Yang mills theory

Advances in Physics Theories and Applications www.iiste.org
ISSN 2224-719X (Paper) ISSN 2225-0638 (Online)
Vol 7, 2012

SOME CONTRIBUTIONS TO YANG MILLS THEORY

FORTIFICATION –DISSIPATION MODELS
1
DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADI

ABSTRACT. We provide a series of Models for the problems that arise in Yang Mills Theory. No claim
is made that the problem is solved. We do factorize the Yang Mills Theory and give a Model for the
values of LHS and RHS of the yang Mills theory. We hope these forms the stepping stone for further
factorizations and solutions to the subatomic denominations at Planck’s scale. Work also throws light on
some important factors like mass acquisition by symmetry breaking, relation between strong interaction
and weak interaction, Lagrangian Invariance despite transformations, Gauge field, Noncommutative
symmetry group of Gauge Theory and Yang Mills Theory itself.

Key Words: Acquisition of mass, Symmetry Breaking, Strong interaction ,Unified Electroweak interaction,
Continuous group of local transformations, Lagrangian Variance, Group generator in Gauge Theory, Vector field or
Gauge field, commutative symmetry group in Gauge Theory, Yang Mills Theory

The outlay of the paper is as follows:

I. INTRODUCTION
II. FORMULATION OF THE PROBLEM
III. STATEMENT OF GOVERNING EQUATIONS
IV. THE SOLUTION-BODY FABRIC OF THE THESIS
V. ACKNOWLEDGEMENTS
VI. REFRENCES

I. INTRODUCTION:

We take in to consideration the following parameters, processes and concepts:

(1) Acquisition of mass
(2) Symmetry Breaking
(3) Strong interaction
(4) Unified Electroweak interaction
(5) Continuous group of local transformations
(6) Lagrangian Variance
(7) Group generator in Gauge Theory
(8) Vector field or Gauge field
(9) Non commutative symmetry group in Gauge Theory
(10) Yang Mills Theory (We repeat the same Bank’s example. Individual
debits and Credits are conservative so also the holistic one. Generalized
theories are applied to various systems which are parameterized. And we live
in ‘measurement world’. Classification is done on the parameters of various
systems to which the Theory is applied. ).
(11) First Term of the Lagrangian of the Yang Mills Theory(LHS)

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(12) RHS of the Yang Mills Theory

II. FORMULATION OF THE PROBLEM

SYMMETRY BREAKING AND ACQUISITION OF MASS:

MODULE NUMBERED ONE

NOTATION :

: CATEGORY ONE OF SYMMETRY BREAKING

: CATEGORY TWO OF SYMMETRY BREAKING

: CATEGORY THREE OF SYMMETRY BREAKING

: CATEGORY ONE OF ACQUISITION OF MASS

: CATEGORY TWO OF ACQUISITION OF MASS

:CATEGORY THREE OF ACQUISITION OF MASS

UNIFIED ELECTROWEAK INTERACTION AND STRONG INTERACTION:

MODULE NUMBERED TWO:

==========================================================================
===

: CATEGORY ONE OF UNIFIED ELECTROWEAK INTERACTION

: CATEGORY TWO OFUNIFIED ELECTROWEAK INTERACTION

: CATEGORY THREE OFUNIFIED ELECTROWEAK IONTERACTION

:CATEGORY ONE OF STRONG INTERACTION

: CATEGORY TWO OF STRONG INTERACTION

: CATEGORY THREE OF STRONG INTERACTION

LAGRANGIAN INVARIANCE AND CONTINOUS GROUP OF LOCAL
TRANSFORMATIONS:

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MODULE NUMBERED THREE:

==========================================================================
===

: CATEGORY ONE OF CONTINUOUS GROUP OF LOCAL TRANSFORMATIONS

:CATEGORY TWO OFCONTINUOUS GROUP OF LOCAL TRANSFORMATIONS

: CATEGORY THREE OF CONTINUOUS GROUP OF LOCAL TRANSFORMATION

: CATEGORY ONE OF LAGRANGIAN INVARIANCE

:CATEGORY TWO OF LAGRANGIAN INVARIANCE

: CATEGORY THREE OF LAGRANGIAN INVARIANCE

GROUP GENERATOR OF GAUGE THEORY AND VECTOR FIELD(GAUGE FIELD):

: MODULE NUMBERED FOUR:

==========================================================================
==

: CATEGORY ONE OF GROUP GENERATOR OF GAUGE THEORY

: CATEGORY TWO OF GROUP GENERATOR OF GAUGE THEORY

: CATEGORY THREE OF GROUP GENERATOR OF GAUGE THEORY

:CATEGORY ONE OF VECTOR FIELD NAMELY GAUGE FIELD

:CATEGORY TWO OF GAUGE FIELD

: CATEGORY THREE OFGAUGE FIELD

YANG MILLS THEORYAND NON COMMUTATIVE SYMMETRY GROUP IN GAUGE
THEORY:

MODULE NUMBERED FIVE:

==========================================================================
===

: CATEGORY ONE OF NON COMMUTATIVE SYMMETRY GROUP OF GAUGE
THEORY

: CATEGORY TWO OF NON COMMUTATIVE SYMMETRY GROUP OPF GAUGE
THEORY

:CATEGORY THREE OFNON COMMUTATIVE SYMMETRY GROUP OF GAUGE

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THEORY

: CATEGORY ONE OFYANG MILLS THEORY (Theory is applied to various subatomic
particle systems and the classification is done based on the parametricization of these systems.
There is not a single system known which is not characterized by some properties)

:CATEGORY TWO OF YANG MILLS THEORY

:CATEGORY THREE OF YANG MILLS THEORY

LHS OF THE YANG MILLS THEORY AND RHS OF THE YANG MILLS THEORY.TAKEN
TO THE OTHER SIDE THE LHS WOULD DISSIPATE THE RHS WITH OR WITHOUT TIME
LAG :

MODULE NUMBERED SIX:

==========================================================================
===

: CATEGORY ONE OF LHS OF YANG MILLS THEORY

: CATEGORY TWO OF LHS OF YANG MILLS THEORY

: CATEGORY THREE OF LHS OF YANG MILLS THEORY

: CATEGORY ONE OF RHS OF YANG MILLS THEORY

: CATEGORY TWO OF RHS OF YANG MILLS THEORY

: CATEGORY THREE OF RHS OF YANG MILLS THEORY (Theory applied to various
characterized systems and the systemic characterizations form the basis for the formulation of the
classification).

==========================================================================
=====

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are Accentuation coefficients

( )( )
( )( )
( )( )
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( )
( )( )
( )( )
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( )
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( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( )
( )( )
( )( )
,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )

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are Dissipation coefficients

III. STATEMENT OF GOVERNING EQUATIONS:

SYMMETRY BREAKING AND ACQUISITION OF MASS: 1

MODULE NUMBERED ONE

The differential system of this model is now (Module Numbered one)

( )( )
[( )( )
( )( ) ( )] 2

( )( )
[( )( )
( )( ) ( )] 3

( )( )
[( )( )
( )( ) ( )] 4

( )( )
[( )( )
( )( ) ( )] 5

( )( )
[( )( )
( )( ) ( )] 6

( )( )
[( )( )
( )( ) ( )] 7

( )( ) ( ) First augmentation factor 8

( )( ) ( ) First detritions factor

UNIFIED ELECTROWEAK INTERACTION AND STRONG INTERACTION: 9

MODULE NUMBERED TWO

The differential system of this model is now ( Module numbered two)

( )( )
[( )( )
( )( ) ( )] 10

( )( )
[( )( )
( )( ) ( )] 11

( )( )
[( )( )
( )( ) ( )] 12

( )( )
[( )( )
( )( ) (( ) )] 13

( )( )
[( )( )
( )( ) (( ) )] 14

( )( )
[( )( )
( )( ) (( ) )] 15


( )( ) (( ) ) First detritions factor 17

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LAGRANGIAN INVARIANCE AND CONTINOUS GROUP OF LOCAL 18
TRANSFORMATIONS:

