Yang–Mills Existence and Mass Gap (Unsolved … · International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 2013 1 ISSN 2250-3153 Yang–Mills

International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 2013 1

ISSN 2250-3153

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Yang Mills Existence and Mass Gap

(Unsolved Problem): Aufklrung La

Altagsgeschichte: Enlightenment of a Micro

History

Dr. K.N.P. Kumar

Post doctoral fellow, Department of mathematics, Kuvempu University, Shimoga, Karnataka, India

Abstract: Yang Mills theory is the (non-Abelian) quantum field theory underlying the Standard Model of

particle physics; \mathbb{R}^4 is Euclidean 4-

particle predicted by the theory. Therefore, the winner must first prove that YangMills theory exists and

that it satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular

constructive quantum field theory, which is referenced in the papers 45 and 35 cited in the official problem

description by Jaffe and Witten. The winner must then prove that the mass of the least massive particle of

the force field predicted by the theory is strictly positive. For example, in the case of G=SU (3) - the strong

nuclear interaction - the winner must prove that glueballs have a lower mass bound, and thus cannot be

arbitrarily light. Biagio Lucini, Michael Teper, Urs Wenger studied Glueballs and k-strings in SU (N) gauge

theories : calculations with improved operators testing a variety of blocking and smearing algorithms for

constructing glueball and string wave-functionals, and find some with much improved overlaps onto the

lightest states. They use these algorithms to obtain improved results on the tensions of k-strings in SU (4),

SU (6), and SU (8) gauge theories. Authors emphasise the major systematic errors that still need to be

controlled in calculations of heavier k-strings, and perform calculations in SU (4) on an anisotropic lattice

in a bid to minimise one of these. All these results point to the k-string tensions lying part-way between the

`MQCD' and `Casimir Scaling' conjectures, with the power in 1/N of the leading correction lying in [1,2].

(See the paper). They also obtain some evidence for the presence of quasi-stable strings in calculations that

do not use sources, and observe some near-degeneracies between (excited) strings in different

representations. We also calculate the lightest glueball masses for N=2... 8, and extrapolate to N=infinity,

obtaining results compatible with earlier work. Biagio Lucini et al show that the N=infinity factorization of

the Euclidean correlators that are used in such mass calculations does not make the masses any less

calculable at large N. JHEP0406:012,2004DOI: 10.1088/1126-6708/2004/06/012 arXiv: hep-

lat/0404008.Quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical

models of subatomic particles in particle physics and quasiparticles in condensed matter physics, by treating

a particle as an excited state of an underlying physical field. These excited states are called field quanta. For

example, quantum electrodynamics (QED) has one electron field and one photon field, quantum

chromodynamics (QCD) has one field for each type of quark, and in condensed matter there is an atomic

displacement field that gives rise to phonon particles. Ed Witten describes QFT as "by far" the most difficult

theory in modern physics. Towards the end of consummation of solution of this long outstanding problem

we make two assumptions that the statements are true or not and the properties is testified by manifested

actions. This bears ample testimony, infallible observatory and impeccable demonstration of the fact that

state mental propositions in either case shall testify the prediction, projection, stability analysis results by

experiments to prove or disprove the theory. In essence the method is that of false princeps and reductio ad

absurdum. Quintessentially it is one model. Towards the end of circumvention of repeated projection of

superscripts and subscripts which is of the order 56, we give the model in two sections. Notwithstanding

variables are all to be taken as different and concatenation is to be done. As said towards the end of


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obtention of felicity of expression and avoiding the extensive superscriptal and subscriptal typing which

might cause systemic errors, model is bifurcated in to two. Section two is only progressive of section one.

INTRODUCTION VARIABLES USED

Source: Wikipedia

The problem is phrased as follows:

Yang Mills Existence and Mass Gap

(1) For any compact simple gauge group G, a non-trivial quantum YangMills theory exists (eb)

on

strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973)

and Osterwalder & Schrader (1975).

(2) For any compact simple gauge group G, a non-trivial quantum YangMills theory does not exist

(eb) on

as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973)

and Osterwalder & Schrader (1975).

(3) In this statement, Yang Mills theory is (=) the (non-Abelian) quantum field theory underlying

the Standard Model of particle physics; is Euclidean 4-space

(4) The mass gap ted by the theory.

(5) Therefore, the winner must first prove that Yang Mills theory exists and that it (eb) satisfies the

standard of rigor that characterizes contemporary mathematical physics, in particular constructive

quantum field theory, which is referenced in the papers 45 and 35 cited in the official problem

description by Jaffe and Witten.

(6) The winner must then prove that the mass of the least massive particle of the force field predicted

by the theory is (=) strictly positive.

(7) For example, in the case of G=SU (3) - the strong nuclear interaction - the winner must prove

that glueballs have (e) a lower mass bound

(8) Thus glueballs cannot (e) be arbitrarily light.

(9) Yang Mills theories are a special example of gauge theory with a non-abelian symmetry group

given by the Lagrangian

with the generators of the Lie algebra corresponding to the F-quantities (the curvature or field-strength form)

satisfying

and the covariant derivative defined as

where I is the identity for the group generators, is the vector potential, and g is the coupling constant.

In four dimensions, the coupling constant g is a pure number and for a SU(N) group one


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has

The relation

can be derived by the commutator

The field has the property of being self-interacting and equations of motion that one obtains are said to be

semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this

theory only by perturbation theory, with small nonlinearities.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor

components is trivial for a indices (e.g.

e.g. to the usual Lorentz signature, .

From the given Lagrangian one can derive the equations of motion given by

Putting , these can be rewritten as

A Bianchi identity holds

which is equivalent to the Jacobi identity

since . Define the dual strength tensor , then

the Bianchi identity can be rewritten as

A source enters into the equations of motion as

Note that the currents must properly change under gauge group transformations.


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We give here some comments about the physical dimensions of the coupling. We note that, in D dimensions,

the field scales as and so the coupling must scale as . This implies

that YangMills theory is not renormalizable for dimensions greater than four. Further, we note that, for D =

4, the coupling is dimensionless and both the field and the square of the coupling have the same dimensions

of the field and the coupling of a massless quartic scalar field theory. So, these theories share the scale

invariance at the classical level.

NOTATION

Module One

For any compact simple gauge group G, a non-trivial quantum YangMills theory exists (eb) on and

has a mass

in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

13 : Category one of For any compact simple gauge group G, a non-trivial quantum YangMills theory

14 : Category two of For any compact simple gauge group G, a non-trivial quantum YangMills theory.

Systemic differentiation. There are various systems to which Yang Mills theory is applicable and mass

gap exists. Characterstics of these systems are taken I to consideration in the consummation of the

diaspora fabric of the classification doxa.

15 : Category three of For any compact simple gauge group G, a non-trivial quantum YangMills theory

13 : Category one of exists (eb) on

axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &

Schrader (1973) and Osterwalder & Schrader (1975).

14 : Category two of exists (eb) on



15 : Category three of exists (eb) on



Module Two

For any compact simple gauge group G, a non-trivial quantum YangMills theory does not exist (eb)

on

those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader

(1975)

16 : Category one of For any compact simple gauge group G, a non-trivial quantum YangMills theory

does not

17: Category two of For any compact simple gauge group G, a non-trivial quantum YangMills theory

does not

18: Category three of For any compact simple gauge group G, a non-trivial quantum YangMills theory

does not

16 : Category one of existence (eb) on 0. Existence includes establishing

axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &Schrader


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(1973) and Osterwalder & Schrader (1975)

17 : Category two of existence (eb) on xistence includes establishing


Schrader (1973) and Osterwalder & Schrader (1975)

18 : Category three of existence (eb) on stence includes establishing


Schrader (1973) and Osterwalder & Schrader (1975)

Module three

In this statement, Yang Mills theory is (=) the (non-Abelian) quantum field theory underlying the Standard

Model of particle physics; is Euclidean 4-space

20 : Category one of(non-Abelian) quantum field theory underlying the Standard Model of particle

physics; is Euclidean 4-space

21 : Category two of(non-Abelian) quantum field theory underlying the Standard Model of particle


22 : Category three of(non-Abelian) quantum field theory underlying the Standard Model of particle


20 : Category one ofYang Mills theory. Systemic differentiation is undertaken for execution. There are

various systems in the world that satisfy the axiomatic predications, postulation alcovishness, and

phenomenological correlates of the Yang mills Theory. Some of them are under experimental observation.

Characterstics of these systems so mentioned in the foregoing and which are under the investigation form the

bastion for the classification scheme.

21 : Category two ofYang Mills theory

22 : Category three ofYang Mills theory

Module four

The mass gap

24 : Category one of mass of the least massive particle predicted by the theory

25 : Category two of mass of the least massive particle predicted by the theory

26 : Category three of mass of the least massive particle predicted by the theory

24 : Category one ofmass gap

under consideration and has mass gap syndrome form the stylobate and sentinel , the fulcrum of the

classification scheme.

25 : Category two ofmass gap

26 : Category three ofmass gap

Module five


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Therefore, the winner must first prove that Yang Mills theory exists and that it (eb) satisfies the standard of

rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory,

which is referenced in the papers 45 and 35 cited in the official problem description by Jaffe and Witten. We

assume the proposition and give the model. Model gives prediction, projection and prognostication of the

variables involved, and in the eventuality of the correctness of the statement it shall remain with the

initial conditions stated in unmistakable terms in the final results in the dovetailed mathematical

exposition.

