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1Dr. K. N. Prasanna Kumar,
2Prof. B. S. Kiranagi,
3 Prof C S Bagewadi
ABSTRACT: A state register that stores the state of the Turing machine, one of finitely many. There is one special start state
with which the state register is initialized. These states, writes Turing, replace the "state of mind" a person performing
computations would ordinarily be in.It is like bank ledger, which has Debits and Credits. Note that in double entry computation, both debits and credits are entered in to the systems, namely the Bank Ledger and balance is posted. Individual
Debits are equivalent to individual credits. On a generalizational and globalist scale, a βGeneral Ledgerβ is written which
records in all its wide ranging manifestation the Debits and Credits. This is also conservative. In other words Assets is
equivalent to Liabilities. True, Profit is distributed among overheads and charges, and there shall be another account in the
General ledger that is the account of Profit. This account is credited with the amount earned as commission, exchange, or
discount of bills. Now when we write the βGeneral Ledgerβ of Turing machine, the Primma Donna and terra firma of the
model, we are writing the General Theory of all the variables that are incorporated in the model. So for every variable, we
have an anti variable. This is the dissipation factor. Conservation laws do hold well in computers. They do not break the
conservation laws. Thus energy is not dissipated in to the atmosphere when computation is being performed. To repeat we
are suggesting a General Theory Of working of a simple Computer and in further papers, we want to extend this theory to
both nanotechnology and Quantum Computation. Turingβs work is the proponent, precursor, primogeniture, progenitor and
promethaleon for the development of Quantum Computers. Computers follow conservation laws. This work is one which formed the primordial concept of diurnal dynamics and hypostasized dynamism of Quantum computers which is the locus of
essence, sense and expression of the present day to day musings and mundane drooling. Verily Turing and Churchill stand
out like connoisseurs, rancouteurs, and cognescenti of eminent persons, who strode like colossus the screen of collective
consciousness of people. We dedicate this paper on the eve of one hundred years of Turing innovation. Model is based on
Hill and Peterson diagram.
INTRODUCTION Turing machine βA beckoning begorra (Extensive excerpts from Wikipedia AND PAGES OF Turing,Churchill,and
other noted personalities-Emphasis is mine) A Turing machine is a device that manipulates symbols on a strip of tape according to a table of rules. Despite its
simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in
explaining the functions of a CPU inside a computer.
The "Turing" machine was described by Alan Turing in 1936 who called it an "a (automatic)-machine". The Turing
machine is not intended as a practical computing technology, but rather as a hypothetical device representing a computing
machine. Turing machines help computer scientists understand the limits of mechanical computation.
Turing gave a succinct and candid definition of the experiment in his 1948 essay, "Intelligent Machinery". Referring to his
1936 publication, Turing wrote that the Turing machine, here called a Logical Computing Machine, consisted of:
...an infinite memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol
could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter
the scanned symbol and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not
affect the behaviour of the machine. However, the tape can be moved back and forth through the machine, this being one of
the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings. (Turing 1948,
p. 61)
A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine (UTM, or simply
a universal machine). A more mathematically oriented definition with a similar "universal" nature was introduced by Alonzo
Church, whose work on calculus intertwined with Turing's in a formal theory of computation known as the ChurchβTuring
thesis. The thesis states that Turing machines indeed capture the informal notion of effective method
in logic and mathematics, and provide a precise definition of an algorithm or 'mechanical procedure'.
In computability theory, the ChurchβTuring thesis (also known as the ChurchβTuring conjecture, Church's thesis, Church's
conjecture, and Turing's thesis) is a combined hypothesis ("thesis") about the nature of functions whose values
are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. In simple
terms, the ChurchβTuring thesis states that "everything algorithmically computable is computable by a Turing machine.β American mathematician Alonzo Church created a method for defining functions called the Ξ»-calculus,
Church, along with mathematician Stephen Kleene and logician J.B. Rosser created a formal definition of a class of
functions whose values could be calculated by recursion.
All three computational processes (recursion, the Ξ»-calculus, and the Turing machine) were shown to be equivalentβall
three approaches define the same class of functions this has led mathematicians and computer scientists to believe that the
concept of computability is accurately characterized by these three equivalent processes. Informally the ChurchβTuring
thesis states that if some method (algorithm) exists to carry out a calculation, then the same calculation can also be carried
out by a Turing machine (as well as by a recursively definable function, and by a Ξ»-function).
The ChurchβTuring thesis is a statement that characterizes the nature of computation and cannot be formally proven. Even
though the three processes mentioned above proved to be equivalent, the fundamental premise behind the thesisβthe
Turing Machine Operation-A Checks and Balances Model
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notion of what it means for a function to be "effectively calculable" (computable)βis "a somewhat vague intuitive
one" Thus, the "thesis" remains a hypothesis.