MODULE NUMBERED THREE

The differential system of this model is now (Module numbered three)

( )( )
[( )( )
( )( ) ( )] 19

( )( )
[( )( )
( )( ) ( )] 20

( )( )
[( )( )
( )( ) ( )] 21

( )( )
[( )( )
( )( ) ( )] 22

( )( )
[( )( )
( )( ) ( )] 23

( )( )
[( )( )
( )( ) ( )] 24

( )( ) ( ) First augmentation factor

( )( ) ( ) First detritions factor 25

26

GROUP GENERATOR OF GAUGE THEORY AND VECTOR FIELD(GAUGE FIELD):

: MODULE NUMBERED FOUR:

==========================================================================
==

The differential system of this model is now (Module numbered Four)

( )( )
[( )( )
( )( ) ( )] 27

( )( )
[( )( )
( )( ) ( )] 28

( )( )
[( )( )
( )( ) ( )] 29

( )( )
[( )( )
( )( ) (( ) )] 30

( )( )
[( )( )
( )( ) (( ) )] 31

( )( )
[( )( )
( )( ) (( ) )] 32



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YANG MILLS THEORYAND NON COMMUTATIVE SYMMETRY GROUP IN GAUGE 35
THEORY:

MODULE NUMBERED FIVE

The differential system of this model is now (Module number five)

( )( )
[( )( )
( )( ) ( )] 36

( )( )
[( )( )
( )( ) ( )] 37

( )( )
[( )( )
( )( ) ( )] 38

( )( )
[( )( )
( )( ) (( ) )] 39

( )( )
[( )( )
( )( ) (( ) )] 40

( )( )
[( )( )
( )( ) (( ) )] 41



LHS OF THE YANG MILLS THEORY AND RHS OF THE YANG MILLS THEORY.TAKEN 44
TO THE OTHER SIDE THE LHS WOULD DISSIPATE THE RHS WITH OR WITHOUT TIME
45
LAG :

MODULE NUMBERED SIX

:

The differential system of this model is now (Module numbered Six)

( )( )
[( )( )
( )( ) ( )] 46

( )( )
[( )( )
( )( ) ( )] 47

( )( )
[( )( )
( )( ) ( )] 48

( )( )
[( )( )
( )( ) (( ) )] 49

( )( )
[( )( )
( )( ) (( ) )] 50

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( )( )
[( )( )
( )( ) (( ) )] 51



HOLISTIC CONCATENATE SYTEMAL EQUATIONS HENCEFORTH REFERRED TO AS “GLOBAL 54
EQUATIONS”

We take in to consideration the following parameters, processes and concepts:

(1) Acquisition of mass
(2) Symmetry Breaking
(3) Strong interaction
(4) Unified Electroweak interaction
(5) Continuous group of local transformations
(6) Lagrangian Variance
(7) Group generator in Gauge Theory
(8) Vector field or Gauge field
(9) Non commutative symmetry group in Gauge Theory
(10) Yang Mills Theory (We repeat the same Bank’s example. Individual
debits and Credits are conservative so also the holistic one. Generalized
theories are applied to various systems which are parameterized. And we live
in ‘measurement world’. Classification is done on the parameters of various
systems to which the Theory is applied. ).
(11) First Term of the Lagrangian of the Yang Mills Theory(LHS)

(12) RHS of the Yang Mills Theory

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 55
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 56
( )( )
[ ]
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( ) ( )( )
( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 57
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

311

Vol 7, 2012

Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 58

( )( )
( ) , ( )( )
( ) , ( )( )
( ) are second augmentation coefficient for category 1, 2 and 3

( )( )
( ) ( )( )
( ) ( )( )
( ) are third augmentation coefficient for category 1, 2 and 3
59

( )( )
( ) , ( )( )
( ) , ( )( )
( ) 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

60

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 61
( )
( ) [ ]
( )( )
( ) ( )( )
( ) ( )( )
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( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 62
( )( )
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( )

( )( )
( )( ) ( ) ( )( )
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( ) 63
( )( )
[ ]
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( ) ( )( )
( ) ( )( )
( )

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 3

65

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 66
( )( )
[ ]
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( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 67
( )( )
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( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 68
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 69

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Vol 7, 2012

( )( )
( ) , ( )( )
( ) , ( )( )
( ) 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
70
( )( )
( ), ( )( )
( ) , ( )( )
( ) are sixth augmentation coefficient for category 1, 2 and
3

71

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 72
( )( )
[ ]
( )( )
( ) ( )( )
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( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 73
( )( )
[ ]
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( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 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

( )( )
( ) ( )( )
( ) , ( )( )
( ) are sixth detritions coefficients for category 1,2 and 3

76

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
( )
( ) [ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

77

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
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( )( )
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( )

78

313

Vol 7, 2012

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

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, 2
and 3

( )( )
( ) ( )( )
( ) ( )( )
( ) are sixth augmentation coefficients for category 1, 2
and 3

81

82

( )( )
( )( ) ( ) –( )( )
( ) –( )( )
( )
( )
( ) [ ]
( )( )
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( ) ( )( )
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83

( )( )
( )( ) ( ) –( )( )
( ) –( )( )
( )
( )( )
[ ]
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( ) ( ) ( ) ( ) ( ) ( )

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, 2
and 3

( )( )
( ) ( )( )
( ) ( )( )
( ) are sixth detritions coefficients for category 1, 2
and 3

86

314

Vol 7, 2012

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 87
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 88
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 89
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

90

( )( ) ( ) ( )( ) ( ) ( )( ) ( )

91
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( ) are fourth augmentation coefficients for category 1, 2,and
3

( )( )
( ), ( )( )
( ) ( )( )
( ) are fifth augmentation coefficients for category 1, 2,and 3

( )( )
( ), ( )( )
( ), ( )( )
( ) are sixth augmentation coefficients for category 1, 2,and 3

92

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 93
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 94
( )( )
[ ]
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( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 95
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 96

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) , ( )( )
( )

( )( )
( ), ( )( )
( ), ( )( )
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–( )( )
( ) –( )( )
( ) –( )( )
( )

315

Vol 7, 2012

97

98
99
( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
( )
( ) [ ]
( )( )
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( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 100
( )( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 101
( )
( ) [ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 102

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( ) are fourth augmentation coefficients for category 1,2,
and 3

( )( )
( ) ( )( )
( ) ( )( )
( ) are fifth augmentation coefficients for category 1,2,and
3

( )( )
( ) ( )( )
( ) ( )( )
( ) are sixth augmentation coefficients for category 1,2, 3

103
104
( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( )
( )
( ) [ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 105
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) ( )( )
( ) –( )( )
( ) 106
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

–( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 107

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( ) are fourth detrition coefficients for category 1,2, and 3

( )( )
( ) ( )( )
( ) ( )( )
( ) are fifth detrition coefficients for category 1,2, and 3

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Vol 7, 2012

–( )( )
( ) , –( )( )
( ) –( )( )
( ) are sixth detrition coefficients for category 1,2, and 3

108
109
( )( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

110
( )( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

111
( )( )

( )( )
( )( ) ( ) ( )( )
( ) ( )( )
( )
[ ]
( )( )
( ) ( )( )
( ) ( )( )
( )

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 112

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( ) - are fourth augmentation coefficients

( )( )
( ) ( )( )
( ) ( )( )
( ) - fifth augmentation coefficients

( )( )
( ), ( )( )
( ) ( )( )
( ) sixth augmentation coefficients

113
114
( )( )
( )( ) ( ) –( )( )
( ) –( )( )
( )
( )
( ) [ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) –( )( )
( ) –( )( )
( ) 115
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( )
( )( ) ( ) –( )( )
( ) –( )( )
( ) 116
( )( )
[ ]
( )( )
( ) ( )( )
( ) –( )( )
( )

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 117

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( )

( )( )
( ) ( )( )
( ) ( )( )
( ) are fourth detrition coefficients for category 1, 2, and 3

317

Vol 7, 2012

( )( )
( ), ( )( )
( ) ( )( )
( ) are fifth detrition coefficients for category 1, 2, and
3

–( )( )
( ) , –( )( )
( ) –( )( )
( ) are sixth detrition coefficients for category 1, 2, and
3

118

Where we suppose 119

(A) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( ) 120

(B) The functions ( )( )
( )( ) are positive continuous increasing and bounded.