28 : Category one ofYang Mills theory exists and that it

29 : Category two ofYang Mills theory exists and that it

30 : Category three ofYang Mills theory exists and that it

28 : Category one ofstandard of rigor that characterizes contemporary mathematical physics, in

particular constructive quantum field theory, which is referenced in the papers 45 and 35 cited in the official

problem description by Jaffe and Witten

29 : Category two ofstandard of rigor that characterizes contemporary mathematical physics, in



T30 : Category three of standard of rigor that characterizes contemporary mathematical physics, in



Module six

The winner must then prove that the mass of the least massive particle of the force field predicted by the

theory is (=) strictly positive. We assume the proposition and delineate and disseminate the model.

Should the correctness exist then the prognostication and prediction formulas given at the end of the

paper should be correct in consistent with the observation of any data or experimental observation.

Lest the converse is true namely, that the force field predicted by the theory is (=) not strictly positive.

32 : Category one of strictly positive

33 : Category two of strictly positive

34 : Category three of strictly positive

T32 : Category one ofmass of the least massive particle of the force field predicted by the theory. Systemic

differentiation. Kindly note that whatever explanation is given of the predicational anteriorities, character

constitution and phenomenological correlates must hold good for all the systems which satisfy the essence of

the statement under question.

33 : Category two ofmass of the least massive particle of the force field predicted by the theory

34 : Category three ofmass of the least massive particle of the force field predicted by the theory

Module seven

For example, in the case of G=SU (3) - the strong nuclear interaction - the winner must prove

that glueballs have (e) a lower mass bound. We assume the proposition and give the model. In the next


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part we assume the inverse and give the results. One of them must hold good.

36 : Category one oflower mass bound

37 : Category two of lower mass bound

38: Category three oflower mass bound

T36 : Category one ofG=SU (3) - the strong nuclear interaction glueballs

37 : Category two ofG=SU (3) - the strong nuclear interaction glueballs

38 : Category three ofG=SU (3) - the strong nuclear interaction glueballs

Module eight

Thus glueballs cannot (e) be arbitrarily light

40 : Category one of arbitrarily light

41 : Category two of arbitrarily light

42 : Category three of arbitrarily light

T40 : Category one ofglueballs

41 : Category two ofglueballs

42 : Category three ofglueballs

Module Nine

Yang Mills theories are a special example of gauge theory with a non-abelian symmetry group given by

the Lagrangian

with the generators of the Lie algebra corresponding to the F-quantities (the curvature or field-strength form)

satisfying

and the covariant derivative defined as

where I is the identity for the group generators, is the vector potential, and g is the coupling constant.

In four dimensions, the coupling constant g is a pure number and for a SU(N) group one

has


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The relation

can be derived by the commutator

The field has the property of being self-interacting and equations of motion that one obtains are said to be

semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this

theory only by perturbation theory, with small nonlinearities.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor

components is trivial for a indices (e.g. rivial, corresponding

e.g. to the usual Lorentz signature, .

From the given Lagrangian one can derive the equations of motion given by

Putting , these can be rewritten as

A Bianchi identity holds

which is equivalent to the Jacobi identity

since . Define the dual strength tensor , then

the Bianchi identity can be rewritten as

A source enters into the equations of motion as

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. We note that, in D dimensions,

the field scales as and so the coupling must scale as . This implies


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that YangMills theory is not renormalizable for dimensions greater than four. Further, we note that, for D =

4, the coupling is dimensionless and both the field and the square of the coupling have the same dimensions

of the field and the coupling of a massless quartic scalar field theory. So, these theories share the scale

invariance at the classical level.

Note: When we write A+B, it means that we are adding B to A until B is exhausted. There may be time

lag or may not be time lag. It is almost like adding water to milk. When we write B+A it means adding

water to milk until water is fully exhausted, which we are familiar. A-B implies removing B from A,

with or without time lag. All these commentaries are true for all additions, subtractions, mappings

and transformations. In the eventuality of multiplication, logarithms can be taken to separate the

variables and hence the terms becomes separate and give results of the prediction for a time t in the

model. As said, there are many systems with phenomenological correlates, differential contiguities,

presuppositional resemblances and ontological consonance and primordial exactitude. Those systems

which are under the scanner can be classified in to three compartments as we have done based on

their characterstics. These statements hold good for the entire monograph. We shall not repeat this

again. We have done this exercise term by term in earlier papers and shall not repeat the same. Kindly

bear with me.

44 : Category one of LHS of all the equations stated in the foregoing (Yang Mills Theory including the

Lagrangian and the Hamiltonian)

45 : Category two of LHS of all the equations stated in the foregoing (Yang Mills Theory including the


46 : Category three of LHS of all the equations stated in the foregoing (Yang Mills Theory including the


T44 : Category one of RHS of all the equations stated in the foregoing (Yang Mills Theory including the


45 : Category two of RHS of all the equations stated in the foregoing (Yang Mills Theory including the


46 : Category three of RHS of all the equations stated in the foregoing (Yang Mills Theory including the


The Coefficients:

131 , 14

1 , 151 , 13

1 , 141 , 15

116

2 , 172 , 18

216

2 , 172 , 18

2 :

203 , 21

3 , 223 ,

203 , 21

3 , 223

244 , 25

4 , 264 , 24

4 , 254 , 26

4 , 285 , 29

5 , 305 ,

285 , 29

5 , 305 , 32

6 , 336 , 34

6 , 326 , 33

6 , 346

367 , 37

7 , 387 , 36

7 , 377 , 38

7

408 , 41

8 , 428 , 40

8 , 418 , 42

8

449 , 45

9 , 469 , 44

9 , 459 , 46

9

are Accentuation coefficients

131 , 14

1 , 151 , 13

1 , 141 , 15

1 , 162 , 17

2 , 182 ,


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162 , 17

2 , 182 , 20

3 , 213 , 22

3 , 203 , 21

3 , 223

244 , 25

4 , 264 , 24

4 , 254 , 26

4 , 285 , 29

5 , 305

285 , 29

5 , 305

, 326 , 33

6 , 346 , 32

6 , 336 , 34

6

367 , 37

7 , 387 , 36

7 , 377 , 38

7 ,

408 , 41

8 , 428 , 40

8 , 418 , 42

8 ,

449 , 45

9 , 469 , 44

9 , 459 , 46

9 ,

are Dissipation coefficients

Module Numbered One

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

13= 13

114 13

1 + 131

14 , 13 1

14= 14

113 14

1 + 141

14 , 14 2

15= 15

114 15

1 + 151

14 , 15 3

13= 13

114 13

113

1 , 13 4

14= 14

113 14

114

1 , 14 5

15= 15

114 15

115

1 , 15 6

+ 131

14 , = First augmentation factor

131 , = First detritions factor

Module Numbered Two

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

16= 16

217 16

2 + 162

17 , 16 7

17= 17

216 17

2 + 172

17 , 17 8

18= 18

217 18

2 + 182

17 , 18 9

16= 16

217 16

216

219 , 16

10

17= 17

216 17

217

219 , 17

11

18= 18

217 18

218

219 , 18

12

+ 162


162


Module Numbered Three

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

20= 20

321 20

3 + 203

21, 20 13

21= 21

320 21

3 + 213

21, 21 14

22= 22

321 22

3 + 223

21, 22 15


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20= 20

321 20

320

323, 20

16

21= 21

320 21

321

323, 21

17

22= 22

321 22

322

323, 22

18

+ 203


203

23, = First detritions factor

Module Numbered Four

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

24= 24

425 24

4 + 244

25, 24 19

25= 25

424 25

4 + 254

25, 25 20

26= 26

425 26

4 + 264

25, 26 21

24= 24

425 24

424

427 , 24

22

25= 25

424 25

425

427 , 25

23

26= 26

425 26

426

427 , 26

24

+ 244


244


Module Numbered Five:

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

28= 28

529 28

5 + 285

29, 28 25

29= 29

528 29

5 + 295

29, 29 26

30= 30

529 30

5 + 305

29, 30 27

28= 28

529 28

528

531 , 28

28

29= 29

528 29

529

531 , 29

29

30= 30

529 30

530

531 , 30

30

+ 285

29, = First augmentation factor

285


Module Numbered Six

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

32= 32

633 32

6 + 326

33, 32 31

33= 33

632 33

6 + 336

33, 33 32


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34= 34

633 34

6 + 346

33, 34 33

32= 32

633 32

632

635 , 32

34

33= 33

632 33

633

635 , 33

35

34= 34

633 34

634

635 , 34

36

+ 326


Module Numbered Seven:

The differential system of this model is now (Seventh Module)