Desultory Bureaucratic burdock or a Driving dromedary? The Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols
which the machine can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of
elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in
state 17, if the symbol seen is 0, write a 1 and change to state 6;" etc. In the original article, Turing imagines not a
mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly (or as
Turing puts it, "in a desultory manner").
The head is always over a particular square of the tape; only a finite stretch of squares is shown. The instruction to be
performed (q4) is shown over the scanned square. (Drawing after Kleene (1952) p.375.)
Here, the internal state (q1) is shown inside the head, and the illustration describes the tape as being infinite and pre-filled with "0", the symbol serving as blank. The system's full state (its configuration) consists of the internal state, the contents of
the shaded squares including the blank scanned by the head ("11B"), and the position of the head. (Drawing after Minsky
(1967) p. 121).
Sequestration dispensation:
A tape which is divided into cells, one next to the other. Each cell contains a symbol from some finite alphabet. The
alphabet contains a special blank symbol (here written as 'B') and one or more other symbols. The tape is assumed to be
arbitrarily extendable to the left and to the right, i.e., the Turing machine is always supplied with as much tape as it needs
for its computation. Cells that have not been written to before are assumed to be filled with the blank symbol. In some
models the tape has a left end marked with a special symbol; the tape extends or is indefinitely extensible to the right.
A head that can read and write symbols on the tape and move the tape left and right one (and only one) cell at a time. In
some models the head moves and the tape is stationary. A state register that stores the state of the Turing machine, one of finitely many. There is one special start state with which
the state register is initialized. These states, writes Turing, replace the "state of mind" a person performing computations
would ordinarily be in.It is like bank ledger, which has Debits and Credits. Note that in double entry computation, both
debits and credits are entered in to the systems, namely the Bank Ledger and balance is posted. Individual Debits are
equivalent to individual Credits. On a generalizational and globalist scale, a βGeneral Ledgerβ is written which records in all
its wide ranging manifestation the Debits and Credits. This is also conservative. In other words Assets is equivalent to
Liabilities. True, Profit is distributed among overheads and charges, and there shall be another account in the General ledger
that is the account of Profit. This account is credited with the amount earned as commission, exchange, or discount of bills.
Now when we write the βGeneral Ledgerβ of Turing machine, the Primma Donna and terra firma of the model, we are
writing the General Theory of all the variables that are incorporated in the model. So for every variable, we have an anti
variable. This is the dissipation factor. Conservation laws do hold well in computers. They do not break the conservation laws. Thus energy is not dissipated in to the atmosphere when computation is being performed. To repeat we are suggesting
a General Theory Of working of a simple Computer and in further papers, we want to extend this theory to both
nanotechnology and Quantum Computation.
A finite table (occasionally called an action table or transition function) of instructions (usually quintuples [5-tuples]:
qiajβqi1aj1dk, but sometimes 4-tuples) that, given the state (qi) the machine is currently in and the symbol (aj) it is reading
on the tape (symbol currently under the head) tells the machine to do the following in sequence (for the 5-tuple models):
Either erase or write a symbol (replacing aj with aj1), and then
Move the head (which is described by dk and can have values: 'L' for one step left or 'R' for one step right or 'N' for staying
in the same place), and then
Assume the same or a new state as prescribed (go to state qi1).
In the 4-tuple models, erasing or writing a symbol (aj1) and moving the head left or right (dk) are specified as separate
instructions. Specifically, the table tells the machine to (ia) erase or write a symbol or (ib) move the head left or right, and then (ii) assume the same or a new state as prescribed, but not both actions (ia) and (ib) in the same instruction. In some
models, if there is no entry in the table for the current combination of symbol and state then the machine will halt; other
models require all entries to be filled.
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Note that every part of the machineβits state and symbol-collectionsβand its actionsβprinting, erasing and tape motionβ
is finite, discrete and distinguishable; Only a virus can act as a predator to it. It is the potentially unlimited amount of tape
that gives it an unbounded amount of storage space.
Quantum mechanical Hamiltonian models of Turing machines are constructed here on a finite lattice of spin-Β½ systems. The models do not dissipate any energy and they operate at the quantum limit in that the system (energy uncertainty) /
(computation speed) is close to the limit given by the time-energy uncertainty principle.
