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The International Institute for Science, Technology and Education (IISTE) Journals Call for paper http://www.iiste.org/Journals

The International Institute for Science, Technology and Education (IISTE) Journals Call for paper http://www.iiste.org/Journals

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  • 1. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012QUANTUM GRAVITY- THE EL DORADO – NAY A NE PLUS ULTRA --THE FINAL FINALE 1 DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADI ABSTRACT: Motivation for quantizing gravity comes from the remarkable success of the quantumtheories of the other three fundamental interactions, and from experimental evidence suggesting thatgravity can be made to show quantum effects Although some quantum gravity theories such as stringtheory and other unified field theories (or theories of everything) attempt to unify gravity with theother fundamental forces, others such as loop quantum gravity make no such attempt; they simplyquantize the gravitational field while keeping it separate from the other forces. Observed physicalphenomena can be described well by quantum mechanics or general relativity, without needing both.This can be thought of as due to an extreme separation of mass scales at which they are important.Quantum effects are usually important only for the "very small", that is, for objects no larger thantypical molecules. General relativistic effects, on the other hand, show up mainly for the "very large"bodies such as collapsed stars. (Planets gravitational fields, as of 2011, are well-describedby linearised except for Mercurys perihelion precession; so strong-field effects—any effects of gravitybeyond lowest nonvanishing order in φ/c2—have not been observed even in the gravitational fieldsof planets and main sequence stars). There is a lack of experimental evidence relating to quantumgravity, and classical physics adequately describes the observed effects of gravity over a range of50 orders of magnitude of mass, i.e., for masses of objects from about 10−23 to 1030 kg.We present acomplete Model which probably explains the positivities and discrepancies and inadequacies of eachmodel. Physics is certainly moving in to the subterranean realm and ceratoid dualism of consciousnessand subject object duality(Freud vouchsafed only at the mother’s breast shall the subject and objectshall be one),like a maverick trying to transcend the boundaries of space time, standing on the thresholdof infinity trying to ponder what lies beyond the veil which separates the scene from unseen? 139
  • 2. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012INTRODUCTION:The following figurative representation is explains in best possible words the model that is proposed. Aconsummate model encompassing all the theories is presented. The theories are there to be applied tovarious physical systems which have different parametric representationalitiesof. Concept of “Theory”isexplained in previous examples. And the bank’s example of conservativeness of individual debits andcredits and the holistic conservativeness of assets and Liability is pronouncedly predominant in this casealso. We shall not repeat in the following the same argument. One more factor that is to be remarked isthat there are possibilities of concatenation of same theory with different theories. That the nameappeared twice in the Model should not foreclose its option for its relationship with others. CLASSICAL MECHANICS AND NEWTONIAN GRAVITY: MODULE NUMBERED ONENOTATION : : CATEGORY ONE OF CLASSICAL MECHANICS : CATEGORY TWO OF CLASICAL MECHANICS : CATEGORY THREE OF CLASSICAL MECHANICS : CATEGORY ONE OF NEWTONIAN GRAVITY 140
  • 3. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 : CATEGORY TWO OFNEWTONIAN GRAVITY :CATEGORY THREE OFNEWTONIAN GRAVITY(WE ARE TALKING OF SYSTEMS;LAW ISTHERE BUT IS APPLICABLE TO VARIOUS SYSTEMS) INVARIANT SU(3),THE PHYSICALPARAMETER STATES QUANTUM MECHANICS AND QUANTUM FIELD THEORY: MODULE NUMBERED TWO:========================================================================== === : CATEGORY ONE OF QUANTUM MECHANICS : CATEGORY TWO OF QUANTUM MECHANICS : CATEGORY THREE OF QUANTUM MECHANICS :CATEGORY ONE OF QUANTUM FIELD THEORY : CATEGORY TWO OF QUANTUM FIELD THEORY : CATEGORY THREE OF QUANTUM FIELD THEORY ELECTROMAGNETISM AND STR(SPECIAL THEORY OF RELATIVITY): MODULE NUMBERED THREE:============================================================================= : CATEGORY ONE OF ELECTROMAGNETISM :CATEGORY TWO OF ELECTROMAGNETIC THEORY : CATEGORY THREE OF ELECTROMAGNETIC THEORY : CATEGORY ONE OF STR :CATEGORY TWO OF STR : CATEGORY THREE OF STR GTR(GENERAL THEORY OF RELATIVITY )ANDQFT(QUANTUM FIELD THEORY)IN CURVED SPACE TIME(BASED ON CERTAIN VARAIBLES OF THE SYSTEM WHICH CONSEQUENTIALLY CLSSIFIABLE ON PARAMETERS) : MODULE NUMBERED FOUR:============================================================================ 141
  • 4. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 : CATEGORY ONE OFGTREVALUATIVE PARAMETRICIZATION OF SITUATIONALORIENTATIONS AND ESSENTIAL COGNITIVE ORIENTATION AND CHOICE VARIABLES OFTHE SYSTEM TO WHICH QFT IS APPLICABLE) : CATEGORY TWO OF GTR : CATEGORY THREE OF GTR :CATEGORY ONE OF QFT IN CURVED SPACE TIME :CATEGORY TWO OF QFT(SYSTEMIC INSTRUMENTAL CHARACTERISATIONS AND ACTIONORIENTATIONS AND FUYNCTIONAL IMPERATIVES OF CHANGE MANIFESTED THEREIN ) : CATEGORY THREE OF QUANTUM FIELD THEORY GTR(GENERAL THEORY OF RELATIVITY(THERE ARE MANY OBSERVES AND GTR IS APPLICABLE TO BILLION SYSTEMS NOTWITHSTANDING THE GENERALISATIONAL NATURE OF THE THEORY) AND QUANTUM GRAVITY MODULE NUMBERED FIVE:============================================================================= : CATEGORY ONE OF GTR : CATEGORY TWO OF QUANTUM GRAVITY :CATEGORY THREE OFQUANTUM GRAVITY(THE FINAL THEORY MUST POSSESS THESAME CHARACTERSTICS OF ITS CONSTITUENTS-IT CANNOT SIT IN IVORY TOWERWITHOUT APPLICABILITY TO VARIOUS SYSTEMS) :CATEGORY ONE OF QUANTUM GRAVITY :CATEGORY TWO OFQUANTUM GRAVITY :CATEGORY THREE OF QUANTUM GRAVITY========================================================================= QFT IN CURVED SPACE TIME AND QUANTUM GRAVITY: MODULE NUMBERED SIX:============================================================================= : CATEGORY ONE OF QFT IN CURVED SPACE AND TIME : CATEGORY TWO OF QFT IN SPACE AND TIME 142
  • 5. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 : CATEGORY THREE OFQFT IN CURVED SPACE AND TIME : CATEGORY ONE OF QUANTUN GRAVITY : CATEGORY TWO OF QUANTUM GRAVITY : CATEGORY THREE OF QUANTUM GRAVITY GTR AND QFT IN CURVED SPACE TIME MODULE NUMBERED SEVEN========================================================================== : CATEGORY ONE OF GTR : CATEGORY TWO OF GTR : CATEGORY THREE OF GTR : CATEGORY ONE OF QFT IN CURVED SPACE TIME : CATEGORY TWO OF : CATEGORY THREE OF QFT IN CURVED SPACEAND TIME===============================================================================( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) : ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Accentuation coefficients( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) , ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Dissipation coefficients CLASSICAL MECHANICS AND NEWTONIAN GRAVITY: 1 MODULE NUMBERED ONEThe differential system of this model is now (Module Numbered one) ( )( ) [( )( ) ( )( ) ( )] 2 143
  • 6. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) [( )( ) ( )( ) ( )] 3 ( )( ) [( )( ) ( )( ) ( )] 4 ( )( ) [( )( ) ( )( ) ( )] 5 ( )( ) [( )( ) ( )( ) ( )] 6 ( )( ) [( )( ) ( )( ) ( )] 7 ( )( ) ( ) First augmentation factor 8 ( )( ) ( ) First detritions factor QUANTUM MECHANICS AND QUANTUM FIELD THEORY: 9 MODULE NUMBERED TWO:The differential system of this model is now ( Module numbered two) ( )( ) [( )( ) ( )( ) ( )] 10 ( )( ) [( )( ) ( )( ) ( )] 11 ( )( ) [( )( ) ( )( ) ( )] 12 ( )( ) [( )( ) ( )( ) (( ) )] 13 ( )( ) [( )( ) ( )( ) (( ) )] 14 ( )( ) [( )( ) ( )( ) (( ) )] 15 ( )( ) ( ) First augmentation factor 16 ( )( ) (( ) ) First detritions factor 17 ELECTROMAGNETISM AND STR(SPECIAL THEORY OF RELATIVITY): 18 MODULE NUMBERED THREE:The differential system of this model is now (Module numbered three) ( )( ) [( )( ) ( )( ) ( )] 19 ( )( ) [( )( ) ( )( ) ( )] 20 ( )( ) [( )( ) ( )( ) ( )] 21 ( )( ) [( )( ) ( )( ) ( )] 22 144
  • 7. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) [( )( ) ( )( ) ( )] 23 ( )( ) [( )( ) ( )( ) ( )] 24 ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factor 25 GTR(GENERAL THEORY OF RELATIVITY )ANDQFT(QUANTUM FIELD THEORY)IN 26 CURVED SPACE TIME(BASED ON CERTAIN VARAIBLES OF THE SYSTEM WHICH CONSEQUENTIALLY CLSSIFIABLE ON PARAMETERS) : MODULE NUMBERED FOUR)The differential system of this model is now (Module numbered Four) ( )( ) [( )( ) ( )( ) ( )] 27 ( )( ) [( )( ) ( )( ) ( )] 28 ( )( ) [( )( ) ( )( ) ( )] 29 ( )( ) [( )( ) ( )( ) (( ) )] 30 ( )( ) [( )( ) ( )( ) (( ) )] 31 ( )( ) [( )( ) ( )( ) (( ) )] 32 ( )( ) ( ) First augmentation factor 33 ( )( ) (( ) ) First detritions factor 34 GTR(GENERAL THEORY OF RELATIVITY(THERE ARE MANY OBSERVES AND GTR IS 35 APPLICABLE TO BILLION SYSTEMS NOTWITHSTANDING THE GENERALISATIONAL NATURE OF THE THEORY) AND QUANTUM GRAVITY MODULE NUMBERED FIVEThe differential system of this model is now (Module number five) ( )( ) [( )( ) ( )( ) ( )] 36 ( )( ) [( )( ) ( )( ) ( )] 37 ( )( ) [( )( ) ( )( ) ( )] 38 ( )( ) [( )( ) ( )( ) (( ) )] 39 145
