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International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 2013 1
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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
axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &
Schrader (1973) and Osterwalder & Schrader (1975).
15 : Category three of exists (eb) on
axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &
Schrader (1973) and Osterwalder & Schrader (1975).
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
axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &
Schrader (1973) and Osterwalder & Schrader (1975)
18 : Category three of existence (eb) on stence includes establishing
axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder &
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
physics; is Euclidean 4-space
22 : Category three of(non-Abelian) quantum field theory underlying the Standard Model of particle
physics; is Euclidean 4-space
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
particular constructive quantum field theory, which is referenced in the papers 45 and 35 cited in the official
problem description by Jaffe and Witten
T30 : Category three of 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
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
Lagrangian and the Hamiltonian)
46 : Category three of LHS of all the equations stated in the foregoing (Yang Mills Theory including the
Lagrangian and the Hamiltonian)
T44 : Category one of RHS of all the equations stated in the foregoing (Yang Mills Theory including the
Lagrangian and the Hamiltonian)
45 : Category two of RHS of all the equations stated in the foregoing (Yang Mills Theory including the
Lagrangian and the Hamiltonian)
46 : Category three of RHS of all the equations stated in the foregoing (Yang Mills Theory including the
Lagrangian and the Hamiltonian)
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
17 , = First augmentation factor
162
19 , = First detritions factor
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
21 , = First augmentation factor
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
25 , = First augmentation factor
244
27 , = First detritions factor
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
31 , = First detritions factor
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
33 , = First augmentation factor
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
37 , = First augmentation factor
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
45 , = First augmentation factor
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
for category 1, 2 and 3
+ 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
for category 1, 2 and 3
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
17 , are first augmentation coefficients for
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
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+ 244,4,4,4,4
25 , , + 254,4,4,4,4
25 , , + 264,4,4,4,4
25 , are fourth augmentation
coefficient for category 1, 2 and 3
+ 285,5,5,5,5
29, , + 295,5,5,5,5
29, , + 305,5,5,5,5
29, are fifth augmentation
coefficient for category 1, 2 and 3
+ 326,6,6,6,6
33 , , + 336,6,6,6,6
33, , + 346,6,6,6,6
33 , are sixth augmentation
coefficient for category 1, 2 and 3
+ 367,7,7
37 , , + 377,7,7
37, , + 387,7,7
37, are seventh augmentation coefficient
for category 1, 2 and 3
+ 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
39, are seventh detrition coefficients for
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
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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
for category 1, 2 and 3
+ 131,1,1,
14 , , + 141,1,1,
14 , , + 151,1,1,
14 , are third augmentation coefficients
for category 1, 2 and 3
+ 244,4,4,4,4,4
25 , , + 254,4,4,4,4,4
25 , , + 264,4,4,4,4,4
25 , are fourth augmentation
coefficients for category 1, 2 and 3
+ 285,5,5,5,5,5
29, , + 295,5,5,5,5,5
29, , + 305,5,5,5,5,5
29, are fifth augmentation
coefficients for category 1, 2 and 3
+ 326,6,6,6,6,6
33 , , + 336,6,6,6,6,6
33 , , + 346,6,6,6,6,6
33, are sixth augmentation
coefficients for category 1, 2 and 3
+ 367,7,7,7
37, , + 377,7,7,7
37 , , + 387,7,7,7
37, are seventh augmentation
coefficients for category 1, 2 and 3
+ 408,8,8,8
41, , + 418,8,8,8
41, , + 428,8,8,8
41, are eight augmentation coefficients
for category 1, 2 and 3
+ 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
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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
27, are fourth detrition
coefficients for category 1, 2 and 3
285,5,5,5,5,5
31, , 295,5,5,5,5,5
31, , 305,5,5,5,5,5
31, are fifth detrition
coefficients for category 1, 2 and 3
326,6,6,6,6,6
35, , 336,6,6,6,6,6
35, , 346,6,6,6,6,6
35, are sixth detrition
coefficients for category 1, 2 and 3
367,7,7,7
39, , 377,7,7,7
39, 387,7,7,7
39, are seventh detrition coefficients for
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
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+ 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
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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
14 , are fourth augmentation
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
21 , are sixth augmentation