36= 36

737 36

7 + 367

37, 36 37

37= 37

736 37

7 + 377

37, 37 38

38= 38

737 38

7 + 387

37, 38 39

36= 36

737 36

736

739 , 36

40

37= 37

736 37

737

739 , 37

41

38= 38

737 38

738

739 , 38

42

+ 367


Module Numbered Eight

The differential system of this model is now

40= 40

841 40

8 + 408

41, 40 43

41= 41

840 41

8 + 418

41, 41 44

42= 42

841 42

8 + 428

41, 42 45

40= 40

841 40

840

843 , 40

46

41= 41

840 41

841

843 , 41

47

42= 42

841 42

842

843 , 42

48

Module Numbered Nine

The differential system of this model is now

44 = 449

45 449 + 44

945, 44 49

45= 45

944 45

9 + 459

45, 45 50

46= 46

945 46

9 + 469

45, 46 51

44= 44

945 44

944

947 , 44

52


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45= 45

944 45

945

947 , 45

53

46= 46

945 46

946

947 , 46

54

+ 449


449

47 , = First detrition factor

13 = 131

14

131 + 13

114 , + 16

2,2,17 , + 20

3,3,21 ,

+ 244,4,4,4,

25 , + 285,5,5,5,

29, + 326,6,6,6,

33,

+ 367,7

37, + 408,8

41 , + 449,9,9,9,9,9,9,9,9

45 ,

13

55

14= 14

113

141 + 14

114 , + 17

2,2,17 , + 21

3,3,21,

+ 254,4,4,4,

25 , + 295,5,5,5,

29, + 336,6,6,6,

33,

+ 377,7

37 , + 418,8

41, + 459,9,9,9,9,9,9,9,9

45,

14

56

15= 15

114

151 + 15

114 , + 18

2,2,17 , + 22

3,3,21,

+ 264,4,4,4,

25, + 305,5,5,5,

29, + 346,6,6,6,

33 ,

+ 387,7

37 , + 428,8

41, + 469,9,9,9,9,9,9,9,9

45,

15

57

Where 131

14 , , 141

14, , 151

14 , are first augmentation coefficients for

category 1, 2 and 3

+ 162,2,

17 , , + 172,2,

17 , , + 182,2,

17 , are second augmentation coefficient for

category 1, 2 and 3

+ 203,3,

21 , , + 213,3,

21 , , + 223,3,

21 , are third augmentation coefficient for

category 1, 2 and 3

+ 244,4,4,4,

25, , + 254,4,4,4,

25, , + 264,4,4,4,

25 , are fourth augmentation

coefficient for category 1, 2 and 3

+ 285,5,5,5,

29, , + 295,5,5,5,

29, , + 305,5,5,5,

29, are fifth augmentation coefficient

for category 1, 2 and 3

+ 326,6,6,6,

33, , + 336,6,6,6,

33 , , + 346,6,6,6,

33, are sixth augmentation coefficient


+ 387,7

37 , + 377,7

37 , + 367,7

37, are seventh augmentation coefficient for 1,2,3

+ 408,8

41, + 418,8

41, + 428,8

41, are eight augmentation coefficient for 1,2,3

+ 449,9,9,9,9,9,9,9,9

45, , + 459,9,9,9,9,9,9,9,9

45, , + 469,9,9,9,9,9,9,9,9

45, are ninth

augmentation coefficient for 1,2,3

13= 13

114

131

131 , 16

2,2,19, 20

3,3,23,

244,4,4,4,

27, 285,5,5,5,

31, 326,6,6,6,

35,

367,7,

39, 408,8

43 , 449,9,9,9,9,9,9,9,9

47,

13

58

14= 14

113

141

141 , 17

2,2,19, 21

3,3,23,

254,4,4,4,

27, 295,5,5,5,

31, 336,6,6,6,

35,

377,7,

39, 418,8

43 , 459,9,9,9,9,9,9,9,9

47,

14

59


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15= 15

114

151

151 , 18

2,2,19, 22

3,3,23,

264,4,4,4,

27, 305,5,5,5,

31, 346,6,6,6,

35,

387,7,

39, 428,8

43 , 469,9,9,9,9,9,9,9,9

47,

15

60

Where 131 , , 14

1 , , 151 , are first detrition coefficients for category 1,

2 and 3

162,2,

19, , 172,2,

19, , 182,2,

19, are second detrition coefficients for

category 1, 2 and 3

203,3,

23, , 213,3,

23, , 223,3,

23, are third detrition coefficients for

category 1, 2 and 3

244,4,4,4,

27, , 254,4,4,4,

27, , 264,4,4,4,

27, are fourth detrition coefficients


285,5,5,5,

31, , 295,5,5,5,

31, , 305,5,5,5,

31, are fifth detrition coefficients for

category 1, 2 and 3

326,6,6,6,

35, , 336,6,6,6,

35, , 346,6,6,6,

35, are sixth detrition coefficients for

category 1, 2 and 3

377,7,

39, , 367,7,

39, , 387,7,

39, are seventh detrition coefficients for

category 1, 2 and 3

408,8

43 , 418,8

43 , 428,8

43 , are eight detrition coefficients for category 1,

2 and 3

449,9,9,9,9,9,9,9,9

47 , , 459,9,9,9,9,9,9,9,9

47 , , 469,9,9,9,9,9,9,9,9

47, are ninth detrition

coefficients for category 1, 2 and 3

16 = 162

17

162 + 16

217 , + 13

1,1,14 , + 20

3,3,321,

+ 244,4,4,4,4

25 , + 285,5,5,5,5

29, + 326,6,6,6,6

33 ,

+ 367,7,7

37, + 408,8,8

41, + 449,9

45,

16

61

17= 17

216

172 + 17

217 , + 14

1,1,14 , + 21

3,3,321 ,

+ 254,4,4,4,4

25, + 295,5,5,5,5

29, + 336,6,6,6,6

33 ,

+ 377,7,7

37, + 418,8,8

41, + 459,9

45,

17

62

18= 18

217

182 + 18

217 , + 15

1,1,14 , + 22

3,3,321 ,

+ 264,4,4,4,4

25, + 305,5,5,5,5

29, + 346,6,6,6,6

33 ,

+ 387,7,7

37, + 428,8,8

41, + 469,9

45,

18

63

Where + 162

17, , + 172

17 , , + 182


category 1, 2 and 3

+ 131,1,

14 , , + 141,1,

14 , , + 151,1,

14 , are second augmentation coefficient for

category 1, 2 and 3

+ 203,3,3

21 , , + 213,3,3

21, , + 223,3,3

21, are third augmentation coefficient for

category 1, 2 and 3


ISSN 2250-3153

www.ijsrp.org

+ 244,4,4,4,4

25 , , + 254,4,4,4,4

25 , , + 264,4,4,4,4



+ 285,5,5,5,5

29, , + 295,5,5,5,5

29, , + 305,5,5,5,5

29, are fifth augmentation


+ 326,6,6,6,6

33 , , + 336,6,6,6,6

33, , + 346,6,6,6,6

33 , are sixth augmentation


+ 367,7,7

37 , , + 377,7,7

37, , + 387,7,7

37, are seventh augmentation coefficient


+ 408,8,8

41, , + 418,8,8

41, , + 428,8,8

41, are eight augmentation coefficient for

category 1, 2 and 3

+ 449,9

45, , + 459,9

45, , + 469,9

45, are ninth augmentation coefficient for

category 1, 2 and 3

16= 16

217

162

162

19, 131,1, , 20

3,3,3,23,

244,4,4,4,4

27, 285,5,5,5,5

31, 326,6,6,6,6

35,

367,7,7

39, 408,8,8

43 , 449,9

47 ,

16

64

17= 17

216

172

172

19, 141,1, , 21

3,3,3,23,

254,4,4,4,4

27, 295,5,5,5,5

31, 336,6,6,6,6

35,

377,7,7

39, 418,8,8

43 , 459,9

47 ,

17

65

18= 18

217

182

182

19, 151,1, , 22

3,3,3,23,

264,4,4,4,4

27, 305,5,5,5,5

31, 346,6,6,6,6

35,

387,7,7

39, 428,8,8

43 , 469,9

47 ,

18

66

where b162 G19, t , b17

2 G19, t , b182 G19, t are first detrition coefficients for

category 1, 2 and 3

131,1, , , 14

1,1, , , 151,1, , are second detrition coefficients for category 1,2

and 3

203,3,3,

23, , 213,3,3,

23, , 223,3,3,

23, are third detrition coefficients for

category 1,2 and 3

244,4,4,4,4

27, , 254,4,4,4,4

27, , 264,4,4,4,4

27, are fourth detrition

coefficients for category 1,2 and 3

285,5,5,5,5

31, , 295,5,5,5,5

31, , 305,5,5,5,5

31, are fifth detrition coefficients

for category 1,2 and 3