  • 8. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) [( )( ) ( )( ) (( ) )] 40 ( )( ) [( )( ) ( )( ) (( ) )] 41 ( )( ) ( ) First augmentation factor 42 ( )( ) (( ) ) First detritions factor 43 44 QFT IN CURVED SPACE TIME AND QUANTUM GRAVITY: 45 MODULE NUMBERED SIX:The differential system of this model is now (Module numbered Six) ( )( ) [( )( ) ( )( ) ( )] 46 ( )( ) [( )( ) ( )( ) ( )] 47 ( )( ) [( )( ) ( )( ) ( )] 48 ( )( ) [( )( ) ( )( ) (( ) )] 49 ( )( ) [( )( ) ( )( ) (( ) )] 50 ( )( ) [( )( ) ( )( ) (( ) )] 51 ( )( ) ( ) First augmentation factor 52 GTR AND QFT IN CURVED SPACE TIME 53 MODULE NUMBERED SEVEN:The differential system of this model is now (SEVENTH MODULE) ( )( ) [( )( ) ( )( ) ( )] 54 ( )( ) [( )( ) ( )( ) ( )] 55 ( )( ) [( )( ) ( )( ) ( )] 56 ( )( ) [( )( ) ( )( ) (( ) )] 57 ( )( ) [( )( ) ( )( ) (( ) )] 58 146
  • 9. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 59 ( )( ) [( )( ) ( )( ) (( ) )] 60 ( )( ) ( ) First augmentation factor 61 ( )( ) (( ) ) First detritions factor 62FIRST MODULE CONCATENATION: ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARESEVENTHAUGMENTATION COEFFICIENTS ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] 147
  • 10. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) –( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH DETRITIONCOEFFICIENTS 63 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 64 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1, 2 and 3SECOND MODULE CONCATENATION: 65 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 66 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) [ ( ) ( ) ] 148
  • 11. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 67 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 68 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ]Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 69 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and3 70 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH DETRITION 71COEFFICIENTS ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 72 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 73 ( )( ) –( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 74 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 75 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1,2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1,2 and 3 149
  • 12. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( )THIRD MODULE CONCATENATION: 76 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 77 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 78 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 79 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2 and 3 80 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are seventh augmentation coefficient 81 82 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( )( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] 83 150
  • 13. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( )( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] 84 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( )( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detritions coefficients for category 1, 2 and 3 85 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detritions coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth detritions coefficients for category 1, 2and 3–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) are seventh detritions coefficients 86====================================================================================FOURTH MODULE CONCATENATION: ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 87 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 88 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 89 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] 90 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 151
  • 14. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 91 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1, 2,and3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2,and 3 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are sixth augmentation coefficients for category 1, 2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH augmentationcoefficients 92 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 93 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 94 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 95 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 96 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( )–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) SEVENTH DETRITIONCOEFFICIENTS 97FIFTH MODULE CONCATENATION: 98 152
  • 15. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 99 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 100 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 101 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 102 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1,2,and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1,2, 3 103 104 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 105 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 106 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 107 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 153
  • 16. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1,2, and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1,2, and 3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1,2, and 3SIXTH MODULE CONCATENATION 108 109 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 110 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 111 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 112 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - are fourth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - fifth augmentation coefficients ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) sixth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SVENTHAUGMENTATION COEFFICIENTS 113 114 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] 154
  • 17. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 115 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) [ –( ) ( ) ] ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 116 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 117 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1, 2, and 3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1, 2, and3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1, 2, and3–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ARE SEVENTH DETRITIONCOEFFICIENTS 118 119SEVENTH MODULE CONCATENATION: 120( )( ) ̇[( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) ( )( ) ( ) ( )( ) ( )] 121 ( )( ) [( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 122 ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ] 123( )( ) 124 ⃛̇[( )( ) ( )( ) ( ) ( ⏟ )( ) ( ) ( )( ) ( ) ( )( ) ( ) 125 155
  • 18. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ] 126( )( ) [( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) )( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) )( )( ) (( ) ) ] ( )( ) 127[( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )( ) (( ) ) ( ) (( ) ) ( )( ) (( ) )( )( ) (( ) ) ]Where we suppose(A) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )(B) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(C) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With 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 ( )( ) ( ) 156
  • 19. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , thefirst augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) :(D) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) :(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) 157
  • 20. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 158
  • 21. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 128( )( ) [( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) ) 129( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) ) 130( )( ) (( ) ) ] 131 132 ( )( ) ( ) First augmentation factor 134(1)( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 135(F) (2) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 136Definition of ( )( ) ( )( ) : 137 ( ) ( )( ) ( ) ( )( ) ( ̂ ) 138 ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) 139(G) (3) ( )( ) ( ) ( )( ) 140 ( )( ) (( ) ) ( )( ) 141Definition of ( ̂ )( ) ( ̂ )( ) : 142Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 143 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 144 )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ 145With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 146and( ) ( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) ( ) then the function ( ) ( ) , the SECOND augmentation coefficient would be absolutelycontinuous.Definition of ( ̂ )( ) (̂ )( ) : 147(H) (4) ( ̂ )( ) (̂ )( ) are positive constants 148 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 149There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) 159
  • 22. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 satisfy the inequalities ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) 150(̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 151( ̂ )( )Where we suppose 152(I) (5) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 153The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) 154 ( )( ) ( ) ( )( ) 155Definition of ( ̂ )( ) ( ̂ )( ) : 156Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 157 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 158 )( ) 159( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 160and( ) ( ( ) ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the THIRD augmentation coefficient, would be absolutelycontinuous.Definition of ( ̂ )( ) (̂ )( ) : 161(J) (6) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with 162( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) 163satisfy the inequalities 164 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) 165( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 166( ̂ )( ) 167 160