Definition of ( )( )
( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

121
( )( ) ( ) ( )( )
( )( )
( ̂ )( )

(C) ( )( ) ( ) ( )( ) 122

( )( ) ( ) ( )( )

Definition of ( ̂ )( )
( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( )
are positive constants and

They satisfy Lipschitz condition: 123

)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 124

)( ) 125
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂

With the Lipschitz condition, we place a restriction on the behavior of functions 126
( )( ) ( ) 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 first augmentation coefficient
WOULD be absolutely continuous.

(̂ )( ) : 127

(D) ( ̂ )( )
(̂ )( )
are positive constants

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) : 128

(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together 129
with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and ( ̂ )( ) and the constants
130
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )

131
satisfy the inequalities

318

Vol 7, 2012

( )( )
( )( )
( ̂ )( )
( ̂ )( ) ( ̂ )( ) 132
( ̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( )
( ̂ )( )


(F) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( ) 135

(G) The functions ( )( )
( )( ) are positive continuous increasing and bounded. 136

( )( ) : 137
( )
( )( ) ( ) ( )( )
( ̂ ) 138

( )( ) ( ) ( )( )
( )( )
( ̂ )( ) 139

(H) ( )( ) ( ) ( )( ) 140

( )( ) (( ) ) ( )( ) 141

( ̂ )( ) : 142

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( ) are positive constants and


)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 144

)( )
( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ 145

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 146
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 SECOND augmentation coefficient would be absolutely
continuous.

(̂ )( ) : 147

(I) ( ̂ )( )
(̂ )( )
are positive constants 148

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) : 149

There exists two constants ( ̂ )( ) and ( ̂ )( ) which together
with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )


( )( )
( )( )
(̂ )( )
( ̂ )( ) ( ̂ )( ) 150
(̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( ) 151
( ̂ )( )

319

Vol 7, 2012


(J) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( ) 153

The functions ( )( )
( )( )
are positive continuous increasing and bounded.

( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) ( ) ( )( )
( )( )
( ̂ )( )

( )( ) ( ) ( )( ) 154

( )( ) ( ) ( )( ) 155

( ̂ )( ) : 156

Where ( ̂ )( )
(̂ )( )
( )( )
( )( )


)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 158

)( ) 159
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂

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 THIRD augmentation coefficient, would be absolutely
continuous.

(̂ )( ) : 161

(K) ( ̂ )( )
(̂ )( )

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with 162
( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 163
164
( ) ( )
( ) ( )
( ̂ ) ( )
( ̂ ) ( )
(̂ ( )
) 165
( ̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( ) 166
( ̂ )( )
167


(L) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( ) 169

(M) The functions ( )( )
( )( ) are positive continuous increasing and bounded.

320

Vol 7, 2012

( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
( ̂ )( )

170

(N) ( )( ) ( ( )( )
)
( )( ) (( ) ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( )


)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂

)( )
( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂

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

Defi174nition of ( ̂ )( )
(̂ )( ) : 174

(O) ( ̂ ) ( )
(̂ )( )
(P)
( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) : 175

(Q) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )


( )( )
( )( )
( ̂ )( )
( ̂ )( ) ( ̂ )( )
( ̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( )
( ̂ )( )


(R) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 177
(S) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )( ) :

321

Vol 7, 2012

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
( ̂ )( )

178

(T) ( )( ) ( ) ( )( )
( )
( ) ( ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( )


)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂

)( )
( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂

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.

(̂ )( ) : 181

(U) ( ̂ )( )
(̂ )( )
( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) : 182

(V) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities

( )( )
( )( )
( ̂ )( )
( ̂ )( ) ( ̂ )( )
( ̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( )
( ̂ )( )


( )( )
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 184
(W) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
(̂ )( )

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Vol 7, 2012

185

(X) ( )( ) ( ) ( )( )
( )
( ) (( ) ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( )


)( )
( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂

)( )
( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂

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 SIXTH augmentation coefficient would be
absolutely continuous.

(̂ )( ) : 188

( ̂ )( )
(̂ )( )
( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) : 189

There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )


( )( )
( )( )
( ̂ )( )
( ̂ )( ) ( ̂ )( )
( ̂ )( )

( )( )
( )( )
(̂ )( )
( ̂ )( )
(̂ )( )
( ̂ )( )

190

Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution 191
satisfying the conditions

Definition of ( ) ( ):
( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

192

193

323

Vol 7, 2012

Definition of ( ) ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

194

195

) ( ̂ )( )
( ) ( ̂ )( , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

Definition of ( ) ( ): 196

( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

197

Definition of ( ) ( ):

( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

198

Definition of ( ) ( ): 199

( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions 200
which satisfy

( ) ( ) ( ̂ )( )
( ̂ )( ) 201

) ( ̂ )( )
( ) ( ̂ )( 202

) ( ̂ )( )
( ) ( ̂ )( 203

By 204

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
205

324

Vol 7, 2012

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
206

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
207

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
208

̅ () ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
209

Where ( ) is the integrand that is integrated over an interval ( )

210

Proof: 211
( )
Consider operator defined on the space of sextuples of continuous functions
which satisfy

( ) ( ) ( ̂ )( )
( ̂ )( ) 212

) ( ̂ )( )
( ) ( ̂ )( 213

) ( ̂ )( )
( ) ( ̂ )( 214

By 215

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
216

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
217

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
218

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
219

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
220


Proof: 221
( )
which satisfy

( ) ( ) ( ̂ )( )
( ̂ )( ) 222

) ( ̂ )( )
( ) ( ̂ )( 223

) ( ̂ )( )
( ) ( ̂ )( 224

By 225

325

Vol 7, 2012

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
226

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
227

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
228

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
229

̅ () ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
230


( ) 231
which satisfy

( ) ( ) ( ̂ )( )
( ̂ )( ) 232

) ( ̂ )( )
( ) ( ̂ )( 233

) ( ̂ )( )
( ) ( ̂ )( 234

By 235

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
236

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
237

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
238

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
239

̅ () ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
240


( ) 241
which satisfy
242

( ) ( ) ( ̂ )( )
( ̂ )( ) 243

) ( ̂ )( )
( ) ( ̂ )( 244

) ( ̂ )( )
( ) ( ̂ )( 245

326

Vol 7, 2012

By 246

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
247

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
248

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
249

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
250

̅ () ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
251


252

( )
which satisfy

( ) ( ) ( ̂ )( )
( ̂ )( ) 253

) ( ̂ )( )
( ) ( ̂ )( 254

) ( ̂ )( )
( ) ( ̂ )( 255

By 256

̅ ( ) ∫ [( )( )
( ( )) (( )( )
)( ) ( ( ( )) ( ) )) ( ( ) )] ( )

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
257

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
258

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
259

̅ ( ) ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
260

̅ () ∫ [( )( )
( ( )) (( )( )
( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )
261


262

(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 263
.Indeed it is obvious that

327

Vol 7, 2012

) ( ̂ )( ) (
( ) ∫ [( )( ) ( ( ̂ )( ) )] ( )

( )( ) ( ̂ )( ) )( )
( ( )( )
) ( (̂ )
( ̂ )( )

From which it follows that 264

(̂ )( )
( )( ) ( )
( ̂ )( ) ̂ )
( ( ) ) [(( ( )
) ( ̂ )( ) ]
( ̂ )( )

( ) is as defined in the statement of theorem 1

Analogous inequalities hold also for 265

(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 266

) ( ̂ )( ) (
( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( ( )( )
) 267
( )( ) ( ̂ )( ) )( )
( (̂ )
( ̂ )( )