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
coefficients for category 1,2, 3
+ 408,8 ,8,8,8,8
41, , + 418,8,8,8,8,8
41, , + 428,8,8,8,8,8
41, are eighth augmentation
coefficients for category 1,2, 3
+ 469,9,9,9,9
45 , , + 459,9,9,9,9
45, , + 449,9,9,9,9
45, are ninth augmentation
coefficients for category 1,2, 3
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
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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
for category 1,2, and 3
367,7,7,7,7,7
39, , 377,7,7,7,7,7
39, , 387,7,7,7,7,7
39, are seventh detrition
coefficients for category 1,2, and 3
428,8,8,8,8,8
43 , , 418,8,8,8,8,8
43 , , 408,8,8,8,8,8
43, are eighth detrition
coefficients for category 1,2, and 3
469,9,9,9,9
47, , 459,9,9,9,9
47 , , 449,9,9,9,9
47 , are ninth detrition coefficients
for category 1,2, and 3
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
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+ 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
19, are fifth detrition
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
23, are sixth detrition
coefficients for category 1, 2, and 3
367,7,7,7,7,7,7
39, , 377,7,7,7,7,7,7
39, , 387,7,7,7,7,7,7
39, are seventh detrition
coefficients for category 1, 2, and 3
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
47, are ninth detrition
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coefficients for category 1, 2, and 3
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
37 , are first augmentation coefficients for
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
coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 285,5,5,5,5,5,5
29, , + 295,5,5,5,5,5,5
29, , + 305,5,5,5,5,5,5
29, are fifth augmentation
coefficient for category 1, 2 and 3
+ 326,6,6,6,6,6,6
33, , + 336,6,6,6,6,6,6
33, , + 346,6,6,6,6,6,6
33 , are sixth augmentation
coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
coefficients for category 1, 2 and 3
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
coefficients for category 1, 2 and 3
244,4,4,4,4,4,4
27, , 254,4,4,4,4,4,4
27, , 264,4,4,4,4,4,4
27, are fourth detrition
coefficients for category 1, 2 and 3
285,5,5,5,5,5,5
31, , 295,5,5,5,5,5,5
31, , 305,5,5,5,5,5,5
31, are fifth detrition
coefficients for category 1, 2 and 3
326,6,6,6,6,6,6
35, , 336,6,6,6,6,6,6
35, , 346,6,6,6,6,6,6
35, are sixth detrition
coefficients for category 1, 2 and 3
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
43, are eighth detrition
coefficients for category 1, 2 and 3
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
coefficients for category 1, 2 and 3
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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for 1,2,3
+ 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
augmentation coefficient for 1,2,3
+ 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
augmentation coefficient for 1,2,3
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
39, are first detrition coefficients for
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
23, are third detrition
coefficients for category 1, 2 and 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
27, are fourth detrition
coefficients for category 1, 2 and 3
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
31, are fifth detrition
coefficients for category 1, 2 and 3
326,6,6,6,
35, , 336,6,6,6,
35, , 151,1,1,1,1,1,1,1 , are sixth detrition coefficients
for category 1, 2 and 3
131,1,1,1,1,1,1,1 , , 14
1,1,1,1,1,1,1,1 , , 387,7,
39, are seventh detrition
coefficients for category 1, 2 and 3
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
39, are eighth detrition
coefficients for category 1, 2 and 3
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
47, are ninth detrition
coefficients for category 1, 2 and 3
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
37 , are first augmentation coefficients for
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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augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for category 1, 2 and 3
+ 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
augmentation coefficient for 1,2,3
+ 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
augmentation coefficient for 1,2,3
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
47, are first detrition coefficients for
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
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
23, are third detrition
coefficients for category 1, 2 and 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
27, are fourth detrition
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coefficients for category 1, 2 and 3
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
31, are fifth detrition
coefficients for category 1, 2 and 3
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
35, are sixth detrition
coefficients for category 1, 2 and 3
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
coefficients for category 1, 2 and 3
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
39, are eighth detrition
coefficients for category 1, 2 and 3
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
detrition coefficients for category 1, 2 and 3
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
The functions 2 , 2 are positive continuousincreasing and bounded.