326,6,6,6,6

35, , 336,6,6,6,6

35, , 346,6,6,6,6

35, are sixth detrition coefficients

for category 1,2 and 3

367,7,7

39, , 377,7,7

39, , 387,7,7


category 1,2 and 3

408,8,8

43 , , 418,8,8

43 , , 428,8,8

43 , are eight detrition coefficients for

category 1,2 and 3

449,9

47 , , 469,9

47 , , 459,9

47 , are ninth detrition coefficients for category


ISSN 2250-3153

www.ijsrp.org

1,2 and 3

20= 20

321

203 + 20

321, + 16

2,2,217 , + 13

1,1,1,14 ,

+ 244,4,4,4,4,4

25, + 285,5,5,5,5,5

29, + 326,6,6,6,6,6

33 ,

+ 367,7,7,7

37 , + 408,8,8,8

41, + 449,9,9

45,

20

67

21= 21

320

213 + 21

321, + 17

2,2,217 , + 14

1,1,1,14 ,

+ 254,4,4,4,4,4

25, + 295,5,5,5,5,5

29, + 336,6,6,6,6,6

33,

+ 377,7,7,7

37 , + 418,8,8,8

41, + 459,9,9

45,

21

68

22= 22

321

223 + 22

321, + 18

2,2,217 , + 15

1,1,1,14 ,

+ 264,4,4,4,4,4

25, + 305,5,5,5,5,5

29, + 346,6,6,6,6,6

33 ,

+ 387,7,7,7

37 , + 428,8,8,8

41, + 469,9,9

45,

22

69

+ 203

21 , , + 213

21, , + 223

21, are first augmentation coefficients for category

1, 2 and 3

+ 162,2,2

17 , , + 172,2,2

17 , , + 182,2,2

17, are second augmentation coefficients


+ 131,1,1,

14 , , + 141,1,1,

14 , , + 151,1,1,

14 , are third augmentation coefficients


+ 244,4,4,4,4,4

25 , , + 254,4,4,4,4,4

25 , , + 264,4,4,4,4,4



+ 285,5,5,5,5,5

29, , + 295,5,5,5,5,5

29, , + 305,5,5,5,5,5



+ 326,6,6,6,6,6

33 , , + 336,6,6,6,6,6

33 , , + 346,6,6,6,6,6

33, are sixth augmentation


+ 367,7,7,7

37, , + 377,7,7,7

37 , , + 387,7,7,7

37, are seventh augmentation


+ 408,8,8,8

41, , + 418,8,8,8

41, , + 428,8,8,8

41, are eight augmentation coefficients


+ 449,9,9

45, , + 459,9,9

45, , + 469,9,9

45 , are ninth augmentation coefficients for

category 1, 2 and 3

20= 20

321

203

203

23, 162,2,2

19, 131,1,1, ,

244,4,4,4,4,4

27, 285,5,5,5,5,5

31, 326,6,6,6,6,6

35,

367,7,7,7

39, 408,8,8,8

43, 449,9,9

47,

20

70

21= 21

320

213

213

23, 172,2,2

19, 141,1,1, ,

254,4,4,4,4,4

27, 295,5,5,5,5,5

31, 336,6,6,6,6,6

35,

377,7,7,7

39, 418,8,8,8

43, 459,9,9

47,

21

71


ISSN 2250-3153

www.ijsrp.org

22= 22

321

223

223

23, 182,2,2

19, 151,1,1, ,

264,4,4,4,4,4

27, 305,5,5,5,5,5

31, 346,6,6,6,6,6

35,

387,7,7,7

39, 428,8,8,8

43, 469,9,9

47,

22

72

203

23, , 213

23, , 223

23, are first detrition coefficients for category 1,

2 and 3

162,2,2

19, , 172,2,2

19, , 182,2,2

19, are second detrition coefficients for

category 1, 2 and 3

131,1,1, , , 14

1,1,1, , , 151,1,1, , are third detrition coefficients for category

1,2 and 3

244,4,4,4,4,4

27, , 254,4,4,4,4,4

27, , 264,4,4,4,4,4



285,5,5,5,5,5

31, , 295,5,5,5,5,5

31, , 305,5,5,5,5,5

31, are fifth detrition


326,6,6,6,6,6

35, , 336,6,6,6,6,6

35, , 346,6,6,6,6,6

35, are sixth detrition


367,7,7,7

39, , 377,7,7,7

39, 387,7,7,7


category 1, 2 and 3

408,8,8,8

43, , 418,8,8,8

43, , 428,8,8,8

43 , are eight detrition coefficients for

category 1, 2 and 3

469,9,9

47 , , 459,9,9

47, , 449,9,9

47, are ninth detrition coefficients for

category 1, 2 and 3

24= 24

425

244 + 24

425, + 28

5,5,29, + 32

6,6,33,

+ 131,1,1,1

14 , + 162,2,2,2

17 , + 203,3,3,3

21,

+ 367,7,7,7,7

37, + 408,8,8,8,8

41 , + 449,9,9,9

45,

24

73

25= 25

424

254 + 25

425 , + 29

5,5,29, + 33

6,633,

+ 141,1,1,1

14 , + 172,2,2,2

17 , + 213,3,3,3

21,

+ 377,7,7,7,7

37, + 418,8,8,8,8

41 , + 459,9,9,9

45,

25

74

26= 26

425

264 + 26

425, + 30

5,5,29, + 34

6,6,33,

+ 151,1,1,1

14 , + 182,2,2,2

17 , + 223,3,3,3

21,

+ 387,7,7,7,7

37, + 428,8,8,8,8

41 , + 469,9,9,9

45,

26

75

244

25 , , 254

25 , , 264

25 ,

1,2 3

+ 285,5,

29, , + 295,5,

29, , + 305,5,

29,

1,2 3

+ 326,6,

33 , , + 336,6,

33 , , + 346,6,

33 ,

1,2 3

+ 131,1,1,1

14 , , + 141,1,1,1

14 , , + 151,1,1,1

14, 1,2 3


ISSN 2250-3153

www.ijsrp.org

+ 162,2,2,2

17 , ,

+ 172,2,2,2

17 , , + 182,2,2,2

17 , 1,2 3

+ 203,3,3,3

21, , + 213,3,3,3

21 , ,

+ 223,3,3,3

21, 1,2 3

+ 367,7,7,7,7

37 , , + 377,7,7,7,7

37, ,

+ 387,7,7,7,7

37 , 1,2 3

+ 408,8,8,8,8

41 , , + 418,8,8,8,8

41, , + 428,8,8,8,8

41,

1,2 3

+ 469,9,9,9

45, , + 459,9,9,9

45, , + 449,9,9,9

45, are ninth detrition coefficients for

category 1 2 3

24= 24

425

244

244

27, 285,5,

31, 326,6,

35,

131,1,1,1 , 16

2,2,2,219, 20

3,3,3,323,

367,7,7,7,7

39, 408,8,8,8,8

43, 449,9,9,9

47 ,

24

76

25= 25

424

254

254

27, 295,5,

31, 336,6,

35,

141,1,1,1 , 17

2,2,2,219, 21

3,3,3,323,

377,7,7,7,7

39, 418,8,8,8,8

43, 459,9,9,9

47 ,

25

77

26= 26

425

264

264

27, 305,5,

31, 346,6,

35,

151,1,1,1 , 18

2,2,2,219, 22

3,3,3,323,

387,7,7,7,7

39, 428,8,8,8,8

43, 469,9,9,9

47 ,

26

78

244

27, , 254

27, , 264

27,

1,2 3

285,5,

31, , 295,5,

31, , 305,5,

31,

1,2 3

326,6,

35, , 336,6,

35, , 346,6,

35,

1,2 3

131,1,1,1 , , 14

1,1,1,1 ,

, 151,1,1,1 , 1,2 3

162,2,2,2

19, , 172,2,2,2

19, ,

182,2,2,2

19, 1,2 3

203,3,3,3

23, , 213,3,3,3

23 , , 223,3,3,3

23, 1,2 3

367,7,7,7,7

39, , 377,7,7,7,7

39,

, 387,7,7,7,7

39, 1,2 3

408,8,8,8,8

43, , 418,8,8,8,8

43 , , 428,8,8,8,8

43 ,

1,2 3

469,9,9,9

47 , , 459,9,9,9

47, , 449,9,9,9

47 , are ninth detrition coefficients for


ISSN 2250-3153

www.ijsrp.org

category 1 2 3

28 = 285

29

285 + 28

529, + 24

4,4,25 , + 32

6,6,633,

+ 131,1,1,1,1

14 , + 162,2,2,2,2

17 , + 203,3,3,3,3

21 ,

+ 367,7,7,7,7,7

37, + 408,8,8,8,8,8

41 , + 449,9,9,9,9

45,

28

79

29= 29

528

295 + 29

529, + 25

4,4,25 , + 33

6,6,633 ,

+ 141,1,1,1,1

14 , + 172,2,2,2,2

17, + 213,3,3,3,3

21,

+ 377,7,7,7,7,7

37, + 418,8,8,8,8,8

41, + 459,9,9,9,9

45,

29

80

30= 30

529

305 + 30

529, + 26

4,4,25 , + 34

6,6,633,

+ 151,1,1,1,1

14 , + 182,2,2,2,2

17 , + 223,3,3,3,3

21 ,

+ 387,7,7,7,7,7

37, + 428,8,8,8,8,8

41, + 469,9,9,9,9

45,

30

81

+ 285

29, , + 295

29, , + 305

29,

1,2 3

+ 244,4,

25 , , + 254,4,

25 , , + 264,4,

25 ,

1,2 3

+ 326,6,6

33 , , + 336,6,6

33, , + 346,6,6

33,

1,2 3

+ 131,1,1,1,1

14 , , + 141,1,1,1,1

14, , + 151,1,1,1,1


coefficients for category 1,2, and 3

+ 162,2,2,2,2

17 , , + 172,2,2,2,2

17, , + 182,2,2,2,2

17 , are fifth augmentation