(̂ )( )
( )( ) ( )
( ̂ )( ) ̂ )(
( ( ) ) ) [((
)
) ( ̂ )( ) ]
( ̂ )(


(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 270

) ( ̂ )( ) (
( ) ∫ [( )( ) ( ( ̂ )( ) )] ( )

( )( ) ( ̂ )( ) )( )
( ( )( )
) ( (̂ )
( ̂ )( )


(̂ )( )
( )( ) ( )
( ̂ )( ) ̂
( ( ) ) ) [(( )( )
) ( ̂ )( ) ]
( ̂ )(


(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 273
̂ )( ) ( )
( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( )

( )( ) ( ̂ )( ) )( )
( ( )( )
) ( (̂ )
( ̂ )( )


(̂ )( )
( )( ) ( )
( ̂ )( ) ̂
( ( ) ) ) [(( )( )
) ( ̂ )( ) ]
( ̂ )(

328

Vol 7, 2012


(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 275
̂ )( ) ( )
( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( )

( )( ) ( ̂ )( ) )( )
( ( )( )
) ( (̂ )
( ̂ )( )


(̂ )( )
( )( ) ( )
( ̂ )( ) ̂
( ( ) ) [(( )( )
) (̂ )( ) ]
( ̂ )( )


(d) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 277

) ( ̂ )( ) (
( ) ∫ [( )( ) ( ( ̂ )( ) )] ( )

( )( ) ( ̂ )( ) )( )
( ( )( )
) ( (̂ )
( ̂ )( )


(̂ )( )
( )( ) ( )
( ̂ )( ) ̂
( ( ) ) [(( )( )
) ( ̂ )( ) ]
( ̂ )( )


Analogous inequalities hold also for

279

280

( )( ) ( )( ) 281
It is now sufficient to take and to choose
( ̂ )( ) ( ̂ )( )

( ̂ )( )
(̂ )( ) large to have
282

(̂ )( ) 283
( )
( )( )
[( ̂ ) ( )
(( ̂ ) ( )
) ] ( ̂ ) ( )
(̂ )( )

(̂ )( ) 284
( )
( )( )
[(( ̂ )( )
) ( ̂ )( ) ] ( ̂ )( )
( ̂ )( )

( ) 285
In order that the operator transforms the space of sextuples of functions satisfying

329

Vol 7, 2012

GLOBAL EQUATIONS into itself
( ) 286
The operator is a contraction with respect to the metric

( ) ( ) ( ) ( )
(( )( ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |

Indeed if we denote 287

Definition of ̃ ̃ :

( ̃ ̃) ( )
( )

It results
( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

( ) ( ) (̂ )( ) 288
| |
(( )( )
( )( )
( ̂ )( )
( ̂ )( ) ( ̂ )( ) ) (( ( ) ( ) ( ) ( )
))
(̂ )( )

And analogous inequalities for . Taking into account the hypothesis the result follows

Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 289
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( ) ( )
( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively of

If 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 ( ) ( ) 290

From 19 to 24 it results

[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] 291
( ) (

( ) ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) : 292

Remark 3: if is bounded, the same property have also . indeed if

330

Vol 7, 2012

( ̂ )( ) 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 293
analogous with the preceding one. An analogous property is true if is bounded from below.

Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) then 294

:

Indeed 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

( )( ) ( )( ) 297
( ̂ )( ) ( ̂ )( )

( ̂ )( )
( ̂ )( ) large to have

(̂ )( ) 298
( )
( )( )
[( ̂ )( )
(( ̂ ) ( )
) ] ( ̂ )( )
( ̂ )( )

299

(̂ )( )
( )
( )( )
[(( ̂ )( )
) ( ̂ )( ) ] ( ̂ )( )
( ̂ )( )

( ) 300
In order that the operator transforms the space of sextuples of functions satisfying
( ) 301

((( )( )
( )( ) ) (( )( )
( )( ) ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |

331

Vol 7, 2012


Definition of ̃ ̃ : ( ̃ ̃ ) ( )
( )

It results 303
( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )

Where ( ) represents integrand that is integrated over the interval 304


)( )
( )( ) | (̂ )( ) 305
|(
(( )( ) ( )( )
( ̂ )( )
(̂ )( )

(̂ ) ( )
( ̂ )( ) ) ((( )( )
( )( )
( )( )
( )( ) ))

And analogous inequalities for . Taking into account the hypothesis the result follows 306

̂ )( ) ̂ )( )
( ̂ )( ) ( ( ̂ )( ) ( respectively of

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 () () 308


[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
() (

() ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 309


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )
310

332

Vol 7, 2012


Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 312

:


( )( )
( )( ) (( )( ) ) () ( )( )

Then ( )( ) ( ) ( )
which leads to 313

( )( ) ( )( )

( )( ) ( )( ) 314
unbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )


315

( )( ) ( )( ) 316
( ̂ )( ) ( ̂ )( )

( ̂ )( )

(̂ )( ) 317
( )
( )( )
[( ̂ )( )
(( ̂ )( )
) ] ( ̂ )( )
( ̂ )( )

(̂ )( ) 318
( )
( )( )
[(( ̂ ) ( )
) ( ̂ ( )
) ] ( ̂ ( )
)
( ̂ )( )

( ) 319
In order that the operator transforms the space of sextuples of functions into itself
( ) 320

((( )( )
( )( ) ) (( )( )
( )( ) ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |


Definition of ̃ ̃ :( (̃) ( ) )
̃ ( )
(( )( ))

It results 322
( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

333

Vol 7, 2012

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) ) 323

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )



( ) ( ) (̂ )( ) 324
| |
(( )( )
( )( )
(̂ )( )
(̂ )( )
( ) ̂
(̂ ) ( )( ) ) ((( )( )
( )( )
( )( )
( )( ) ))


( ) ( )
( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively of




[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 327


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:
330

334

Vol 7, 2012

( )( )
( )( ) (( )( ) ) ( ) ( )( )

Then ( )( ) ( ) ( )
which leads to 331

( )( ) ( )( )

( )( ) ( )( )
( ) ( )
unbounded. The same property holds for if ( ) (( )( ) ) ( )


332

( )( ) ( )( ) 333
( ̂ )( ) ( ̂ )( )

( ̂ )( )

(̂ )( ) 334
( )
( )( )
[( ̂ )( )
(( ̂ ) ( )
) ] ( ̂ ) ( )
( ̂ )( )

(̂ )( ) 335
( )
( )( )
[(( ̂ ) ( )
) ( ̂ )( ) ] ( ̂ )( )
( ̂ )( )

( ) 336
In order that the operator transforms the space of sextuples of functions satisfying IN to
itself

( ) 337

((( )( )
( )( ) ) (( )( )
( )( ) ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |

Indeed if we denote

Definition of (̃) ( ) : ( (̃) ( ) )
̃ ̃ ( )
(( )( ))

It results

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )


335

Vol 7, 2012


338

)( )
( )( ) | (̂ )( ) 339
|(
(( )( )
( )( )
(̂ )( )
(̂ )( )

( ̂ )( ) ( ̂ )( ) ) ((( )( )
( )( )
( )( )
( )( ) ))


Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 340
as not conformal with the reality, however we have put this hypothesis ,in order that we can
postulate condition necessary to prove the uniqueness of the solution bounded by
( ) ( )
( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively of

( ) ( )



[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 342


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:


336

Vol 7, 2012

( )( )
( )( ) (( )( ) ) ( ) ( )( )

Then ( )( ) ( )( )
which leads to 345

( )( ) ( )( )

( )( ) ( )( )

We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS
inequalities hold also for

346

( )( ) ( )( ) 347
( ̂ )( ) ( ̂ )( )

( ̂ )( )

(̂ )( ) 348
( )
( )( )
[( ̂ ( )
) (( ̂ )( )
) ] ( ̂ )( )
( ̂ )( )

(̂ )( ) 349
( )
( )( )
[(( ̂ ) ( )
) ( ̂ )( ) ] ( ̂ )( )
( ̂ )( )

( ) 350

( ) 351

((( )( )
( )( ) ) (( )( )
( )( ) ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |

Indeed if we denote

Definition of (̃) ( ) : ( (̃) ( ) )
̃ ̃ ( )
(( )( ))

It results

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) )

337

Vol 7, 2012

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )


352

)( )
( )( ) | (̂ )( ) 353
|(
(( )( )
( )( )
(̂ )( )
(̂ )( )

( ̂ )( ) ( ̂ )( ) ) ((( )( )
( )( )
( )( )
( )( ) ))

And analogous inequalities for . Taking into account the hypothesis (35,35,36) the result
follows

( ) ( )
( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively of

( ) ( )


From GLOBAL EQUATIONS it results

[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 356


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




338

Vol 7, 2012

:


359
( )( )
( )( ) (( )( ) ) ( ) ( )( )

Then ( )( ) ( ) ( )
which leads to 360

( )( ) ( )( )

( )( ) ( )( )


Analogous inequalities hold also for

361

( )( ) ( )( ) 362
( ̂ )( ) ( ̂ )( )

( ̂ )( )

(̂ )( ) 363
( )
( )( )
[( ̂ ) ( )
(( ̂ )( )
) ] ( ̂ ) ( )
(̂ )( )

(̂ )( ) 364
( )
( )( )
[(( ̂ ) ( )
) ( ̂ )( )
] ( ̂ )( )
( ̂ )( )

( ) 365

( ) 366

((( )( )
( )( ) ) (( )( )
( )( ) ))

( )
( ) ( )
( )| (̂ )( ) ( )
( ) ( )
( )| (̂ )( )
| |

Indeed if we denote

Definition of (̃) ( ) : ( (̃) ( ) )
̃ ̃ ( )
(( )( ))

It results

339

Vol 7, 2012

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( ) )| | ) )

( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (
( )( ) ( ( )) ( )( ) ( ( ))
) )
( )

367


)( )
( )( ) | (̂ )( ) 368
|(
(( )( )
( )( )
(̂ )( )
(̂ )( )

( ̂ )( ) ( ̂ )( ) ) ((( )( )
( )( )
( )( )
( )( ) ))


( ) ( )
( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively of




[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

Definition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 371


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


340

Vol 7, 2012



:


( )( )
( )( ) (( )( ) ) ( ) ( )( )

374

Then ( )( ) ( )( )
which leads to 375

( )( ) ( )( )

( )( ) ( )( )
unbounded. The same property holds for if ( )( ) (( )( ) ( ) ) ( )( )


376

Behavior of the solutions 377

If we denote and define

( )( )
( )( )
( )( ) :

(a) )( )
( )( )
( )( )
( )( )
four constants satisfying

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( ) ( ) ( )
: 378

(b) By ( )( )
( )( )
and respectively ( )( )
( )( )
the roots of the
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
equations ( ) ( ) ( ) ( ) and ( ) ( ) ( )
( )( )

Definition of ( ̅ )( )
( ̅ )( )
( ̅ )( )
( ̅ )( ) : 379

By ( ̅ )( )
( ̅ )( )
and respectively ( ̅ )( )
( ̅ )( )
the roots of the equations
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) and ( ) ( ) ( )( ) ( )
( )( )

( )( )
( )( )
( )( )
( )( ) :- 380

(c) If we define ( )( )
( )( )
( )( )
( )( )
by

341

Vol 7, 2012

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

and analogously 381

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )
382
are defined respectively

Then the solution satisfies the inequalities 383

(( )( ) ( )( ) ) ( )( )
( )

where ( )( ) is defined

(( )( ) ( )( ) ) ( )( )
( )
( )( ) ( )( )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 384
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )( ) ( )( ) ( )( ) ( )( )
)
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) 385
( )

( )( ) (( )( ) ( )( ) ) 386
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 387
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 388

Where ( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

342

Vol 7, 2012



( )( )
( )( )
( )( ) : 390

(d) )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( ) 391

( )( )
( )( )
( )( )
( )( ) (( ) ) ( )( ) (( ) ) ( )( ) 392

( )( )
( )( )
( )( ) : 393

By ( )( )
( )( )
( )( )
the roots 394

(e) of the equations ( )( ) ( ( )
) ( )( ) ( )
( )( ) 395

and ( )( ) ( ( )
) ( )( ) ( )
( )( )
and 396

( ̅ )( )
( ̅ )( )
( ̅ )( ) : 397

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the 398

roots of the equations ( )( ) ( ( )
) ( )( ) ( )
( )( ) 399

and ( )( ) ( ( )
) ( )( ) ( )
( )( ) 400

( )( )
( )( )
( )( ) :- 401

(f) If we define ( )( )
( )( )
( )( )
( )( )
by 402

( )( )
( )( )
( )( )
( )( )
( )( )
( )( ) 403

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( ) 404

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( ) 405

and analogously 406

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( ) 407


(( )( ) ( )( ) ) ( ) ( )( )

( )( ) is defined 409

343

Vol 7, 2012

(( )( ) ( )( ) ) ( )( ) 410
( )
( )( ) ( )( )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 411
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )
( ) ( )( ) ( )( ) ( )( )
)
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) 412
( )

( )( ) (( )( ) ( )( ) ) 413
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 414
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 415

Where ( )( )
( )( ) ( )( )
( )( ) 416

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( ) 417

( )( )
( )( )
( )( )

418



( )( )
( )( )
( )( ) :

(a) )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) (( ) ) ( )( )

( )( )
( )( )
( )( ) : 420

(b) By ( )( )
( )( )
( )( )
the roots of the
( ) ( ) ( ) ( ) ( )
equations ( ) ( ) ( ) ( )

and ( )( ) ( ( )
) ( )( ) ( )
( )( )
and

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the

roots of the equations ( )( ) ( ( )
) ( )( ) ( )
( )( )

and ( )( ) ( ( )
) ( )( ) ( )
( )( )

( )( )
( )( )
( )( ) :- 421

344

Vol 7, 2012

(c) If we define ( )( )
( )( )
( )( )
( )( )
by

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

and analogously 422

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )
and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

Then the solution satisfies the inequalities

(( )( ) ( )( ) ) ( )( )
( )

( )( ) is defined 423

(( )( ) ( )( ) ) ( )( ) 424
( )
( )( ) ( )( )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 425
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )( ) ( )( ) ( )( ) ( )( )
)
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) 426
( )

( )( ) (( )( ) ( )( ) ) 427
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 428
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 429

Where ( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

430

431


345

Vol 7, 2012


( )( )
( )( )
( )( ) :

(d) ( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) (( ) ) ( )( ) (( ) ) ( )( )

( )( )
( )( )
( )( ) ( ) ( )
: 433

(e) By ( )( )
( )( )
( )( )
the roots of the
( ) ( ) ( ) ( ) ( )
equations ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( ) and

( ̅ )( )
( ̅ )( )
( ̅ )( ) : 434
435
By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
roots of the equations ( ) ( ) ( ) ( )
and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 436
Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-

(f) If we define ( )( )
( )( )
( )( )
( )( )
by

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

and analogously 437
438
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( )
( )( )
where ( )( )
( ̅ )( )

are defined by 59 and 64 respectively

440
(( )( ) ( )( ) ) ( ) ( )( ) 441
442
where ( )( ) is defined 443
444
445

346

Vol 7, 2012

(( )( ) ( )( ) ) ( ) ( )( ) 446
( )( ) ( )( )
447
( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 448
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )( ) ( )( ) ( )( ) ( )( )
[ ] )
( )( ) (( )( ) ( )( ) )

( )( ) ( ) (( )( ) ( )( ) ) 449

( )( ) (( )( ) ( )( ) ) 450
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 451
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 452

Where ( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( ) 453


( )( )
( )( )
( )( ) :

(g) ( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) (( ) ) ( )( ) (( ) ) ( )( )

( )( )
( )( )
( )( ) ( ) ( )
: 455

(h) By ( )( )
( )( )
( )( )
the roots of the
( ) ( ) ( ) ( ) ( )
equations ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( ) and

( ̅ )( )
( ̅ )( )
( ̅ )( ) : 456

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( )
Definition of ( )( ) ( )( ) ( )( ) ( )( )
( )( ) :-