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 )
(2) , ( ) 2 , ( ) 2 are positive constants and = 16,17,18
106
They satisfy Lipschitz condition:
|( ) 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) ,are positive constants
( ) 2
( 16 )(2)
,( ) 2
( 16 )(2)
< 1
109
Definition of ( 13 )(2) , ( 13 )
(2) :
There exists two constants( 16 )(2) and ( 16 )
(2)which together
with ( 16 )(2) , ( 16 )
(2) , ( 16)(2) ( 16 )
(2)and the
constants( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , ( ) 2 , = 16,17,18,
satisfy the inequalities
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
The functions 3 , 3 are positive continuousincreasing and bounded.
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 )
(3) , ( ) 3 , ( ) 3 are positive constants and = 20,21,22
113
They satisfy Lipschitz condition:
|( ) 3 21 , ( )3
21 , | ( 20 )(3) | 21 21 |
( 20 )(3)
|( ) 3 23 , ( )3
23, | < ( 20 )(3) || 23 23 ||
( 20 )(3)
114
With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 3 21 ,
and( ) 3 21 , . 21 , And 21 , are points belonging to the interval ( 20 )(3) , ( 20 )
(3) . It is to
be noted that ( ) 3 21 , is uniformly continuous. In the eventuality of the fact, that if ( 20 )(3) = 1
then the function ( ) 3 21 , , the first augmentation coefficient attributable to the system, would
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be absolutely continuous.
Definition of ( 20 )(3) , ( 20 )
(3) :
( 20 )(3) ,( 20 )
(3) ,are positive constants
( ) 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,
satisfy the inequalities
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
The functions 4 , 4 are positive continuousincreasing and bounded.
Definition of( ) 4 , ( ) 4 :
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 )
(4) , ( ) 4 , ( ) 4 are positive constants and = 24,25,26
118
They satisfy Lipschitz condition:
|( ) 4 25 , ( )4
25 , | ( 24 )(4) | 25 25 |
( 24 )(4)
|( ) 4 27 , ( )4
27 , | < ( 24 )(4) || 27 27 ||
( 24 )(4)
119
With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 4 25 ,
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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be absolutely continuous.
Definition of ( 24 )(4) , ( 24 )
(4) :
( 24 )(4) ,( 24 )
(4) ,are positive constants
( ) 4
( 24 )(4)
,( ) 4
( 24 )(4)
< 1
120
Definition of ( 24 )(4) , ( 24 )
(4) :
There exists two constants( 24 )(4) and ( 24 )
(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
The functions 5 , 5 are positive continuousincreasing and bounded.
Definition of( ) 5 , ( ) 5 :
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 )
(5) , ( ) 5 , ( ) 5 are positive constants and = 28,29,30
123
They satisfy Lipschitz condition:
|( ) 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
be absolutely continuous.
Definition of ( 28 )(5) , ( 28 )
(5) :
( 28 )(5) ,( 28 )
(5) ,are positive constants
( ) 5
( 28 )(5)
,( ) 5
( 28 )(5)
< 1
125
Definition of ( 28 )(5) , ( 28 )
(5) :
There exists two constants( 28 )(5) and ( 28 )
(5)which together
with ( 28 )(5) , ( 28 )
(5) , ( 28)(5) ( 28 )
(5)and the
constants( ) 5 , ( ) 5 , ( ) 5 , ( ) 5 , ( ) 5 , ( ) 5 , = 28,29,30,satisfy the inequalities
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
The functions 6 , 6 are positive continuousincreasing and bounded.