coefficients for category 1,2,and 3

+ 203,3,3,3,3

21 , , + 213,3,3,3,3

21 , , + 223,3,3,3,3


coefficients for category 1,2, 3

+ 367,7,7,7,7,7

37 , , + 377,7,7,7,7,7

37, , + 387,7,7,7,7,7

37, are seventh augmentation


+ 408,8 ,8,8,8,8

41, , + 418,8,8,8,8,8

41, , + 428,8,8,8,8,8

41, are eighth augmentation


+ 469,9,9,9,9

45 , , + 459,9,9,9,9

45, , + 449,9,9,9,9

45, are ninth augmentation


28= 28

529

285

285

31, 244,4,

27, 326,6,6

35 ,

131,1,1,1,1 , 16

2,2,2,2,219, 20

3,3,3,3,323,

367,7,7,7,7,7

39, 408,8,8,8,8,8

43 , 449,9,9,9,9

47 ,

28

82

29= 29

528

295

295

31, 254,4,

27, 336,6,6

35,

141,1,1,1,1 , 17

2,2,2,2,219, 21

3,3,3,3,323,

377,7,7,7,7,7

39, 418,8,8,8,8,8

43, 459,9,9,9,9

47,

29

83


ISSN 2250-3153

www.ijsrp.org

30= 30

529

305

305

31, 264,4,

27, 346,6,6

35 ,

151,1,1,1,1, , 18

2,2,2,2,219, 22

3,3,3,3,323,

387,7,7,7,7,7

39, 428,8,8,8,8,8

43 , 469,9,9,9,9

47 ,

30

84

285

31, , 295

31, , 305

31, 1,2 3

244,4,

27, , 254,4,

27, , 264,4,

27,

1,2 3

326,6,6

35, , 336,6,6

35, , 346,6,6

35,

1,2 3

131,1,1,1,1 , , 14

1,1,1,1,1 , , 151,1,1,1,1, , are fourth detrition coefficients for

category 1,2, and 3

162,2,2,2,2

19, , 172,2,2,2,2

19, , 182,2,2,2,2

19, are fifth detrition coefficients

for category 1,2, and 3

203,3,3,3,3

23, , 213,3,3,3,3

23, , 223,3,3,3,3

23, are sixth detrition coefficients


367,7,7,7,7,7

39, , 377,7,7,7,7,7

39, , 387,7,7,7,7,7

39, are seventh detrition


428,8,8,8,8,8

43 , , 418,8,8,8,8,8

43 , , 408,8,8,8,8,8

43, are eighth detrition


469,9,9,9,9

47, , 459,9,9,9,9

47 , , 449,9,9,9,9

47 , are ninth detrition coefficients


32= 32

633

326 + 32

633, + 28

5,5,529, + 24

4,4,4,25,

+ 131,1,1,1,1,1

14, + 162,2,2,2,2,2

17 , + 203,3,3,3,3,3

21 ,

+ 367,7,7,7,7,7,7

37 , + 408,8,8,8,8,8,8

41, + 449,9,9,9,9,9

45,

32

85

33= 33

632

336 + 33

633 , + 29

5,5,529, + 25

4,4,4,25 ,

+ 141,1,1,1,1,1

14, + 172,2,2,2,2,2

17 , + 213,3,3,3,3,3

21 ,

+ 377,7,7,7,7,7,7

37 , + 418,8,8,8,8,8,8

41, + 459,9,9,9,9,9

45,

33

86

34= 34

633

346 + 34

633, + 30

5,5,529, + 26

4,4,4,25,

+ 151,1,1,1,1,1

14, + 182,2,2,2,2,2

17 , + 223,3,3,3,3,3

21 ,

+ 387,7,7,7,7,7,7

37 , + 428,8,8,8,8,8,8

41, + 469,9,9,9,9,9

45,

34

87

+ 326

33 , , + 336

33 , , + 346

33,

1,2 3

+ 285,5,5

29, , + 295,5,5

29, , + 305,5,5

29,

1,2 3

+ 244,4,4,

25, , + 254,4,4,

25, , + 264,4,4,

25 ,

1,2 3

+ 131,1,1,1,1,1

14 , , + 141,1,1,1,1,1

14 , , + 151,1,1,1,1,1

14 , - are fourth augmentation

coefficients


ISSN 2250-3153

www.ijsrp.org

+ 162,2,2,2,2,2

17 , , + 172,2,2,2,2,2

17 , , + 182,2,2,2,2,2

17 , - fifth augmentation

coefficients

+ 203,3,3,3,3,3

21 , , + 213,3,3,3,3,3

21 , , + 223,3,3,3,3,3

21, sixth augmentation

coefficients

+ 367,7,7,7,7,7,7

37, , + 377,7,7,7,7,7,7

37, ,

+ 387,7,7,7,7,7,7

37, seventh augmentation coefficients

+ 408,8,8,8,8,8,8

41, , + 418,8,8,8,8,8,8

41, , + 428,8,8,8,8,8,8

41 ,

Eighth augmentation coefficients

+ 449,9,9,9,9,9

45 , , + 459,9,9,9,9,9

45, , + 469,9,9,9,9,9

45, ninth augmentation

coefficients

32= 32

633

326

326

35, 285,5,5

31, 244,4,4,

27,

131,1,1,1,1,1 , 16

2,2,2,2,2,219, 20

3,3,3,3,3,323,

367,7,7,7,7,7,7

39, 408,8,8,8,8,8,8

43 , 449,9,9,9,9,9

47,

32

88

33= 33

632

336

336

35, 295,5,5

31, 254,4,4,

27,

141,1,1,1,1,1 , 17

2,2,2,2,2,219, 21

3,3,3,3,3,323,

377,7,7,7,7,7,7

39, 418,8,8,8,8,8,8

43 , 459,9,9,9,9,9

47,

33

89

34= 34

633

346

346

35, 305,5,5

31, 264,4,4,

27,

151,1,1,1,1,1 , 18

2,2,2,2,2,219, 22

3,3,3,3,3,323,

387,7,7,7,7,7,7

39, 428,8,8,8,8,8,8

43 , 469,9,9,9,9,9

47,

34

90

326

35, , 336

35, , 346

35,

1,2 3

285,5,5

31, , 295,5,5

31, , 305,5,5

31,

1,2 3

244,4,4,

27, , 254,4,4,

27, , 264,4,4,

27,

1,2 3

131,1,1,1,1,1 , , 14

1,1,1,1,1,1 , , 151,1,1,1,1,1 , are fourth detrition coefficients

for category 1, 2, and 3

162,2,2,2,2,2

19, , 172,2,2,2,2,2

19, , 182,2,2,2,2,2


coefficients for category 1, 2, and 3

203,3,3,3,3,3

23, , 213,3,3,3,3,3

23, , 223,3,3,3,3,3



367,7,7,7,7,7,7

39, , 377,7,7,7,7,7,7

39, , 387,7,7,7,7,7,7



408,8,8,8,8,8,8

43 , , 418,8,8,8,8,8,8

43, , 428,8,8,8,8,8,8

43,

are eighth detrition coefficients for category 1, 2, and 3

469,9,9,9,9,9

47, , 459,9,9,9,9,9

47 , , 449,9,9,9,9,9



ISSN 2250-3153

www.ijsrp.org


36= 36

737

367 + 36

737 , + 16

2,2,2,2,2,2,217 , + 20

3,3,3,3,3,3,321 ,

+ 244,4,4,4,4,4,4

25 , + 285,5,5,5,5,5,5

29, + 326,6,6,6,6,6,6

33,

+ 131,1,1,1,1,1,1

14 , + 408,8,8,8,8,8,8,8,

41, + 449,9,9,9,9,9,9

45,

13

91

37= 37

736

377 + 37

737, + 17

2,2,2,2,2,2,217 , + 21

3,3,3,3,3,3,321 ,

+ 254,4,4,4,4,4,4

25 , + 295,5,5,5,5,5,5

29, + 336,6,6,6,6,6,6

33,

+ 131,1,1,1,1,1,1

14 , + 418,8,8,8,8,8,8,8

41, + 459,9,9,9,9,9,9

45,

14

92

38= 38

737

387 + 38

737, + 18

2,2,2,2,2,2,217 , + 22

3,3,3,3,3,3,321 ,

+ 264,4,4,4,4,4,4

25, + 305,5,5,5,5,5,5

29, + 346,6,6,6,6,6,6

33,

+ 151,1,1,1,1,1,1

14 , + 428,8,8,8,8,8,8,8

41, + 469,9,9,9,9,9,9

45,

15

93

Where 367

37 , , 377

37 , , 387


category 1, 2 and 3

+ 162,2,2,2,2,2,2

17 , , + 172,2,2,2,2,2,2

17 , , + 182,2,2,2,2,2,2

17 , are second

augmentation coefficient for category 1, 2 and 3

+ 203,3,3,3,3,3,3

21, , + 213,3,3,3,3,3,3

21 , , + 223,3,3,3,3,3,3

21 , are third augmentation


+ 244,4,4,4,4,4,4

25, , + 254,4,4,4,4,4,4

25 , , + 264,4,4,4,4,4,4

25, are fourth


+ 285,5,5,5,5,5,5

29, , + 295,5,5,5,5,5,5

29, , + 305,5,5,5,5,5,5



+ 326,6,6,6,6,6,6

33, , + 336,6,6,6,6,6,6

33, , + 346,6,6,6,6,6,6



+ 131,1,1,1,1,1,1

14 , , + 131,1,1,1,1,1,1

14 , , + 151,1,1,1,1,1,1

14 , are seventh


+ 428,8,8,8,8,8,8,8

41 , , + 418,8,8,8,8,8,8,8

41, , + 408,8,8,8,8,8,8,8,

41,

are eighth augmentation coefficient for 1,2,3

+ 469,9,9,9,9,9,9

45, , + 459,9,9,9,9,9,9

45, , + 449,9,9,9,9,9,9

45, are ninth augmentation

coefficient for 1,2,3

36= 36

737

367

367

39, 162,2,2,2,2,2,2

19, 203,3,3,3,3,3,3

23,

244,4,4,4,4,4,4

27, 285,5,5,5,5,5,5

31, 326,6,6,6,6,6,6

35,

131,1,1,1,1,1,1 , 40

8,8,8,8,8,8,8,843 , 44

9,9,9,9,9,9,947 ,

13

94


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37= 37

736

377

377