  • 23. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 Where we suppose 168(K) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 169 (L) (7) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 170 (M) (8) ( )( ) ( ) ( )( ) ( ) ( ) (( ) ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 171 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 172 and( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the FOURTH augmentation coefficient WOULD be absolutely continuous. 173 Definition of ( ̂ )( ) (̂ )( ) : 174(N) ( ̂ ) ( ) (̂ )( ) are positive constants(O) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 175 (P) (9) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) 161
  • 24. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 Where we suppose 176(Q) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 177 (R) (10) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 178 (S) (11) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 179 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 180 and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ ) ( ) then the function ( )( ) ( ) , theFIFTH augmentation coefficient attributable would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) : 181(T) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 182(U) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose 183 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 184 (12) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 162
  • 25. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) (̂ )( ) 185(13) ( )( ) ( ) ( )( ) ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 186 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 187and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ ) ( ̂ )( ) ] . It is ( )to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ )( ) then the function ( )( ) ( ) , the SIXTH augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 188( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 189There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Where we suppose 190 163
  • 26. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ) ( ) ( ) 191(V) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )(W) The functions ( ) ( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( ) ( ) ( ) ( )( ) ( ̂ )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ̂ )( ) 192 ( )(X) ( ) ( ) ( )( ) ( ) ( ) (( ) ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 193 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( )( )) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 194and( )( ) ( ) .( ) 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 ( ̂ )( ) (̂ )( ) : 195 164
  • 27. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(Y) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 196(Z) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) 197Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 198Definition of ( ) ( ): 199 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )===================================================================================Definition of ( ) ( ): 165
  • 28. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Proof: Consider operator ( ) defined on the space of sextuples of continuous functions 200 which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 201 ) ( ̂ )( ) ( ) ( ̂ )( 202 ) ( ̂ )( ) ( ) ( ̂ )( 203By 204 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 205 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 206̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 207̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 208̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 209Where ( ) is the integrand that is integrated over an interval ( ) 210if the conditions IN THE FOREGOING above are fulfilled, there exists a solution satisfying the conditions Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ( )Consider operator defined on the space of sextuples of continuous functions which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( 166
  • 29. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ) ( ̂ )( ) ( ) ( ̂ )( By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) Where ( ) is the integrand that is integrated over an interval ( ) 167
  • 30. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 168
  • 31. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 211 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 212 ) ( ̂ )( ) ( ) ( ̂ )( 213 ) ( ̂ )( ) ( ) ( ̂ )( 214By 215 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 216 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 217̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 218̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 219̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 220Where ( ) is the integrand that is integrated over an interval ( ) 221 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 222 ) ( ̂ )( ) ( ) ( ̂ )( 223 ) ( ̂ )( ) ( ) ( ̂ )( 224By 225 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 226 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 227̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 228 169
  • 32. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 229̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 230Where ( ) is the integrand that is integrated over an interval ( ) ( ) 231Consider operator defined on the space of sextuples of continuous functions which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 232 ) ( ̂ )( ) ( ) ( ̂ )( 233 ) ( ̂ )( ) ( ) ( ̂ )( 234By 235 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 236 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 237̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 238̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 239̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 240Where ( ) is the integrand that is integrated over an interval ( ) ( ) 241Consider operator defined on the space of sextuples of continuous functionswhich satisfy 242 ( ) ( ) ( ̂ )( ) ( ̂ )( ) 243 ) ( ̂ )( ) ( ) ( ̂ )( 244 ) ( ̂ )( ) ( ) ( ̂ )( 245By 246 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 247 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 248 170
  • 33. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 249̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 250̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 251Where ( ) is the integrand that is integrated over an interval ( ) 252 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 253 ) ( ̂ )( ) ( ) ( ̂ )( 254 ) ( ̂ )( ) ( ) ( ̂ )( 255By 256 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 257 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 258̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 259̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 260̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 261Where ( ) is the integrand that is integrated over an interval ( ): if the conditions IN THE FOREGOING are fulfilled, there exists a solution satisfying the conditions 262 Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Proof: ( )Consider operator defined on the space of sextuples of continuous functions 171
  • 34. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 263 ) ( ̂ )( ) ( ) ( ̂ )( 264 ) ( ̂ )( ) ( ) ( ̂ )( 265 By 266 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 267 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 268 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 269 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) 270 ( ) ( ) ( ) ∫ [( ) ( ( )) (( ) ( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ () 271 ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 172
  • 35. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 Where ( ) is the integrand that is integrated over an interval ( )Analogous inequalities hold also for 272(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 273 .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 274 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ) ( ) is as defined in the statement of theorem 1(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 275 .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 276 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) (̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theorem 1(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 277 .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 278 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ) ( ) is as defined in the statement of theorem1 173
  • 36. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Analogous inequalities hold also for ( ) 279 (d) The operator maps the space of functions satisfying 37,35,36 into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( ) 280From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ( ) is as defined in the statement of theorem 7 ( )( ) ( )( ) 281It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 282 (̂ )( ) 283 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 284 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ) ( ) ] ( ̂ ) ( )( ̂ )( )In order that the operator ( ) transforms the space of sextuples of functions satisfying 285GLOBAL EQUATIONS into itself ( ) 286The operator is a contraction with respect to the metric ( ) ( ) ( ) ( ) (( )( )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 287Definition of ̃ ̃ : ( ̃ ̃) ( ) ( )It results 174