(i) If we define ( )( )
( )( )
( )( )
( )( )
by

347

Vol 7, 2012

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

and analogously 457

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( ) ( )( ) ( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )



(( )( ) ( )( ) ) ( )( )
( )

(( )( ) ( )( ) ) ( )( ) 459
( )
( )( ) ( )( )

460
( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 461
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )( ) ( )( ) ( )( ) ( )( )
[ ] )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) 462
( )

( )( ) (( )( ) ( )( ) ) 463
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 464
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 465

Where ( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )


348

Vol 7, 2012


( )( )
( )( )
( )( ) :

(j) ( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( )( ) ( ) ( )( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) (( ) ) ( )( ) (( ) ) ( )( )

( )( )
( )( )
( )( ) ( ) ( )
: 467

(k) By ( )( )
( )( )
( )( )
the roots of the
( ) ( ) ( ) ( ) ( )
equations ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( ) and

( ̅ )( )
( ̅ )( )
( ̅ )( ) : 468

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )
Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-

(l) If we define ( )( )
( )( )
( )( )
( )( )
by

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

470
( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )

and analogously 471

( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )

and ( )( )

( )( ) ( )( ) ( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )



(( )( ) ( )( ) ) ( )( )
( )

(( )( ) ( )( ) ) ( )( ) 473
( )
( )( ) ( )( )

349

Vol 7, 2012

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 474
( [ ] ( )
( )( ) (( )( ) ( )( ) ( )( ) )
( )( ) ( )( ) ( )( ) ( )( )
[ ] )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) 475
( )

( )( ) (( )( ) ( )( ) ) 476
( )
( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( ) 477
[ ] ( )
( )( ) (( )( ) ( )( ) )

( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )
[ ]
( )( ) (( )( ) ( )( ) ( )( ) )

( )( )
( )( )
( )( ) :- 478

Where ( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

( )( )
( )( ) ( )( )
( )( )

( )( )
( )( )
( )( )

479

Proof : From GLOBAL EQUATIONS we obtain 480
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( ) 481
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )

From which one obtains

( )( ) :-

(a) For ( )( )
( )( )
( ̅ )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( ) (( )( ) ( )( ) ) ]
, ( )( )
[ ( ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )

In the same manner , we get 482

350

Vol 7, 2012

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( )((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
[ ( ( )( ) (̅ )( )
( ̅ )( )

From which we deduce ( )( ) ( )
( ) ( ̅ )( )

(b) If ( )( )
( )( )
( ̅ )( )
we find like in the previous case, 483

[ ( )( ) (( )( ) ( )( ) ) ]
( )( ) ( )( ) ( )( )
( )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )
( )
( )( )

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( )((̅ )( ) (̅ )( ) ) ]
( ̅ )( )
( ̅ )( )

484
(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
( )

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

Now, using this result and replacing it in GLOBAL E486QUATIONS we get easily the result stated
in the theorem.

Particular case : 485

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 ( )( )

486

we obtain 487

351

Vol 7, 2012

( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( ) 488
Definition of :-

It follows 489
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )

( )( ) )

From which one obtains 490

( )( ) :-

(d) For ( )( )
( )( )
( ̅ )( )

[ ( )( )(( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( )(( )( ) ( )( ) ) ]
, ( )( )
[ ( ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )


[ ( )( ) ((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( ) ((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
[ ( ( )( ) (̅ )( )
( ̅ )( )

( ) ( ̅ )( ) 492

(e) If ( )( )
( )( )
( ̅ )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )( ) ( )( ) ( )( )
( )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )
( )
( )( )

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( )((̅ )( ) (̅ )( ) ) ]
( ̅ )( )
( ̅ )( )

(f) If ( )( )
( ̅ )( )
( )( )
, we obtain 494

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
( )( ) ( )
( ) [ ( )( )((̅ )( ) (̅ )( ) ) ]
( )( )
( ̅ )( )

( )
Definition of ( ) :- 495

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

In a completely analogous way, we obtain 496
( )

352

Vol 7, 2012

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

. 497

Particular case : 498

( )( ) ( )( ) then ( )
( ) ( )( ) and as a consequence ( ) ( )( ) ( )

( )( )
( )( )
( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( )
( ) This is an important

499

From GLOBAL EQUATIONS we obtain 500
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

Definition of ( )
:- ( ) 501

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )

( )( ) )

502

(a) For ( )( )
( )( )
( ̅ )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( ) (( )( ) ( )( ) ) ]
, ( )( )
[ ( ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )


[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( )((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
[ ( ( )( ) (̅ )( )
( ̅ )( )

Definition of ( ̅ )( ) :-

( ) ( ̅ )( )

(b) If ( )( )
( )( )
( ̅ )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )
( ) )( ) ((
( )
[ ( )( ) ( )( ) ) ]
( )( )

353

Vol 7, 2012

[ ( )( ) ((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( ) ((̅ )( ) (̅ )( ) ) ]
( ̅ )( )
( ̅ )( )

(c) If ( )( )
( ̅ )( )
( )( )
, we obtain 505
[ ( )( ) ((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
( )( ) ( )
( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ]
( )( )
( ̅ )( )

( )

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

( )

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in
the theorem.

Particular case :

( )( ) ( )( ) then ( )
( ) ( )( ) and as a consequence ( ) ( )( ) ( )

( )( )
( )( )
( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( )
( ) This is an important

506

: From GLOBAL EQUATIONS we obtain 507

( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :- 508

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )

( )( ) )

( )( ) :-

(d) For ( )( )
( )( )
( ̅ )( )

354

Vol 7, 2012

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( ) ((
, ( )( )
[ ( )( ) ( )( ) ) ] ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )


[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( )((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
( ̅ )( )
[ ( ( )( ) (̅ )( )

( ) ( ̅ )( )

(e) If ( )( )
( )( )
( ̅ )( ) we find like in the previous case, 510

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )
( ) )( ) ((
( )
[ ( )( ) ( )( ) ) ]
( )( )

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( )((̅ )( ) (̅ )( ) ) ]
( ̅ )( )
( ̅ )( )
511
( ) ( ) ( ) 512
(f) If ( ) ( ̅ ) ( ) , 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 :

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

355

Vol 7, 2012

From GLOBAL EQUATIONS we obtain 515

( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )


( )( ) :-

(g) For ( )( )
( )( )
( ̅ )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( ) (( )( ) ( )( ) ) ]
, ( )( )
[ ( ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )


[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( )((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
[ ( ( )( ) (̅ )( )
( ̅ )( )

( ) ( ̅ )( )

(h) If ( )( )
( )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )
( ) )( ) ((
( )
[ ( )( ) ( )( ) ) ]
( )( )

[ ( )( )((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( )((̅ )( ) (̅ )( ) ) ]
( ̅ )( )
( ̅ )( )
518
(i) If ( )( )
( ̅ )( )
( )( )
, we obtain

[ ( )( ) ((̅ )( ) (̅ )( )) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
( )( ) ( )
( ) [ ( )( ) ((̅ )( ) (̅ )( )) ]
( )( )
( ̅ )( )
519

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

356

Vol 7, 2012

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

theorem.

Particular case :

( ) ( ) ( )
( )

( ) ( ) ( )
consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )
( ) ( )

520
we obtain 521

( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )

( )( ) )


( )( ) :-

(j) For ( )( )
( )( )
( ̅ )( )

[ ( )( ) (( )( ) ( )( ) ) ]
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
( ) )( ) (( )( ) ( )( ) ) ]
, ( )( )
[ ( ( )( ) ( )( )
( )( )

( )( ) ( )
( ) ( )( )


[ ( )( )((̅ )( ) (̅ )( ) ) ] 523
(̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( )
( )
( ) )( )((̅ )( ) (̅ )( ) ) ]
, ( ̅ )( )
[ ( ( )( ) (̅ )( )
( ̅ )( )

( ) ( ̅ )( )

(k) If ( )( )
( )( )

357

Vol 7, 2012

[ ( )( ) (( )( ) ( )( ) ) ]
( )( ) ( )( ) ( )( )
( )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )
( )
( )( )

[ ( )( ) ((̅ )( ) (̅ )( )) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
[ ( )( ) ((̅ )( ) (̅ )( )) ]
( ̅ )( )
( ̅ )( )
525
(l) If ( )( )
( ̅ )( )
( )( )
, we obtain

[ ( )( ) ((̅ )( ) (̅ )( )) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
( )( ) ( )
( ) [ ( )( ) ((̅ )( ) (̅ )( )) ]
( )( )
( ̅ )( )


( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )

theorem.