39, 172,2,2,2,2,2,2

19, 213,3,3,3,3,3,3

23,

254,4,4,4,4,4,4

27, 295,5,5,5,5,5,5

31, 336,6,6,6,6,6,6

35,

141,1,1,1,1,1,1 , 41

8,8,8,8,8,8,8,843 , 45

9,9,9,9,9,9,947 ,

14

38= 38

737

387

387

39, 182,2,2,2,2,2,2

19, 223,3,3,3,3,3,3

23,

264,4,4,4,4,4,4

27, 305,5,5,5,5,5,5

31, 346,6,6,6,6,6,6

35,

151,1,1,1,1,1,1 , 42

8,8,8,8,8,8,8,843 , 46

9,9,9,9,9,9,947 ,

15

Where 367

39, , 377

39, , 387

39, are first detrition coefficients for

category 1, 2 and 3

162,2,2,2,2,2,2

19, , 172,2,2,2,2,2,2

19, , 182,2,2,2,2,2,2

19, are second detrition


203,3,3,3,3,3,3

23, , 213,3,3,3,3,3,3

23, , 223,3,3,3,3,3,3

23, are third detrition


244,4,4,4,4,4,4

27, , 254,4,4,4,4,4,4

27, , 264,4,4,4,4,4,4



285,5,5,5,5,5,5

31, , 295,5,5,5,5,5,5

31, , 305,5,5,5,5,5,5



326,6,6,6,6,6,6

35, , 336,6,6,6,6,6,6

35, , 346,6,6,6,6,6,6



151,1,1,1,1,1,1 , , 14

1,1,1,1,1,1,1 , , 131,1,1,1,1,1,1 ,

are seventh detrition coefficients for category 1, 2 and 3

408,8,8,8,8,8,8,8

43, , 418,8,8,8,8,8,8,8

43 , , 428,8,8,8,8,8,8,8



469,9,9,9,9,9,9

47, , 459,9,9,9,9,9,9

47 , , 449,9,9,9,9,9,9

47 , are ninth detrition


40

= 408

41

408 + 40

841, + 16

2,2,2,2,2,2,2,217 , + 20

3,3,3,3,3,3,3,321 ,

+ 244,4,4,4,4,4,4,4

25 , + 285,5,5,5,5,5,5,5

29, + 326,6,6,6,6,6,6,6

33,

+ 131,1,1,1,1,1,1,1

14 , + 367,7,7,7,7,7,7,7

37 , + 449,9,9,9,9,9,9,9

45 ,

13

95

41

= 418

40

418 + 41

841, + 17

2,2,2,2,2,2,2,217 , + 21

3,3,3,3,3,3,3,321,

+ 254,4,4,4,4,4,4,4

25 , + 295,5,5,5,5,5,5,5

29, + 336,6,6,6,6,6,6,6

33 ,

+ 131,1,1,1,1,1,1,1

14 , + 377,7,7,7,7,7,7,7

37 , + 459,9,9,9,9,9,9,9

45,

14


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42

= 428

41

428 + 42

841, + 18

2,2,2,2,2,2,2,217 , + 22

3,3,3,3,3,3,3,321 ,

+ 264,4,4,4,4,4,4,4

25 , + 305,5,5,5,5,5,5,5

29, + 346,6,6,6,6,6,6,6

33,

+ 151,1,1,1,1,1,1,1

14 , + 387,7,7,7,7,7,7,7

37 , + 469,9,9,9,9,9,9,9

45 ,

15

Where + 408

41 , , + 418

41, , + 428

41, are first augmentation coefficients for

category 1, 2 and 3

+ 162,2,2,2,2,2,2,2

17, , + 172,2,2,2,2,2,2,2

17 , , + 182,2,2,2,2,2,2,2

17 , are second


+ 203,3,3,3,3,3,3,3

21 , , + 213,3,3,3,3,3,3,3

21, , + 223,3,3,3,3,3,3,3

21 , are third


+ 244,4,4,4,4,4,4,4

25 , , + 254,4,4,4,4,4,4,4

25 , , + 264,4,4,4,4,4,4,4

25, are fourth


+ 285,5,5,5,5,5,5,5

29, , + 295,5,5,5,5,5,5,5

29, , + 305,5,5,5,5,5,5,5

29, are fifth


+ 326,6,6,6,6,6,6,6

33 , , + 336,6,6,6,6,6,6,6

33 , , + 346,6,6,6,6,6,6,6

33 , are sixth


+ 131,1,1,1,1,1,1,1

14, + 141,1,1,1,1,1,1,1

14 , + 151,1,1,1,1,1,1,1

14 , are seventh


+ 367,7,7,7,7,7,7,7

37 , , + 377,7,7,7,7,7,7,7

37 , , + 387,7,7,7,7,7,7,7

37 , are eighth


+ 469,9,9,9,9,9,9,9

45, , + 459,9,9,9,9,9,9,9

45, , + 449,9,9,9,9,9,9,9

45, are ninth


40

= 408

41

408

408

43, 162,2,2,2,2,2,2,2

19, 203,3,3,3,3,3,3,3

23,

244,4,4,4,4,4,4,4

27, 285,5,5,5,5,5,5,5

31, 326,6,6,6,6,6,6,6

35,

131,1,1,1,1,1,1,1 , 36

7,7,7,7,7,7,7,739, 44

9,9,9,9,9,9,9,947,

13

41

= 418

40

418

418

43, 172,2,2,2,2,2,2,2

19, 213,3,3,3,3,3,3,3

23,

254,4,4,4,4,4,4,4

27, 295,5,5,5,5,5,5,5

31, 336,6,6,6,6,6,6,6

35,

141,1,1,1,1,1,1,1 , 37

7,7,7,7,7,7,7,739, 45

9,9,9,9,9,9,9,947,

14

42

= 428

41

428

428

43, 182,2,2,2,2,2,2,2

19, 223,3,3,3,3,3,3,3

23,

264,4,4,4,4,4,4,4

27, 305,5,5,5,5,5,5,5

31, 346,6,6,6,6,6,6,6

35,

151,1,1,1,1,1,1,1 , 38

7,7,7,7,7,7,7,739, 46

9,9,9,9,9,9,9,947,

15


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Where 367

39, , 377

39, , 387


category 1, 2 and 3

162,2,2,2,2,2,2,2

19, , 172,2,2,2,2,2,2,2

19, , 182,2,2,2,2,2,2,2

19, are second

detrition coefficients for category 1, 2 and 3

203,3,3,3,3,3,3,3

23, , 213,3,3,3,3,3,3,3

23, , 223,3,3,3,3,3,3,3



244,4,4,4,4,4,4,4

27, , 254,4,4,4,4,4,4,4

27, , 264,4,4,4,4,4,4,4



285,5,5,5,5,5,5,5

31, , 295,5,5,5,5,5,5,5

31, , 305,5,5,5,5,5,5,5



326,6,6,6,

35, , 336,6,6,6,

35, , 151,1,1,1,1,1,1,1 , are sixth detrition coefficients


131,1,1,1,1,1,1,1 , , 14

1,1,1,1,1,1,1,1 , , 387,7,



367,7,7,7,7,7,7,7

39, , 377,7,7,7,7,7,7,7

39, , 387,7,7,7,7,7,7,7



449,9,9,9,9,9,9,9

47, , 459,9,9,9,9,9,9,9

47, , 469,9,9,9,9,9,9,9



44

= 449

45

449 + 44

945, + 16

2,2,2,2,2,2,2,2,217, + 20

3,3,3,3,3,3,3,3,321 ,

+ 244,4,4,4,4,4,4,4,4

25 , + 285,5,5,5,5,5,5,5,5

29, + 326,6,6,6,6,6,6,6,6

33 ,

+ 131,1,1,1,1,1,1,1,1

14, + 367,7,7,7,7,7,7,7,7

37 , + 408,8,8,8,8,8,8,8,8

41,

13

96

45

= 459

44

459 + 45

945, + 17

2,2,2,2,2,2,2,2,217 , + 21

3,3,3,3,3,3,3,3,321,

+ 254,4,4,4,4,4,4,4,4

25 , + 295,5,5,5,5,5,5,5,5

29, + 336,6,6,6,6,6,6,6,6

33 ,

+ 141,1,1,1,1,1,1,1,1

14 , + 377,7,7,7,7,7,7,7,7

37, + 418,8,8,8,8,8,8,8,8

41 ,

14

46

= 469

45

469 + 46

937, + 18

2,2,2,2,2,2,2,2,217 , + 22

3,3,3,3,3,3,3,3,321,

+ 264,4,4,4,4,4,4,4,4

25 , + 305,5,5,5,5,5,5,5,5

29, + 346,6,6,6,6,6,6,6,6

33 ,

+ 151,1,1,1,1,1,1,1,1

14, + 387,7,7,7,7,7,7,7,7

37 , + 428,8,8,8,8,8,8,8,8

41,

15

Where + 449

45, , + 459

45, , + 469


category 1, 2 and 3

+ 162,2,2,2,2,2,2,2,2

17 , , + 172,2,2,2,2,2,2,2,2

17 , , + 182,2,2,2,2,2,2,2,2

17 , are second


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+ 203,3,3,3,3,3,3,3,3

21 , , + 213,3,3,3,3,3,3,3,3

21 , , + 223,3,3,3,3,3,3,3,3

21 , are third


+ 244,4,4,4,4,4,4,4,4

25 , , + 254,4,4,4,4,4,4,4,4

25, , + 264,4,4,4,4,4,4,4,4

25 , are fourth


+ 285,5,5,5,5,5,5,5,5

29, , + 295,5,5,5,5,5,5,5,5

29, , + 305,5,5,5,5,5,5,5,5

29, are fifth


+ 326,6,6,6,6,6,6,6,6

33 , , + 336,6,6,6,6,6,6,6,6

33 , , + 346,6,6,6,6,6,6,6,6

33, are sixth


+ 131,1,1,1,1,1,1,1,1

14 , , + 141,1,1,1,1,1,1,1,1

14 , , + 151,1,1,1,1,1,1,1,1

14 , are Seventh


+ 387,7,7,7,7,7,7,7,7

37 , + 377,7,7,7,7,7,7,7,7

37 , + 367,7,7,7,7,7,7,7,7

37, are eighth


+ 408,8,8,8,8,8,8,8,8

41, , + 428,8,8,8,8,8,8,8,8

41, , + 418,8,8,8,8,8,8,8,8

41, are ninth


44

= 449

45

449

449

47, 162,2,2,2,2,2,2,2,2

19, 203,3,3,3,3,3,3,3,3

23,

244,4,4,4,4,4,4,4,4

27, 285,5,5,5,5,5,5,5,5

31, 326,6,6,6,6,6,6,6,6

35,

131,1,1,1,1,1,1,1,1 , 36

7,7,7,7,7,7,7,7,739, 40

8,8,8,8,8,8,8,8,843,

13

45

= 459

44

459

459

47 , 172,2,2,2,2,2,2,2

19, 213,3,3,3,3,3,3,3

23,

254,4,4,4,4,4,4,4