  • 37. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 288| | (( ) ( ) ( ) ( ) (̂ ) ( ) (̂ ) ( ) (̂ ) ( ) ) (( ( ) ( ) ( ) ( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 289not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition 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 thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on ( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 290From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] 291 ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) : 292Remark 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 is 293analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) then 294Definition of ( )( ) : 175
  • 38. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Indeed let be so that for( )( ) ( )( ) ( ( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 295 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is ( ) ( )unbounded. The same property holds for if ( ) ( ( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutions 296 ( )( ) ( )( ) 297It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( ̂ )( ) large to have (̂ )( ) 298 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) 299 (̂ )( ) ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 300In order that the operator transforms the space of sextuples of functions satisfying ( ) 301The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 302Definition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results 303 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 304 176
  • 39. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012From the hypotheses it follows )( ) ( )( ) | (̂ )( ) 305|( (( )( ) ( ) ( ) (̂ ) ( )(̂ )( )(̂ ) ( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result follows 306Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 307not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition 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 and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where () () 308From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] () ( () ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 309Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( ) 310If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 311analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 312Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) () ( )( )Then ( )( ) ( ) ( ) which leads to 313 ( )( ) ( )( ) ( )( ) If we take such that it results 177
  • 40. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) 314 ( ) 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 315 ( )( ) ( )( ) 316It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 317 ( ) ( )( ) [( ̂ ) ( ) (( ̂ )( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 318 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ ) ( )( ̂ )( ) ( ) 319In order that the operator transforms the space of sextuples of functions into itself ( ) 320The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 321Definition of ̃ ̃ :( (̃) ( ) ) ̃ ( ) (( )( ))It results 322 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) 323 ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 324| | (( ) ( ) ( ) ( ) (̂ ) ( )(̂ )( ) ( ) ̂(̂ ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 325 178
  • 41. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition 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 thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 326From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 327Remark 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 is 328analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 329Definition of ( )( ) : 330Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 331 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is ( ) ( )unbounded. The same property holds for if ( ) (( )( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutions 332 ( )( ) ( )( ) 333It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( ) 179
  • 42. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( ̂ )( ) (̂ )( ) large to have (̂ )( ) 334 ( ) ( )( ) [( ̂ ) ( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 335 ( ) ( )( ) ) [(( ̂ ) ( ) ) ( ̂ ( ) ) ] ( ̂ ) ( )(̂ )( ( ) 336In order that the operator transforms the space of sextuples of functions satisfying IN toitself ( ) 337The 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 338 )( ) ( )( ) | (̂ )( ) 339|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 340as not 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 of 180
  • 43. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If 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 and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 341From GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 342Remark 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 is 343analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 344Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 345 ( )( ) ( )( ) ( )( ) 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 ANALOGOUSinequalities hold also for 346 181
  • 44. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) 347 It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 348 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 349 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ ) ( )(̂ )( ) ( ) 350In order that the operator transforms the space of sextuples of functions into itself ( ) 351The 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 352 )( ) ( )( ) | (̂ )( ) 353|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis (35,35,36) the resultfollows 182
  • 45. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 354as not 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 thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 355From GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 356Remark 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 is 357analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 358Definition of ( )( ) :Indeed let be so that for 359 ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 360 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is 183
  • 46. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012unbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutionsAnalogous inequalities hold also for 361 ( )( ) ( )( ) 362It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 363 ( ) ( )( ) [( ̂ ( ) ) (( ̂ ( ) ) ) ] ( ̂ ) ( )( ̂ )( ) (̂ )( ) 364 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ( ) ) ] ( ̂ ) ( )( ̂ )( ) ( ) 365In order that the operator transforms the space of sextuples of functions into itself ( ) 366The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 367From the hypotheses it follows (1) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )(2)The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 184
  • 47. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(3) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With 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 ( )( ) ( ) isuniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the firstaugmentation coefficient attributable to terrestrial organisms, would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) :(AA) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) :(BB) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( )Analogous inequalities hold also for 368 ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( ) 185
  • 48. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( ̂ )( ) (̂ )( ) large to have (̂ )( ) 369 ( ) ( )( ) [( ̂ ) ( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )( ̂ )( ) 370 (̂ )( ) ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 371In order that the operator transforms the space of sextuples of functions satisfying37,35,36 into itself ( ) 372The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ̃ ( (̃) ( ) ) ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( ) 186
  • 49. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows 373 )( ) ( )( ) | (̂ )( ) |( (( )( ) ( )( ) (̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) )) 374And analogous inequalities for . Taking into account the hypothesis (37,35,36) the resultfollowsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 375not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition 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 and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition. 376Remark 2: There does not exist any where ( ) ( )From 79 to 36 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ))} )] ( ) ( 187
  • 50. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 377Remark 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 7: If bounded, from below, the same property holds for The proof is 378analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 379Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( ) 188
  • 51. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Then ( )( ) ( ) ( ) which leads to 380 ( )( ) ( )( ) ( )( ) 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 37to 72In order that the operator ( ) transforms the space of sextuples of functions satisfying 381GLOBAL EQUATIONS AND ITS CONCOMITANT CONDITIONALITIES into itself 382 ( ) 383The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ̃ ( (̃) ( ) ) ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( ) )| | ) ) 189