Particular case :

( ) ( ) ( )
( )
( ) ( ) ( )
consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )
( ) ( )

526
527 527

We can prove the following 528

Theorem 3: If ( )( )
( )( ) are independent on , and the conditions

( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
,

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( )
( )( ) as defined, then the system
529

If ( )( )
( )( ) are independent on , and the conditions 530.

358

Vol 7, 2012

( )( ) ( )( )
( )( ) ( )( ) 531

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) 532

( )( ) ( )( )
( )( ) ( )( )
, 533

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) 534

( )( )
( )( ) as defined are satisfied , then the system

If ( )( )
( )( ) are independent on , and the conditions 535

( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
,

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( )

If ( )( )

( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
,

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( )

If ( )( )

( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
,

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( )
( )( ) as defined satisfied , then the system

If ( )( )

( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )

( )( ) ( )( )
( )( ) ( )( )
,
539
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

359

Vol 7, 2012

( )( )

( )( )
[( )( )
( )( ) ( )] 540

( )( )
[( )( )
( )( ) ( )] 541

( )( )
[( )( )
( )( ) ( )] 542

( )( )
( )( )
( )( ) ( ) 543

( )( )
( )( )
( )( ) ( ) 544

( )( )
( )( )
( )( ) ( ) 545

has a unique positive solution , which is an equilibrium solution for the system 546

( )( )
[( )( )
( )( ) ( )] 547

( )( )
[( )( )
( )( ) ( )] 548

( )( )
[( )( )
( )( ) ( )] 549

( )( )
( )( )
( )( ) ( ) 550

( )( )
( )( )
( )( ) ( ) 551

( )( )
( )( )
( )( ) ( ) 552

has a unique positive solution , which is an equilibrium solution for 553

( )( )
[( )( )
( )( ) ( )] 554

( )( )
[( )( )
( )( ) ( )] 555

( )( )
[( )( )
( )( ) ( )] 556

( )( )
( )( )
( )( ) ( ) 557

( )( )
( )( )
( )( ) ( ) 558

( )( )
( )( )
( )( ) ( ) 559

has a unique positive solution , which is an equilibrium solution 560

( )( )
[( )( )
( )( ) ( )] 561

( )( )
[( )( )
( )( ) ( )] 563

( )( )
[( )( )
( )( ) ( )] 564

( )( )
( )( )
( )( ) (( )) 565

( )( )
( )( )
( )( ) (( )) 566

( )( )
( )( )
( )( ) (( )) 567

360

Vol 7, 2012


( )( )
[( )( )
( )( ) ( )] 569

( )( )
[( )( )
( )( ) ( )] 570

( )( )
[( )( )
( )( ) ( )] 571

( )( )
( )( )
( )( ) ( ) 572

( )( )
( )( )
( )( ) ( ) 573

( )( )
( )( )
( )( ) ( ) 574


( )( )
[( )( )
( )( ) ( )] 576

( )( )
[( )( )
( )( ) ( )] 577

( )( )
[( )( )
( )( ) ( )] 578

( )( )
( )( )
( )( ) ( ) 579

( )( )
( )( )
( )( ) ( ) 580

( )( )
( )( )
( )( ) ( ) 584


583

584

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )( ) ( )( )( ) ( )

585


( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )
( ) ( )( )( ) ( ) 586

361

Vol 7, 2012

587


( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )
( ) ( )( )( ) ( )

588


( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )( )( )( ) ( )
( )

589


( )
( )
( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )( ) ( )( )( )( )

560


( ) ( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )
( )( )( )( ) ( )
( )

Definition and uniqueness of :- 561

After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows
that there exists a unique for which ( ) . With this value , we obtain from the three first
equations

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]


that there exists a unique for which ( ) . With this value , we obtain from the three first
equations

( )( ) ( )( ) 563
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]


After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows that
there exists a unique for which ( ) . With this value , we obtain from the three first
equations

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

362

Vol 7, 2012

565


that there exists a unique for which ( ) . With this value , we obtain from the three
first equations

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]


first equations

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]


first equations

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

(e) 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 that
is a decreasing function in taking into account the hypothesis ( ) ( ) it follows
that there exists a unique such that ( )

(f) 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
that there exists a unique such that (( ) )

(g) By the same argument, the concatenated equations admit solutions if 572

( ) ( )( ) ( )( )
( )( ) ( )( )

[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )

363

Vol 7, 2012

Where in ( ) must be replaced by their values from 96. It is easy to see that
that there exists a unique such that (( ) )
573

(h) 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 see
that is a decreasing function in taking into account the hypothesis ( ) ( ) it
follows that there exists a unique such that (( ) )

(i) 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 see
that is a decreasing function in taking into account the hypothesis ( ) ( ) it
follows that there exists a unique such that (( ) )

(j) By the same argument, the equations (modules) admit solutions if 578

579
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
580
( ) ( ) ( ) ( ) ( ) ( )
[( ) ( ) ( ) ( ) ( ) ( )] ( ) ( )( ) ( )
581
Where in ( )( ) must be replaced by their values It is easy to see that is a
decreasing function in taking into account the hypothesis ( ) ( ) it follows that
there exists a unique such that ( )

Finally we obtain the unique solution of 89 to 94 582

( ) , ( ) and

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( )( )]

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution 583

(( )) , ( ) and 584

364

Vol 7, 2012

( )( ) ( )( ) 585
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( ) 586
,
[( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]

Obviously, these values represent an equilibrium solution 587


(( )) , ( ) and

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]



( ) , ( ) and

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( ) 590
,
[( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]



(( )) , ( ) and

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( ) 592
,
[( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]



(( )) , ( ) and

( )( ) ( )( )
,
[( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]

( )( ) ( )( ) 594
,
[( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]


ASYMPTOTIC STABILITY ANALYSIS 595

365

Vol 7, 2012

Theorem 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: Denote

Definition of :-

,
596
( )( ) ( ) ( )( )
( ) ( ) , ( )

Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597

(( )( )
( )( ) ) ( )( )
( )( ) 598

(( )( )
( )( ) ) ( )( )
( )( ) 599

(( )( )
( )( ) ) ( )( )
( )( ) 600

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 601

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 602

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 603

If the conditions of the previous theorem are satisfied and if the functions ( )( )
( )( ) 604
Belong to ( ) ( ) then the above equilibrium point is asymptotically stable

Denote 605

Definition of :-

, 606

( )( ) ( )( ) 607
( ) ( )( )
, (( ) )

taking into account equations (global)and neglecting the terms of power 2, we obtain 608

(( )( )
( )( ) ) ( )( )
( )( ) 609

(( )( )
( )( ) ) ( )( )
( )( ) 610

(( )( )
( )( ) ) ( )( )
( )( ) 611

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 612

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 613

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 614

366

Vol 7, 2012

( )( ) 615
Belong to ( ) ( ) then the above equilibrium point is asymptotically stabl

Denote

Definition of :-

,

( )( ) ( )( )
( ) ( )( )
, (( ) )

616


(( )( )
( )( ) ) ( )( )
( )( ) 618

(( )( )
( )( ) ) ( )( )
( )( ) 619

(( )( )
( )( ) ) ( )( )
( )( ) 6120

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 621

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 622

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 623

( )( ) 624
Belong to ( ) ( ) then the above equilibrium point is asymptotically stabl

Denote


,

( )( ) ( )( )
( ) ( )( )
, (( ) )