27, 295,5,5,5,5,5,5,5

31, 336,6,6,6,6,6,6,6

35,

141,1,1,1,1,1,1,1 , 37

7,7,7,7,7,7,7,739, 41

8,8,8,8,8,8,8,8,843 ,

14

46

= 469

45

469

469

47 , 182,2,2,2,2,2,2,2

19, 223,3,3,3,3,3,3,3

23,

264,4,4,4,4,4,4,4

27, 305,5,5,5,5,5,5,5

31, 346,6,6,6,6,6,6,6

35,

151,1,1,1,1,1,1,1 , 38

7,7,7,7,7,7,7,739, 42

8,8,8,8,8,8,8,8,843 ,

15

Where 449

47, , 459

47 , , 469


category 1, 2 and 3

162,2,2,2,2,2,2,2,2

19, , 172,2,2,2,2,2,2,2,2

19, , 182,2,2,2,2,2,2,2,2

19, are second


203,3,3,3,3,3,3,3

23, , 213,3,3,3,3,3,3,3

23, , 223,3,3,3,3,3,3,3



244,4,4,4,4,4,4,4

27, , 254,4,4,4,4,4,4,4

27, , 264,4,4,4,4,4,4,4



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285,5,5,5,5,5,5,5

31, , 295,5,5,5,5,5,5,5

31, , 305,5,5,5,5,5,5,5



326,6,6,6,6,6,6,6

35, , 336,6,6,6,6,6,6,6

35, , 346,6,6,6,6,6,6,6



151,1,1,1,1,1,1,1 , , 14

1,1,1,1,1,1,1,1 , , 131,1,1,1,1,1,1,1,1 , are seventh detrition


377,7,7,7,7,7,7,7

39, , 367,7,7,7,7,7,7,7

39, , 387,7,7,7,7,7,7,7



428,8,8,8,8,8,8,8,8

43 , , 418,8,8,8,8,8,8,8,8

43, , 408,8,8,8,8,8,8,8,8

43 , are ninth


Where we suppose

1 , 1 , 1 , 1 , 1 , 1 > 0, , = 13,14,15

The functions 1 , 1 are positive continuousincreasing and bounded.

Definition of( ) 1 , ( ) 1 :

1 ( 14 , ) ( )1 ( 13 )

(1)

1 ( , ) ( ) 1 ( ) 1 ( 13 )(1)

97

2

114, = ( )

1

limG

1 , = ( ) 1

Definition of( 13 )(1) , ( 13 )

(1) :

Where ( 13 )(1) , ( 13 )

(1) , ( ) 1 , ( ) 1 are positive constants and = 13,14,15

98

They satisfy Lipschitz condition:

|( ) 1 14 , ( )1

14 , | ( 13 )(1) | 14 14 |

( 13 )(1)

|( ) 1 , ( ) 1 , | < ( 13 )(1) || || ( 13 )

(1)

99

With the Lipschitz condition, we place a restriction on the behavior of functions

( ) 1 14 , and( )1

14 , . 14, and 14 , are points belonging to the interval

( 13 )(1) , ( 13 )

(1) . It is to be noted that ( ) 1 14 , is uniformly continuous. In the eventuality of

the fact, that if ( 13 )(1) = 1 then the function ( ) 1 14 , , the first augmentation coefficient

attributable to the system, would be absolutely continuous.

Definition of ( 13 )(1) , ( 13 )

(1) :

( 13 )(1) , ( 13 )

(1) ,are positive constants

100


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( ) 1

( 13 )(1)

,( ) 1

( 13 )(1)

< 1

Definition of( 13 )(1) , ( 13 )

(1) :

There exists two constants( 13 )(1) and ( 13 )

(1)which together With ( 13 )(1) , ( 13 )

(1) , ( 13)(1) and

( 13 )(1)and the constants( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , = 13,14,15,

satisfy the inequalities

1

( 13 )(1)

[ ( ) 1 + ( ) 1 + ( 13 )(1) + ( 13 )

(1) ( 13 )(1) ] < 1

1

( 13 )(1)

[ ( ) 1 + ( ) 1 + ( 13 )(1) + ( 13 )

(1) ( 13 )(1) ] < 1

101

Where we suppose

2 , 2 , 2 , 2 , 2 , 2 > 0, , = 16,17,18


Definition of(pi )2 , (r i )

2 :

217 , ( )

216

2 102

2 ( 19, ) ( )2 ( ) 2 ( 16 )

(2) 103

lim2

217 , = ( )

2 104

lim 2 19 , = ( )2 105

Definition of( 16 )(2) , ( 16 )

(2) :

Where ( 16 )(2) , ( 16 )


106


|( ) 2 17 , ( )2

17 , | ( 16 )(2) | 17 17 |

( 16 )(2)

107

|( ) 2 19 , ( )2

19 , | < ( 16 )(2) || 19 19 ||

( 16 )(2)

108

With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 2 17 ,

and( ) 2 17 , . 17, and 17, are points belonging to the interval ( 16 )(2) , ( 16 )

(2) . It is to

be noted that ( ) 2 17 , is uniformly continuous. In the eventuality of the fact, that if ( 16 )(2) = 1

then the function ( ) 2 17 , , the first augmentation coefficient attributable to the system, would

be absolutely continuous.

Definition of ( 16 )(2) , ( 16 )

(2) :


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( 16 )(2) , ( 16 )


( ) 2

( 16 )(2)

,( ) 2

( 16 )(2)

< 1

109

Definition of ( 13 )(2) , ( 13 )

(2) :


(2)which together

with ( 16 )(2) , ( 16 )

(2) , ( 16)(2) ( 16 )

(2)and the

constants( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , = 16,17,18,


1

( 16 )(2)

[ ( ) 2 + ( ) 2 + ( 16 )(2) + ( 16 )

(2) ( 16 )(2) ] < 1

110

1

( 16 )(2)

[ ( ) 2 + ( ) 2 + ( 16 )(2) + ( 16 )

(2) ( 16 )(2) ] < 1

111

Where we suppose

3 , 3 , 3 , 3 , 3 , 3 > 0, , = 20,21,22


Definition of( ) 3 , (r i )3 :

3 ( 21, ) ( )3 ( 20 )

(3)

3 ( 23, ) ( )3 ( ) 3 ( 20 )

(3)

112

2

321, = ( )

3

limG

323, = ( )

3

Definition of( 20 )(3) , ( 20 )

(3) :

Where ( 20 )(3) , ( 20 )


113


|( ) 3 21 , ( )3

21 , | ( 20 )(3) | 21 21 |

( 20 )(3)

|( ) 3 23 , ( )3

23, | < ( 20 )(3) || 23 23 ||

( 20 )(3)

114


and( ) 3 21 , . 21 , And 21 , are points belonging to the interval ( 20 )(3) , ( 20 )

(3) . It is to




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Definition of ( 20 )(3) , ( 20 )

(3) :

( 20 )(3) ,( 20 )


( ) 3

( 20 )(3)

,( ) 3

( 20 )(3)

< 1

115

There exists two constantsThere exists two constants( 20 )(3) and ( 20 )

(3)which together

with ( 20 )(3) , ( 20 )

(3) , ( 20)(3) ( 20 )

(3)and the

constants( ) 3 , ( ) 3 , ( ) 3 , ( ) 3 , ( ) 3 , ( ) 3 , = 20,21,22,


1

( 20 )(3)

[ ( ) 3 + ( ) 3 + ( 20 )(3) + ( 20 )

(3)( 20 )(3) ] < 1

1

( 20 )(3)

[ ( ) 3 + ( ) 3 + ( 20 )(3) + ( 20 )

(3) ( 20 )(3) ] < 1

116

Where we suppose

4 , 4 , 4 , 4 , 4 , 4 > 0, , = 24,25,26



4 ( 25, ) ( )4 ( 24 )

(4)

427 , ( )

4 ( ) 4 ( 24 )(4)

117

2

425, = ( )

4

limG

427 , = ( )

4

Definition of( 24 )(4) , ( 24 )

(4) :

Where ( 24 )(4) , ( 24 )


118


|( ) 4 25 , ( )4

25 , | ( 24 )(4) | 25 25 |

( 24 )(4)

|( ) 4 27 , ( )4

27 , | < ( 24 )(4) || 27 27 ||

( 24 )(4)

119


and( ) 4 25 , . 25 , and 25 , are points belonging to the interval ( 24 )(4) , ( 24 )

(4) . It is to

be noted that ( ) 4 25 , is uniformly continuous. In the eventuality of the fact, that if ( 24 )(4) =

1 then the function ( ) 4 25, , the first augmentation coefficient attributable to the system, would


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Definition of ( 24 )(4) , ( 24 )

(4) :

( 24 )(4) ,( 24 )


( ) 4

( 24 )(4)

,( ) 4

( 24 )(4)

< 1

120

Definition of ( 24 )(4) , ( 24 )

(4) :


(4)which together

with ( 24 )(4) , ( 24 )

(4) , ( 24)(4) ( 24 )

(4)and the

constants( ) 4 , ( ) 4 , ( ) 4 , ( ) 4 , ( ) 4 , ( ) 4 , = 24,25,26,satisfy the inequalities

1

( 24 )(4)

[ ( ) 4 + ( ) 4 + ( 24 )(4) + ( 24 )

(4)( 24 )(4) ] < 1

1

( 24 )(4)

[ ( ) 4 + ( ) 4 + ( 24 )(4) + ( 24 )