  • 52. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows 384 )( ) ( )( ) | (̂ )( ) |( (( )( ) ( )( ) (̂ )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 385as not 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 thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 386From CONCATENATED GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 387Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain 190
  • 53. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 7: If bounded, from below, the same property holds for The proof is 388analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 389Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 390 ( )( ) ( )( ) ( )( ) 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 ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 391 ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) 392Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 393By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots 394(a) of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 395 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and 396Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 397By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the 398roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 399and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 400Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 401(b) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 402 191
  • 54. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 403( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) 404and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 405and analogously 406( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 407Then the solution satisfies the inequalities 408 (( )( ) ( )( ) ) ( ) ( )( )( )( ) is defined 409 (( )( ) ( )( ) ) ( )( ) 410 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 411( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 412 ( ) ( )( ) (( )( ) ( )( ) ) 413 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 414 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) ) Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 415 Where ( )( ) ( )( ) ( )( ) ( )( ) 416 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 417 ( )( ) ( )( ) ( )( ) 418Behavior of the solutions 419 192
  • 55. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 420(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 421(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 422 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )( )( ) is defined 423 (( )( ) ( )( ) ) ( )( ) 424 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 425( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) 193
  • 56. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 426 ( ) ( )( ) (( )( ) ( )( ) ) 427 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 428 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 429 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 430 431 432If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 433(e) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 434 435 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) 436Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 194
  • 57. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 437 438 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 439 440 (( )( ) ( )( ) ) ( ) ( )( ) 441 442 where ( )( ) is defined 443 444 445 (( )( ) ( )( ) ) ( ) ( )( ) 446 ( )( ) ( )( ) 447 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 448( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) ( ) (( )( ) ( )( ) ) 449 ( )( ) (( )( ) ( )( ) ) 450 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 451 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 452 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 453 195
  • 58. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Behavior of the solutions 454If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(g) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 455(h) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 456 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(i) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 457 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 458 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 459 ( ) ( )( ) ( )( ) 196
  • 59. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 460 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 461( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 462 ( ) ( )( ) (( )( ) ( )( ) ) 463 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 464 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 465 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions 466If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(j) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 467(k) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 468 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(l) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 197
  • 60. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 470 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ̅ ) ( ) ( ) ( ̅ ) ( ) and ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 471 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 472 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 473 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 474( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 475 ( ) ( )( ) (( )( ) ( )( ) ) 476 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 477 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 478 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 479 198
  • 61. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(m) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 480(n) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) 481 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 482 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(o) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 483 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) 199
  • 62. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 484 (( )( ) ( )( ) ) ( )( ) ( ) where ( )( ) is defined 485 (( )( ) ( )( ) ) ( )( ) 486 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 487 ( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 488 ( ) ( )( ) (( )( ) ( )( ) ) 489 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 490 [ ] ( ) ( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ] ( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 491 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 200
  • 63. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) From GLOBAL EQUATIONS we obtain 492 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 493 201
  • 64. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 494 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 495 (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 GLOBAL EQUATIONS we get easily the result stated in the theorem. 202
  • 65. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Particular 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 ( )( )We can prove the following 496If ( )( ) ( )( ) are independent on , and the conditions 496A 496B( )( ) ( )( ) ( )( ) ( )( ) 496C( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 497C 497D ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) , 497E 497F( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 497G ( )( ) ( )( ) as defined are satisfied , then the system WITH THE SATISFACTION OF THEFOLLOWING PROPERTIES HAS A SOLUTION AS DERIVED BELOW.. 497Particular case : 498If ( )( ) ( )( ) ( )( ) ( )( ) 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 ( ̅ )( ) 499From GLOBAL EQUATIONS we obtain 500 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) 203
  • 66. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 Definition of ( ) :- ( ) 501 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) 502 From which one obtains(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 503 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) Definition of ( ̅ )( ) :- From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 504 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 505 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ) ( ) (̅ )( ) ( ̅ )( ) (̅ )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( ) 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 ( ) :- 204
  • 67. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. 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 ( ̅ )( ) 506 : From GLOBAL EQUATIONS we obtain 507 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- 508 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( , ( )( ) [ ( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 509 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) ( ̅ )( ) [ ( ( )( ) (̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 510 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) 205
  • 68. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 511 512 (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 ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) 513 ( ) ( ) 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 ( )( ) ( ) ( ) 514 From GLOBAL EQUATIONS we obtain 515 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(g) For ( )( ) ( )( ) ( ̅ )( ) 206
  • 69. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 516 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(h) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 517 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 518 (i) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) 519 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 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 ( )( ) 520 we obtain 521 207