Definition of( ) 6 , ( ) 6 :
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
They satisfy Lipschitz condition:
|( ) 6 33 , ( )6
33 , | ( 32 )(6) | 33 33 |
( 32 )(6)
|( ) 6 35 , ( )6
35 , | < ( 32 )(6) || 35 35 ||
( 32 )(6)
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With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 6 33 ,
and( ) 6 33 , . 33 , and 33 , are points belonging to the interval ( 32 )(6) , ( 32 )
(6) . It is to
be noted that ( ) 6 33 , is uniformly continuous. In the eventuality of the fact, that if ( 32 )(6) = 1
then the function ( ) 6 33 , , the first augmentation coefficient attributable to the system, would
be absolutely continuous.
Definition of ( 32 )(6) , ( 32 )
(6) :
( 32 )(6) ,( 32 )
(6) ,are positive constants
( ) 6
( 32 )(6)
,( ) 6
( 32 )(6)
< 1
129
Definition of ( 32 )(6) , ( 32 )
(6) :
There exists two constants( 32 )(6) and ( 32 )
(6)which together
with ( 32 )(6) , ( 32 )
(6) , ( 32)(6) ( 32 )
(6)and the
constants( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , ( ) 6 , = 32,33,34,
satisfy the inequalities
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.
Definition of( ) 7 , ( ) 7 :
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 )
(7) , ( ) 7 , ( ) 7 are positive constants and = 36,37,38
132
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They satisfy Lipschitz condition:
|( ) 7 37 , ( )7
37 , | ( 36 )(7) | 37 37 |
( 36 )(7)
|( ) 7 39 , ( )7
39 , | < ( 36 )(7) || 39 39 ||
( 36 )(7)
133
With the Lipschitz condition, we place a restriction on the behavior of functions ( ) 7 37 ,
and( ) 7 37 , . 37 , and 37 , are points belonging to the interval ( 36 )(7) , ( 36 )
(7) . It is to
be noted that ( ) 7 37 , is uniformly continuous. In the eventuality of the fact, that if ( 36 )(7) = 1
then the function ( ) 7 37 , , the first augmentation coefficient attributable to the system, would
be absolutely continuous.
Definition of ( 36 )(7) , ( 36 )
(7) :
(E) ( 36 )(7) ,( 36 )
(7) ,are positive constants
( ) 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
constants( ) 7 , ( ) 7 , ( ) 7 , ( ) 7 , ( ) 7 , ( ) 7 , = 36,37,38,satisfy the inequalities
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
Definition of( ) 8 , ( ) 8 :
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) , ( ) 8 , ( ) 8 are positive constants and = 40,41,42
They satisfy Lipschitz condition:
|( ) 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
be absolutely continuous.
Definition of ( 40 )(8) , ( 40 )
(8) :
( 40 )(8) ,( 40 )
(8) ,are positive constants
( ) 8
( 40 )(8)
,( ) 8
( 40 )(8)
< 1
144
Definition of ( 40 )(8) , ( 40 )
(8) :
There exists two constants( 40 )(8) and ( 40 )
(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
The functions 9 , 9 are positive continuousincreasing and bounded.
Definition of( ) 9 , ( ) 9 :
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) , ( ) 9 , ( ) 9 are positive constants and = 44,45,46
They satisfy Lipschitz condition:
|( ) 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
be noted that ( ) 9 45 , is uniformly continuous. In the eventuality of the fact, that if ( 44 )(9) = 1
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) ,are positive constants
( ) 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
Theorem 3 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
( 20 )(3) ( 20 )
(3) , 0 = 0 > 0
( ) ( 20 )(3) ( 20 )
(3) , 0 = 0 > 0
149
Theorem 4 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of 0 , 0 :
244
244
, 0 = 0 > 0
( ) ( 24 )(4) ( 24 )
(4) , 0 = 0 > 0
150
Theorem 5 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of 0 , 0 :
285
285
, 0 = 0 > 0
( ) ( 28 )(5) ( 28 )
(5) , 0 = 0 > 0
151
Theorem 6 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of 0 , 0 :
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