(( )( )
( )( ) ) ( )( )
( )( ) 627

(( )( )
( )( ) ) ( )( )
( )( ) 628

(( )( )
( )( ) ) ( )( )
( )( ) 629

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 630

367

Vol 7, 2012

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 631

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 632

633

( )( )


Denote


,

( )( ) ( )( )
( ) ( )( )
, (( ) )


(( )( )
( )( ) ) ( )( )
( )( ) 636

(( )( )
( )( ) ) ( )( )
( )( ) 637

(( )( )
( )( ) ) ( )( )
( )( ) 638

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 639

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 640

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 641

( )( ) 642

Denote


,

( )( ) ( )( )
( ) ( )( )
, (( ) )

Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644

(( )( )
( )( ) ) ( )( )
( )( ) 645

(( )( )
( )( ) ) ( )( )
( )( ) 646

368

Vol 7, 2012

(( )( )
( )( ) ) ( )( )
( )( ) 647

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 648

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 649

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) ) 650

651

The characteristic equation of this system is 652

(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]
653
((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) )

369

Vol 7, 2012

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

370

Vol 7, 2012

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

[((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)]

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( )
( )( )
( )( ) )( )( )
( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) )

371

Vol 7, 2012

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( ) ( ) )

((( )( ) ) (( )( )
( )( )
( )( )
( )( ) ) ( )( ) ) ( )( )

(( )( )
( )( )
( )( ) ) (( )( ) ( )( )
( )( ) ( )( ) ( )( )
)

((( )( )
( )( )
( )( ) ) ( )( ) ( )( )
( )( ) )

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part,
and this proves the theorem.

IV. Acknowledgments:
=======================================================================
The introduction is a collection of information from various articles, Books, News
Paper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,Article
Abstracts, Research papers, Abstracts Of Research Papers, Stanford
Encyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors,
the internet including Wikipedia. We acknowledge all authors who have contributed
to the same. In the eventuality of the fact that there has been any act of omission on
the part of the authors, we regret with great deal of compunction, contrition, regret,
trepidiation and remorse. As Newton said, it is only because erudite and eminent
people allowed one to piggy ride on their backs; probably an attempt has been made
to look slightly further. Once again, it is stated that the references are only
illustrative and not comprehensive

V. REFERENCES
================================================================
=========

1. Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi - MEASUREMENT
DISTURBS EXPLANATION OF QUANTUM MECHANICAL STATES-A HIDDEN VARIABLE
THEORY - published at: "International Journal of Scientific and Research Publications, Volume 2,
Issue 5, May 2012 Edition".
2. DR K N PRASANNA KUMAR, PROF B S KIRANAGI and PROF C S BAGEWADI -
CLASSIC 2 FLAVOUR COLOR SUPERCONDUCTIVITY AND ORDINARY NUCLEAR
MATTER-A NEW PARADIGM STATEMENT - published at: "International Journal of Scientific
and Research Publications, Volume 2, Issue 5, May 2012 Edition".
3. 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

4. FRTJOF CAPRA: “The web of life” Flamingo, Harper Collins See "Dissipative structures”
pages 172-188

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5. 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

6. 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

7. 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, 980-1006

8. 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

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(9)^ a b c Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhalt
abhängig?", 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 and
Company, 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, and
that the velocity is not a Minkowski four-vector, in contrast to the quantity , where is the differential
of 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 distinction
between 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.ISBN
0-7167-2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclear
bombs, 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

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≡ 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-dominated
average 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
1.5
prototype, which are ± micrograms.
2

(24)^ [3] Article on Earth rotation energy. Divided by c^2.

(25)^ a b Earth's gravitational self-energy is 4.6 × 10-10 that of Earth's total mass, or 2.7 trillion metric
tons. 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 be
made to couple to gravity in many different ways. By general scaling arguments, the correct answer
at everyday distances, which are long compared to the quantum gravity scale, should be minimal
coupling, 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 Yang
Mills 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 Physics
51:189–241 (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.

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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, Princeton
University Press, p. 51, ISBN 0-691-01017-X

(35)^ Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts of
mass",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 Einstein's special theory of relativity. Emergence (1905) and
early 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 Optical
Society 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), "Einstein's first derivation of mass-energy
equivalence", 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 of
Science Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.002

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(45)^ Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal of
Physics 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 of
Physics 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 nuclear
spin". Nature 455 (7216): 1085–1088. Bibcode 2008Natur.455.1085M.DOI:10.1038/nature07295.

(50)^ S. Weisner (1983). "Conjugate coding". Association of Computing Machinery, Special Interest
Group 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)^ a b Straumann, N (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in
1953". ArXiv: gr-qc/0012054 [gr-qc].

(54)^ See Abraham Pais' account of this period as well as L. Susskind's "Superstrings, Physics World
on the first non-abelian gauge theory" where Susskind wrote that Yang–Mills was "rediscovered"
only because Pauli had chosen not to publish.

(55)^ Reifler, N (2007). "Conditions for exact equivalence of Kaluza-Klein and Yang-Mills
theories". ArXiv: gr-qc/0707.3790 [gr-qc].

(56)^ Yang, C. N.; Mills, R. (1954). "Conservation of Isotopic Spin and Isotopic Gauge
Invariance". Physical Review 96 (1): 191–

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195. Bibcode 1954PhRv...96...191Y.DOI:10.1103/PhysRev.96.191.

(57)^ Caprini, I.; Colangelo, G.; Leutwyler, H. (2006). "Mass and width of the lowest resonance in
QCD". Physical Review Letters 96 (13): 132001. ArXiv: hep-
ph/0512364.Bibcode 2006PhRvL..96m2001C. DOI:10.1103/PhysRevLett.96.132001.

(58)^ Yndurain, F. J.; Garcia-Martin, R.; Pelaez, J. R. (2007). "Experimental status of the ππ
isoscalar S wave at low energy: f0 (600) pole and scattering length". Physical Review D76 (7):
074034. ArXiv:hep-
ph/0701025. Bibcode 2007PhRvD..76g4034G.DOI:10.1103/PhysRevD.76.074034.

(59)^ Novikov, V. A.; Shifman, M. A.; A. I. Vainshtein, A. I.; Zakharov, V. I. (1983). "Exact Gell-
Mann-Low Function of Supersymmetric Yang-Mills Theories From Instanton Calculus”.
Nuclear 229 (2): 381–393. Bibcode 1983NuPhB.229..381N.DOI:10.1016/0550-3213(83)90338-3.

(60)^ Ryttov, T.; Sannino, F. (2008). "Super symmetry Inspired QCD Beta Function". Physical
Review D 78 (6): 065001. Bibcode 2008PhRvD..78f5001R.DOI:10.1103/PhysRevD.78.065001

(61)^ Bogolubsky, I. L.; Ilgenfritz, E.-M.; A. I. Müller-Preussker, M.; Sternbeck, A. (2009). "Lattice
gluodynamics computation of Landau-gauge Green's functions in the deep infrared". Physics Letters
B 676 (1-3): 69–73. Bibcode 2009PhLB..676...69B.DOI:10.1016/j.physletb.2009.04.076.

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First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics,
Political Science. Thesis was based on Mathematical Modeling. He was recently awarded D.litt. for his work on
‘Mathematical Models in Political Science’--- Department of studies in Mathematics, Kuvempu University,
Shimoga, Karnataka, India Corresponding Author:drknpkumar@gmail.com

Second 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 guided
over 25 students and he has received many encomiums and laurels for his contribution to Co homology Groups
and Mathematical Sciences. Known for his prolific writing, and one of the senior most Professors of the
country, he has over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his credit
several books on Lie Groups, Co Homology Groups, and other mathematical application topics, and excellent
publication history.-- UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri,
University of Mysore, Karnataka, India

Third Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Department
of Studies in Computer Science and has guided over 25 students. He has published articles in both national and
international journals. Professor Bagewadi specializes in Differential Geometry and its wide-ranging
ramifications. 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 Computer
science, Jnanasahyadri Kuvempu University, Shankarghatta, Shimoga district, Karnataka, India

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