  • 70. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(j) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 522 [ ( )( )((̅ )( ) (̅ )( ) ) ] 523 (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(k) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 524 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ̅ )( ) ( ̅ )( ) 525 (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 208
  • 71. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )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 ( )( ) ( ) ( ) 526Behavior of the solutions 527If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(p) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 528(q) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and 529Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 530. By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 209
  • 72. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(r) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 531 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined by 59 and 67 respectivelyThen the solution of GLOBAL EQUATIONS satisfies the inequalities 532 (( )( ) ( )( ) ) ( )( ) ( ) where ( )( ) is defined 210
  • 73. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) 533 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 534 ( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 535 ( ) ( )( ) (( )( ) ( )( ) ) 536 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 537 [ ] ( ) ( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ] ( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 538 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 539 ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) From CONCATENATED GLOBAL EQUATIONS we obtain 540 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :- It follows 211
  • 74. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(m) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 541 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(n) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 542 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 543 (o) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ) ( ) (̅ )( ) ( ̅ )( ) (̅ )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- 212
  • 75. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in CONCATENATED GLOBAL EQUATIONS we get easilythe 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 importantconsequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( )( )( ) ( )( ) ( )( ) ( ) 544( )( ) ( )( ) ( )( ) ( ) 545has a unique positive solution , which is an equilibrium solution for the system 546( )( ) [( )( ) ( )( ) ( )] 547( )( ) [( )( ) ( )( ) ( )] 548( )( ) [( )( ) ( )( ) ( )] 549( )( ) ( )( ) ( )( ) ( ) 550( )( ) ( )( ) ( )( ) ( ) 551( )( ) ( )( ) ( )( ) ( ) 552has a unique positive solution , which is an equilibrium solution for 553( )( ) [( )( ) ( )( ) ( )] 554 213
  • 76. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) [( )( ) ( )( ) ( )] 555( )( ) [( )( ) ( )( ) ( )] 556( )( ) ( )( ) ( )( ) ( ) 557( )( ) ( )( ) ( )( ) ( ) 558( )( ) ( )( ) ( )( ) ( ) 559has a unique positive solution , which is an equilibrium solution 560( )( ) [( )( ) ( )( ) ( )] 561( )( ) [( )( ) ( )( ) ( )] 563( )( ) [( )( ) ( )( ) ( )] 564( )( ) ( )( ) ( )( ) (( )) 565( )( ) ( )( ) ( )( ) (( )) 566( )( ) ( )( ) ( )( ) (( )) 567has a unique positive solution , which is an equilibrium solution for the system 568( )( ) [( )( ) ( )( ) ( )] 569( )( ) [( )( ) ( )( ) ( )] 570( )( ) [( )( ) ( )( ) ( )] 571( )( ) ( )( ) ( )( ) ( ) 572( )( ) ( )( ) ( )( ) ( ) 573( )( ) ( )( ) ( )( ) ( ) 574has a unique positive solution , which is an equilibrium solution for the system 575( )( ) [( )( ) ( )( ) ( )] 576( )( ) [( )( ) ( )( ) ( )] 577( )( ) [( )( ) ( )( ) ( )] 578( )( ) ( )( ) ( )( ) ( ) 579 214
  • 77. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) ( )( ) ( )( ) ( ) 580( )( ) ( )( ) ( )( ) ( ) 584has a unique positive solution , which is an equilibrium solution for the system 582( )( ) [( )( ) ( )( ) ( )] 583( )( ) [( )( ) ( )( ) ( )] 584( )( ) [( )( ) ( )( ) ( )] 585 586( )( ) ( )( ) ( )( ) ( ) 587( )( ) ( )( ) ( )( ) ( ) 588( )( ) ( )( ) ( )( ) ( ) 589has a unique positive solution , which is an equilibrium solution for the system 560(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )Definition and uniqueness of :- 561After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](e) By the same argument, the equations( SOLUTIONAL) admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( ) 215
  • 78. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see 562that is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that ( )Finally we obtain the unique solution OF THE SYSTEM (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 563 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Definition and uniqueness of :- 564After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] 565Definition and uniqueness of :- 566After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 567After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 568After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations 216
  • 79. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](f) By the same argument, the equations 92,93 admit solutions if 569 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) 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 ( )(g) By the same argument, the equations 92,93 admit solutions if 570 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that 571 is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ) )(a) By the same argument, the concatenated equations admit solutions if 572 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )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 (( ) ) 573(b) By the same argument, the equations of modules admit solutions if 574 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) )(c) By the same argument, the equations (modules) admit solutions if 575 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) ) 217
  • 80. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(d) By the same argument, the equations (modules) admit solutions if 578 579 ( ) ( )( ) ( )( ) ( )( ) ( )( ) 580 ( ) ( ) ( ) ( ) ( ) ( )[( ) ( ) ( ) ( ) ( ) ( )] ( ) ( )( ) ( ) 581Where in ( )( ) must be replaced by their values It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows thatthere exists a unique such that ( )Finally we obtain the unique solution of 89 to 94 582 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( )( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 583 (( )) , ( ) and 584 ( )( ) ( )( ) 585 , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 586 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )] Obviously, these values represent an equilibrium solution 587Finally we obtain the unique solution 588 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 589 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 590 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )] 218
  • 81. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 591 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 592 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 593 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 594 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionASYMPTOTIC STABILITY ANALYSIS 595Theorem 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 :- , 596 ( )( ) ( ) ( )( ) ( ) ( ) , ( )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597 (( )( ) ( )( ) ) ( )( ) ( )( ) 598 (( )( ) ( )( ) ) ( )( ) ( )( ) 599 (( )( ) ( )( ) ) ( )( ) ( )( ) 600 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 601 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 602 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 603If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 604 219
  • 82. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )Belong to ( ) then the above equilibrium point is asymptotically stableDenote 605Definition of :- , 606 ( )( ) ( )( ) 607 ( ) ( )( ) , (( ) )taking into account equations (global)and neglecting the terms of power 2, we obtain 608 (( )( ) ( )( ) ) ( )( ) ( )( ) 609 (( )( ) ( )( ) ) ( )( ) ( )( ) 610 (( )( ) ( )( ) ) ( )( ) ( )( ) 611 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 612 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 613 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 614If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 615Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) ) 616Then taking into account equations (global) and neglecting the terms of power 2, we obtain 617 (( )( ) ( )( ) ) ( )( ) ( )( ) 618 (( )( ) ( )( ) ) ( )( ) ( )( ) 619 (( )( ) ( )( ) ) ( )( ) ( )( ) 6120 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 621 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 622 220
  • 83. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 623If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 624Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- 625 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 626 (( )( ) ( )( ) ) ( )( ) ( )( ) 627 (( )( ) ( )( ) ) ( )( ) ( )( ) 628 (( )( ) ( )( ) ) ( )( ) ( )( ) 629 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 630 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 631 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 632 633 If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( )Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 634 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 635 (( )( ) ( )( ) ) ( )( ) ( )( ) 636 (( )( ) ( )( ) ) ( )( ) ( )( ) 637 (( )( ) ( )( ) ) ( )( ) ( )( ) 638 221
