 students. He has published articles in both national andinternational journals. Professor Bagewadi specializes in Differential Geometry and its wide-rangingramifications. He has to his credit more than 159 research papers. Several Books on Differential Geometry,Differential Equations are coauthored by him--- Chairman, Department of studies in Mathematics and Computerscience, Jnanasahyadri Kuvempu university, Shankarghatta, Shimoga district, Karnataka, India 233
  • 59. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. Prospective authors ofIISTE journals can find the submission instruction on the following page:http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar

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