(4) ( 24 )(4) ] < 1

121

Where we suppose

5 , 5 , 5 , 5 , 5 , 5 > 0, , = 28,29,30



5 ( 29, ) ( )5 ( 28 )

(5)

531 , ( )

5 ( ) 5 ( 28 )(5)

122

2

529, = ( )

5

limG

531, = ( )

5

Definition of( 28 )(5) , ( 28 )

(5) :

Where ( 28 )(5) , ( 28 )


123


|( ) 5 29, ( )5

29, | ( 28 )(5) | 29 29|

( 28 )(5)

|( ) 5 31 , ( )5

31 , | < ( 28 )(5) || 31 31 ||

( 28 )(5)

124

With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 5 29,

and( ) 5 29, . 29, and 29, are points belonging to the interval ( 28 )(5) , ( 28 )

(5) . It is to


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be noted that ( ) 5 29, is uniformly continuous. In the eventuality of the fact, that if ( 28 )(5) = 1

then the function ( ) 5 29, , the first augmentation coefficient attributable to the system, would


Definition of ( 28 )(5) , ( 28 )

(5) :

( 28 )(5) ,( 28 )


( ) 5

( 28 )(5)

,( ) 5

( 28 )(5)

< 1

125

Definition of ( 28 )(5) , ( 28 )

(5) :


(5)which together

with ( 28 )(5) , ( 28 )

(5) , ( 28)(5) ( 28 )

(5)and the


1

( 28 )(5)

[ ( ) 5 + ( ) 5 + ( 28 )(5) + ( 28 )

(5)( 28 )(5) ] < 1

1

( 28 )(5)

[ ( ) 5 + ( ) 5 + ( 28 )(5) + ( 28 )

(5) ( 28 )(5) ] < 1

126

Where we suppose

6 , 6 , 6 , 6 , 6 , 6 > 0, , = 32,33,34



6 ( 33, ) ( )6 ( 32 )

(6)

6 ( 35 , ) ( )6 ( ) 6 ( 32 )

(6)

127

2

633, = ( )

6

limG

635 , = ( )

6

Definition of( 32 )(6) , ( 32 )

(6) :

Where ( 32 )(6) , ( 32 )

(6) , ( ) 6 , ( ) 6 are positive constantsand = 32,33,34

128


|( ) 6 33 , ( )6

33 , | ( 32 )(6) | 33 33 |

( 32 )(6)

|( ) 6 35 , ( )6

35 , | < ( 32 )(6) || 35 35 ||

( 32 )(6)


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(6) . It is to




Definition of ( 32 )(6) , ( 32 )

(6) :

( 32 )(6) ,( 32 )


( ) 6

( 32 )(6)

,( ) 6

( 32 )(6)

< 1

129

Definition of ( 32 )(6) , ( 32 )

(6) :


(6)which together

with ( 32 )(6) , ( 32 )

(6) , ( 32)(6) ( 32 )

(6)and the

constants( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , = 32,33,34,


1

( 32 )(6)

[ ( ) 6 + ( ) 6 + ( 32 )(6) + ( 32 )

(6)( 32 )(6) ] < 1

1

( 32 )(6)

[ ( ) 6 + ( ) 6 + ( 32 )(6) + ( 32 )

(6) ( 32 )(6) ] < 1

130

Where we suppose

(A) 7 , 7 , 7 , 7 , 7 , 7 > 0, , = 36,37,38

(B) The functions 7 , 7 are positive continuousincreasing and bounded.


7 ( 37, ) ( )

7 ( 36 )(7)

7 ( 39, ) ( )

7 ( ) 7 ( 36 )(7)

131

(C) lim2

737, = ( )

7

(D)

limG

739 , = ( )

7

Definition of( 36 )(7) , ( 36 )

(7) :

Where ( 36 )(7) , ( 36 )


132


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|( ) 7 37 , ( )7

37 , | ( 36 )(7) | 37 37 |

( 36 )(7)

|( ) 7 39 , ( )7

39 , | < ( 36 )(7) || 39 39 ||

( 36 )(7)

133



(7) . It is to




Definition of ( 36 )(7) , ( 36 )

(7) :

(E) ( 36 )(7) ,( 36 )


( ) 7

( 36 )(7)

,( ) 7

( 36 )(7)

< 1

134

Definition of ( 36 )(7) , ( 36 )

(7) :

(F) There exists two constants( 36 )(7) and ( 36 )

(7)which together

with ( 36 )(7) , ( 36 )

(7) , ( 36)(7) ( 36 )

(7)and the


1

( 36 )(7)

[ ( ) 7 + ( ) 7 + ( 36 )(7) + ( 36 )

(7)( 36 )(7) ] < 1

1

( 36 )(7)

[ ( ) 7 + ( ) 7 + ( 36 )(7) + ( 36 )

(7)( 36 )(7) ] < 1

135

Where we suppose

8 , 8 , 8 , 8 , 8 , 8 > 0, , = 40,41,42

136

The functions 8 , 8 are positive continuousincreasing and bounded


137

8 ( 41, ) ( )8 ( 40 )

(8)

138

8 ( 43 , ) ( )8 ( ) 8 ( 40 )

(8) 139

lim2

841, = ( )

8

140


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lim 8 43 , = ( )8 141

Definition of( 40 )(8) , ( 40 )

(8) :

Where ( 40 )(8) , ( 40 )



|( ) 8 41, ( )8

41, | ( 40 )(8) | 41 41 |

( 40 )(8)

142

|( ) 8 43 , ( )8

43 , | < ( 40 )(8) || 43 43 ||

( 40 )(8)

143

With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 8 41 , and

( ) 8 41, . 41, and 41, are points belonging to the interval ( 40 )(8) , ( 40 )

(8) . It is to be

noted that ( ) 8 41 , is uniformly continuous. In the eventuality of the fact, that if ( 40 )(8) = 1

then the function ( ) 8 41, , the first augmentation coefficient attributable to the system, would


Definition of ( 40 )(8) , ( 40 )

(8) :

( 40 )(8) ,( 40 )


( ) 8

( 40 )(8)

,( ) 8

( 40 )(8)

< 1

144

Definition of ( 40 )(8) , ( 40 )

(8) :


(8)which together with( 40 )(8) , ( 40 )

(8) , ( 40)(8)

( 40 )(8)and the constants( ) 8 , ( ) 8 , ( ) 8 , ( ) 8 , ( ) 8 , ( ) 8 , = 40,41,42,

Satisfy the inequalities

1

( 40 )(8)

[ ( ) 8 + ( ) 8 + ( 40 )(8) + ( 40 )

(8) ( 40 )(8) ] < 1

145

1

( 40 )(8)

[ ( ) 8 + ( ) 8 + ( 40 )(8) + ( 40 )

(8)( 40 )(8) ] < 1

146

Where we suppose

9 , 9 , 9 , 9 , 9 , 9 > 0, , = 44,45,46



9 ( 45 , ) ( )9 ( 44 )

(9)

9 ( 47 , ) ( )9 ( ) 9 ( 44 )

(9)

146A


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2

945, = ( )

9

limG

947 , = ( )

9

Definition of( 44 )

(9) , ( 44 )(9) :

Where ( 44 )(9) , ( 44 )



|( ) 9 45, ( )9

45, | ( 44 )(9) | 45 45 |

( 44 )(9)

|( ) 9 47 , ( )9

47 , | < ( 44 )(9) || 47 47 ||

( 44 )(9)

With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 9 45,

and( ) 9 45, . 45 , and 45, are points belonging to the interval ( 44 )(9) , ( 44 )

(9) . It is to


then the function ( ) 9 45 , , the first augmentation coefficient attributable to the system, would be absolutely continuous.

Definition of ( 44 )(9) , ( 44 )

(9) :

( 44 )(9) , ( 44 )


( ) 9

( 44 )(9)

,( ) 9

( 44 )(9)

< 1

Definition of ( 44 )(9) , ( 44 )

(9) : There exists two constants( 44 )

(9) and ( 44 )(9)which together

with ( 44 )(9) , ( 44 )

(9) , ( 44)(9) ( 44 )

(9)and the

constants( ) 9 , ( ) 9 , ( ) 9 ,( ) 9 ,( ) 9 , ( ) 9 , = 44,45,46, satisfy the inequalities

1

( 44 )(9)

[ ( ) 9 + ( ) 9 + ( 44 )(9) + ( 44 )

(9) ( 44 )(9) ] < 1

1

( 44 )(9)

[ ( ) 9 + ( ) 9 + ( 44 )(9) + ( 44 )

(9) ( 44 )(9) ] < 1

Theorem 1: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of 0 , 0 :

131

131

, 0 = 0 > 0

( ) ( 13 )(1) ( 13 )

(1) , 0 = 0 > 0

147


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Theorem 2 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of 0 , 0

( 16 )(2) ( 16 )

(2) , 0 = 0 > 0

( ) ( 16 )(2) ( 16 )

(2) , 0 = 0 > 0

148


( 20 )(3) ( 20 )

(3) , 0 = 0 > 0

( ) ( 20 )(3) ( 20 )

(3) , 0 = 0 > 0

149



244

244

, 0 = 0 > 0

( ) ( 24 )(4) ( 24 )

(4) , 0 = 0 > 0

150



285

285

, 0 = 0 > 0

( ) ( 28 )(5) ( 28 )

(5) , 0 = 0 > 0

151



326

326

, 0 = 0 > 0

( ) ( 32 )(6) ( 32 )

(6) , 0 = 0 > 0

152

Theorem 7: if the conditions above are fulfilled, there exists a solution satisfying the condi

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Yang–Mills Existence and Mass Gap (Unsolved … · International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 2013 1 ISSN 2250-3153 Yang–Mills