  • 84. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 639 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 640 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 641If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 642Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 643 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644 (( )( ) ( )( ) ) ( )( ) ( )( ) 645 (( )( ) ( )( ) ) ( )( ) ( )( ) 646 (( )( ) ( )( ) ) ( )( ) ( )( ) 647 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 648 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 649 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 650Obviously, these values represent an equilibrium solution of 79,20,36,22,23, 651 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 :- 652 , 653 222
  • 85. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations(SOLUTIONAL) and neglecting the terms of power 2, we obtain 654 655 (( )( ) ( )( ) ) ( )( ) ( )( ) 656 (( )( ) ( )( ) ) ( )( ) ( )( ) 657 (( )( ) ( )( ) ) ( )( ) ( )( ) 658 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 659 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 660 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 6612. 662 The characteristic equation of this system is ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ( ) ( ) ((( )( ) ) (( ) ( ) ( )( ) ( )( ) ) ( )( ) ) 223
  • 86. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) 224
  • 87. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ( ) ( ) ((( )( ) ) (( ) ( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) 225
  • 88. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + ( ) ( ) ( ) (( )( ) ( ) ( )( ) ) (( )( ) ( ) ( ) ) ( ) ( ) ( ) ( ) [((( )( ) ( ) ( ) )( ) ( )( ) ( ) )] ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ( ) ( ) ((( )( ) ) (( ) ( ) ( ) ( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) 226
  • 89. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ((( )( ) ( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) + (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ) [((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )] ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) REFERENCES ============================================================================ (1) A HAIMOVICI: “On the growth of a two species ecological system divided on age groups”. Tensor, Vol 37 (1982),Commemoration volume dedicated to Professor Akitsugu Kawaguchi on his 80th birthday (2)FRTJOF CAPRA: “The web of life” Flamingo, Harper Collins See "Dissipative structures” pages 172-188 (3)HEYLIGHEN F. (2001): "The Science of Self-organization and Adaptivity", in L. D. Kiel, (ed) . Knowledge Management, Organizational Intelligence and Learning, and Complexity, in: The Encyclopedia of Life Support Systems ((EOLSS), (Eolss Publishers, Oxford) [http://www.eolss.net (4)MATSUI, T, H. Masunaga, S. M. Kreidenweis, R. A. Pielke Sr., W.-K. Tao, M. Chin, and Y. J Kaufman (2006), “Satellite-based assessment of marine low cloud variability associated with aerosol, atmospheric stability, and the diurnal cycle”, J. Geophys. Res., 111, D17204, doi:10.1029/2005JD006097 (5)STEVENS, B, G. Feingold, W.R. Cotton and R.L. Walko, “Elements of the microphysical structure of numerically simulated nonprecipitating stratocumulus” J. Atmos. Sci., 53, 980- 227
  • 90. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 1006 (6)FEINGOLD, G, Koren, I; Wang, HL; Xue, HW; Brewer, WA (2010), “Precipitation- generated oscillations in open cellular cloud fields” Nature, 466 (7308) 849-852, doi: 10.1038/nature09314, Published 12-Aug 2010 (7)HEYLIGHEN F. (2001): "The Science of Self-organization and Adaptivity", in L. D. Kiel, (ed) . Knowledge Management, Organizational Intelligence and Learning, and Complexity, in: The Encyclopedia of Life Support Systems ((EOLSS), (Eolss Publishers, Oxford) [http://www.eolss.net (8)MATSUI, T, H. Masunaga, S. M. Kreidenweis, R. A. Pielke Sr., W.-K. Tao, M. Chin, and Y. J Kaufman (2006), “Satellite-based assessment of marine low cloud variability associated with aerosol, atmospheric stability, and the diurnal cycle”, J. Geophys. Res., 111, D17204, doi:10.1029/2005JD006097(8A)STEVENS, B, G. Feingold, W.R. Cotton and R.L. Walko, “Elements of the microphysicalstructure of numerically simulated nonprecipitating stratocumulus” J. Atmos. Sci., 53, 980-1006 (8B)FEINGOLD, G, Koren, I; Wang, HL; Xue, HW; Brewer, WA (2010), “Precipitation- generated oscillations in open cellular cloud fields” Nature, 466 (7308) 849-852, doi: 10.1038/nature09314, Published 12-Aug 2010 228
  • 91. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(9)^ a b c Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhaltabhängig?", Annalen der Physik 18:639 Bibcode 1905AnP...323..639E,DOI:10.1002/andp.19053231314. See also the English translation.(10)^ a b Paul Allen Tipler, Ralph A. Llewellyn (2003-01), Modern Physics, W. H. Freeman andCompany, pp. 87–88, ISBN 0-7167-4345-0(11)^ a b Rainville, S. et al. World Year of Physics: A direct test of E=mc2. Nature 438, 1096-1097(22 December 2005) | doi: 10.1038/4381096a; Published online 21 December 2005.(12)^ In F. Fernflores. The Equivalence of Mass and Energy. Stanford Encyclopedia of Philosophy(13)^ Note that the relativistic mass, in contrast to the rest mass m0, is not a relativistic invariant, andthat the velocity is not a Minkowski four-vector, in contrast to the quantity , where is the differentialof the proper time. However, the energy-momentum four-vector is a genuine Minkowski four-vector,and the intrinsic origin of the square-root in the definition of the relativistic mass is the distinctionbetween dτ and dt.(14)^ Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0(15)^ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8(16)^ Hans, H. S.; Puri, S. P. (2003). Mechanics (2 ed.). Tata McGraw-Hill. p. 433. ISBN 0-07-047360-9., Chapter 12 page 433(17)^ E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., NY. 1992.ISBN0-7167-2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclearbombs, until heat is allowed to escape.(18)^ Mould, Richard A. (2002). Basic relativity (2 ed.). Springer. p. 126. ISBN 0-387-95210-1., Chapter 5 page 126(19)^ Chow, Tail L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones &Bartlett Learning. p. 392. ISBN 0-7637-3827-1., Chapter 10 page 392(20)^ [2] Cockcroft-Walton experiment(21)^ a b c Conversions used: 1956 International (Steam) Table (IT) values where one calorie 229
  • 92. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers conversion value of one gram TNT≡ 1000 calories used.(22)^ Assuming the dam is generating at its peak capacity of 6,809 MW.(23)^ Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominatedaverage Cp of 25.8, 5.134 moles of metal, and 132 J.K-1 for the prototype. A variation of± picograms is of course, much smaller than the actual uncertainty in the mass of the international 1.5prototype, which are ± micrograms. 2(24)^ [3] Article on Earth rotation energy. Divided by c^2.(25)^ a b Earths gravitational self-energy is 4.6 × 10-10 that of Earths total mass, or 2.7 trillion metrictons. Citation: The Apache Point Observatory Lunar Laser-Ranging Operation (APOLLO), T. W.Murphy, Jr. et al. University of Washington, Dept. of Physics (132 kB PDF, here.).(26)^ There is usually more than one possible way to define a field energy, because any field can bemade to couple to gravity in many different ways. By general scaling arguments, the correct answerat everyday distances, which are long compared to the quantum gravity scale, should be minimalcoupling, which means that no powers of the curvature tensor appear. Any non-minimal couplings,along with other higher order terms, are presumably only determined by a theory of quantum gravity,and within string theory, they only start to contribute to experiments at the string scale.(27)^ G. t Hooft, "Computation of the quantum effects due to a four-dimensional pseudoparticle",Physical Review D14:3432–3450 (1976).(28)^ A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin, "Pseudoparticle Solutions to YangMills Equations", Physics Letters 59B:85 (1975).(29)^ F. Klinkhammer, N. Manton, "A Saddle Point Solution in the Weinberg Salam Theory",Physical Review D 30:2212.(30)^ Rubakov V. A. "Monopole Catalysis of Proton Decay", Reports on Progress in Physics51:189–241 (1988).(31)^ S.W. Hawking "Black Holes Explosions?" Nature 248:30 (1974).(32)^ Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper." (PDF), Annalen der Physik 17:891–921, Bibcode 1905AnP...322...891E,DOI:10.1002/andp.19053221004. English translation. 230
  • 93. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(33)^ See e.g. Lev B.Okun, The concept of Mass, Physics Today 42 (6), June 1969, p. 31–36, http://www.physicstoday.org/vol-42/iss-6/vol42no6p31_36.pdf(34)^ Max Jammer (1999), Concepts of mass in contemporary physics and philosophy, PrincetonUniversity Press, p. 51, ISBN 0-691-01017-X(35)^ Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts ofmass",Foundations of Physics (Springer) 6: 115–124, Bibcode 1976FoPh....6..115E,DOI:10.1007/BF00708670(36)^ a b Jannsen, M., Mecklenburg, M. (2007), From classical to relativistic mechanics:Electromagnetic models of the electron., in V. F. Hendricks, et al., , Interactions: Mathematics,Physics and Philosophy (Dordrecht: Springer): 65–134(37)^ a b Whittaker, E.T. (1951–1953), 2. Edition: A History of the theories of aether and electricity,vol. 1: The classical theories / vol. 2: The modern theories 1900–1926, London: Nelson(38)^ Miller, Arthur I. (1981), Albert Einsteins special theory of relativity. Emergence (1905) andearly interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2(39)^ a b Darrigol, O. (2005), "The Genesis of the theory of relativity." (PDF), Séminaire Poincaré1:1–22(40)^ Philip Ball (Aug 23, 2011). "Did Einstein discover E = mc2?” Physics World.(41)^ Ives, Herbert E. (1952), "Derivation of the mass-energy relation", Journal of the OpticalSociety of America 42 (8): 540–543, DOI:10.1364/JOSA.42.000540(42)^ Jammer, Max (1961/1997). Concepts of Mass in Classical and Modern Physics. New York:Dover. ISBN 0-486-29998-8.(43)^ Stachel, John; Torretti, Roberto (1982), "Einsteins first derivation of mass-energyequivalence", American Journal of Physics 50 (8): 760–763, Bibcode1982AmJPh..50..760S, DOI:10.1119/1.12764(44)^ Ohanian, Hans (2008), "Did Einstein prove E=mc2?", Studies In History and Philosophy ofScience Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.002 231
  • 94. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(45)^ Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal ofPhysics 79 (6): 591–600, Bibcode 2011AmJPh..79..591H, DOI:10.1119/1.3549223(46)^ Rohrlich, Fritz (1990), "An elementary derivation of E=mc2", American Journal ofPhysics 58 (4): 348–349, Bibcode 1990AmJPh..58..348R, DOI:10.1119/1.16168(47) (1996). Lise Meitner: A Life in Physics. California Studies in the History of Science. 13.Berkeley: University of California Press. pp. 236–237. ISBN 0-520-20860-(48)^ UIBK.ac.at(49)^ J. J. L. Morton; et al. (2008). "Solid-state quantum memory using the 31P nuclearspin". Nature 455 (7216): 1085–1088. Bibcode 2008Natur.455.1085M.DOI:10.1038/nature07295.(50)^ S. Weisner (1983). "Conjugate coding". Association of Computing Machinery, Special InterestGroup in Algorithms and Computation Theory 15: 78–88.(51)^ A. Zelinger, Dance of the Photons: From Einstein to Quantum Teleportation, Farrar, Straus &Giroux, New York, 2010, pp. 189, 192, ISBN 0374239665(52)^ B. Schumacher (1995). "Quantum coding". Physical Review A 51 (4): 2738–2747. Bibcode 1995PhRvA..51.2738S. DOI:10.1103/PhysRevA.51.2738.(53)Delamotte, Bertrand; A hint of renormalization, American Journal of Physics 72 (2004) pp. 170–184. Beautiful elementary introduction to the ideas, no prior knowledge of field theory beingnecessary. Full text available at: hep-th/0212049(54)Baez, John; Renormalization Made Easy, (2005). A qualitative introduction to the subject.(55)Blechman, Andrew E. ; Renormalization: Our Greatly Misunderstood Friend, (2002). Summaryof a lecture; has more information about specific regularization and divergence-subtraction schemes.(56)Cao, Tian Yu & Schweber, Silvian S. ; The Conceptual Foundations and the PhilosophicalAspects of Renormalization Theory, Synthese, 97(1) (1993), 33–108.(57)Shirkov, Dmitry; Fifty Years of the Renormalization Group, C.E.R.N. Courrier 41(7) (2001). Full 232
  • 95. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012text available at: I.O.P Magazines.(58)E. Elizalde; Zeta regularization techniques with Applications.(59)N. N. Bogoliubov, D. V. Shirkov (1959): The Theory of Quantized Fields. New York,Interscience. The first text-book on the renormalization group theory.(60)Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), ISBN 0-521-33859-X Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particlephysics.(61)Zee, Anthony; Quantum Field Theory in a Nutshell, Princeton University Press (2003) ISBN 0-691-01019-6. Another excellent textbook on Q.F.T.(62)Weinberg, Steven; The Quantum Theory of Fields (3 volumes) Cambridge University Press(1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.(63)Pokorski, Stefan; Gauge Field Theories, Cambridge University Press (1987) ISBN 0-521-47816-2.(64)t Hooft, Gerard; The Glorious Days of Physics – Renormalization of Gauge theories, lecturegiven at Erice (August/September 1998) by the Nobel laureate 1999 . Full text available at: hep-th/9812203.(65)Rivasseau, Vincent; An introduction to renormalization, PoincaréSeminar (Paris, Oct. 12, 2002),published in: Duplantier, Bertrand; Rivasseau, Vincent (Eds.) ; PoincaréSeminar 2002, Progress inMathematical Physics 30, Birkhäuser (2003) ISBN 3-7643-0579-7. Full text available in PostScript.(66) Rivasseau, Vincent; From perturbative to constructive renormalization, Princeton UniversityPress (1991) ISBN 0-691-08530-7. Full text available in PostScript.(67) Iagolnitzer, Daniel & Magnen, J. ; Renormalization group analysis, Encyclopaedia ofMathematics, Kluwer Academic Publisher (1996). Full text available in PostScript and pdfhere.(68) Scharf, Günter; Finite quantum electrodynamics: The casual approach, Springer Verlag BerlinHeidelberg New York (1995) ISBN 3-540-60142-2.(69)A. S. Švarc (Albert Schwarz), Математические основы квантовой теории поля, (Mathematicalaspects of quantum field theory), Atomizdat, Moscow, 1975. 368 pp. 233
  • 96. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(70) A. N. Vasilev The Field Theoretic Renormalization Group in Critical Behavior Theory andStochastic Dynamics (Routledge Chapman & Hall 2004); ISBN 978-0-415-31002-4(71) Nigel Goldenfeld ; Lectures on Phase Transitions and the Renormalization Group, Frontiers inPhysics 85, West view Press (June, 1992) ISBN 0-201-55409-7. Covering the elementary aspects ofthe physics of phase’s transitions and the renormalization group, this popular book emphasizesunderstanding and clarity rather than technical manipulations.(72) Zinn-Justin, Jean; Quantum Field Theory and Critical Phenomena, Oxford University Press (4thedition – 2002) ISBN 0-19-850923-5. A masterpiece on applications of renormalization methods tothe calculation of critical exponents in statistical mechanics, following Wilsons ideas (KennethWilson was Nobel laureate 1982).(73)Zinn-Justin, Jean; Phase Transitions & Renormalization Group: from Theory to Numbers,PoincaréSeminar (Paris, Oct. 12, 2002), published in: Duplantier, Bertrand; Rivasseau, Vincent(Eds.); PoincaréSeminar 2002, Progress in Mathematical Physics 30, Birkhäuser (2003) ISBN 3-7643-0579-7. Full text available in PostScript.(74 )Domb, Cyril; The Critical Point: A Historical Introduction to the Modern Theory of CriticalPhenomena, CRC Press (March, 1996) ISBN 0-7484-0435-X.(75)Brown, Laurie M. (Ed.) ; Renormalization: From Lorentz to Landau (and Beyond), Springer-Verlag (New York-1993) ISBN 0-387-97933-6.(76) Cardy, John; Scaling and Renormalization in Statistical Physics, Cambridge University Press(1996) ISBN 0-521-49959-3.(77)Shirkov, Dmitry; The Bogoliubov Renormalization Group, JINR Communication E2-96-15(1996). Full text available at: hep-th/9602024(78)Zinn Justin, Jean; Renormalization and renormalization group: From the discovery of UVdivergences to the concept of effective field theories, in: de Witt-Morette C., Zuber J.-B. (eds),Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective, June 15–26,1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, 375–388 (1999).Full text available in PostScript.(79) Connes, Alain; Symmetries Galoisiennes & Renormalisation, PoincaréSeminar (Paris, Oct. 12,2002), published in: Duplantier, Bertrand; Rivasseau, Vincent (Eds.) ; PoincaréSeminar 2002, 234
  • 97. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Progress in Mathematical Physics 30, Birkhäuser (2003) ISBN 3-7643-0579-7. Frenchmathematician Alain Connes (Fields medallist 1982) describes the mathematical underlying structure(the Hopf algebra) of renormalization, and its link to the Riemann-Hilbert problem. Full text (inFrench) available at math/0211199v1.(80)^ http://www.ingentaconnect.com/con/klu/math/1996/00000037/00000004/00091778(81)^ F. J. Dyson, Phys. Rev. 85 (1952) 631.(82)^ A. W. Stern, Science 116 (1952) 493.(83)^ P.A.M. Dirac, "The Evolution of the Physicists Picture of Nature," in Scientific American,May 1963, p. 53.(84) ^ Kragh, Helge ; Dirac: A scientific biography, CUP 1990, p. 184(85) ^ Feynman, Richard P. ; QED, The Strange Theory of Light and Matter, Penguin 1990, p. 128(86)^ C.J.Isham, A.Salam and J.Strathdee, `Infinity Suppression Gravity Modified QuantumElectrodynamics, Phys. Rev. D5, 2548 (1972)(87)^ Ryder, Lewis. Quantum Field Theory, page 390 (Cambridge University Press 1996).Acknowledgments:The introduction is a collection of information from various articles, Books, NewsPaper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,ArticleAbstracts, Research papers, Abstracts Of Research Papers, StanfordEncyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors,the internet including Wikipedia. We acknowledge all authors who have contributedto the same. In the eventuality of the fact that there has been any act of omission onthe part of the authors, we regret with great deal of compunction, contrition, regret,trepidation and remorse. As Newton said, it is only because erudite and eminentpeople allowed one to piggy ride on their backs; probably an attempt has been madeto look slightly further. Once again, it is stated that the references are onlyillustrative and not comprehensive 235
  • 98. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012First 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============================================================================== 236
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