Regarding finite state machines as Markov chains facilitates the application of probabilistic methods to very large logic
synthesis and formal verification problems.Variational concepts and exegetic evanescence of the subject matter is done by
Hachtel, G.D. Macii, E. ; Pardo, A. ; Somenzi, F. with symbolic algorithms to compute the steady-state probabilities for
very large finite state machines (up to 1027 states). These algorithms, based on Algebraic Decision Diagrams (ADD's) -an
extension of BDD's that allows arbitrary values to be associated with the terminal nodes of the diagrams-determine the
steady-state probabilities by regarding finite state machines as homogeneous, discrete-parameter Markov chains with finite
state spaces, and by solving the corresponding Chapman-Kolmogorov equations. Finite state machines with state graphs
composed of a single terminal strongly connected component systems authors have used two solution techniques: One is
based on the Gauss-Jacobi iteration, the other one is based on simple matrix multiplication. Extension of the treatment is
done to the most general case of systems which can be modeled as finite state machines with arbitrary transition structures; until a certain temporal point, having no relevant options and effects for the decision maker beyond that point. Structural
morphology and easy decomposition is resorted to towards the consummation of results. Conservations Laws powerhouse
performance and no breakage is done with heterogeneous synthesis of conditionalities. Accumulation. Formulation and
experimentation are by word and watch word.
Logistics of misnomerliness and anathema:
In any scientific discipline there are many reasons to use terms that have precise definitions. Understanding the terminology
of a discipline is essential to learning a subject and precise terminology enables us to communicate ideas clearly with other
people. In computer science the problem is even more acute: we need to construct software and hardware components that
must smoothly interoperate across interfaces with clients and other components in distributed systems. The definitions of
these interfaces need to be precisely specified for interoperability and good systems performance. Using the term "computation" without qualification often generates a lot of confusion. Part of the problem is that the nature
of systems exhibiting computational behavior is varied and the term computation means different things to different people
depending on the kinds of computational systems they are studying and the kinds of problems they are investigating. Since
computation refers to a process that is defined in terms of some underlying model of computation, we would achieve clearer
communication if we made clear what the underlying model is.
Rather than talking about a vague notion of "computation," suggestion is to use the term in conjunction with a well-defined
model of computation whose semantics is clear and which matches the problem being investigated. Computer science
already has a number of useful clearly defined models of computation whose behaviors and capabilities are well understood.
We should use such models as part of any definition of the term computation. However, for new domains of investigation
where there are no appropriate models it may be necessary to invent new formalisms to represent the systems under study.
Courage of conviction and will for vindication: We consider computational thinking to be the thought processes involved in formulating problems so their solutions can be
represented as computational steps and algorithms. An important part of this process is finding appropriate models of
computation with which to formulate the problem and derive its solutions. A familiar example would be the use of finite
automata to solve string pattern matching problems. A less familiar example might be the quantum circuits and order finding
formulation that Peter Schor used to devise an integer-factoring algorithm that runs in polynomial time on a quantum
computer. Associated with the basic models of computation in computer science is a wealth of well-known algorithm-design
and problem-solving techniques that can be used to solve common problems arising in computing.
However, as the computer systems we wish to build become more complex and as we apply computer science abstractions to
new problem domains, we discover that we do not always have the appropriate models to devise solutions. In these cases,
computational thinking becomes a research activity that includes inventing appropriate new models of computation.
Corrado Priami and his colleagues at the Centre for Computational and Systems Biology in Trento, Italy have been using process calculi as a model of computation to create programming languages to simulate biological processes. Priami states
"the basic feature of computational thinking is abstraction of reality in such a way that the neglected details in the model
make it executable by a machine." [Priami, 2007]
As we shall see, finding or devising appropriate models of computation to formulate problems is a central and often
nontrivial part of computational thinking.
Hero or Zero?
In the last half century, what we think of as a computational system has expanded dramatically. In the earliest days of
computing, a computer was an isolated machine with limited memory to which programs were submitted one at a time to be
compiled and run. Today, in the Internet era, we have networks consisting of millions of interconnected computers and as
we move into cloud computing, many foresee a global computing environment with billions of clients having universal on-demand access to computing services and data hosted in gigantic data centers located around the planet. Anything from a
PC or a phone or a TV or a sensor can be a client and a data center may consist of hundreds of thousands of servers.
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Needless to say, the models for studying such a universally accessible, complex, highly concurrent distributed system are
very different from the ones for a single isolated computer. In fact, our aim is to build the model for infinite number of
interconnected ness of computers.
Another force at play is that because of heat dissipation considerations the architecture of computers is changing. An ordinary PC today has many different computing elements such as multicore chips and graphics processing units, and an
exascale supercomputer by the end of this decade is expected to be a giant parallel machine with up to a million nodes each
with possibly a thousand processors. Our understanding of how to write efficient programs for these machines is limited.
Good models of parallel computation and parallel algorithm design techniques are a vital open research area for effective
parallel computing.
In addition, there is increasing interest in applying computation to studying virtually all areas of human endeavor. One
fascinating example is simulating the highly parallel biological processes found in human cells and organs for the purposes
of understanding disease and drug design. Good computational models for biological processes are still in their infancy. And
it is not clear we will ever be able to find a computational model for the human brain that would account for emergent
phenomena such as consciousness or intelligence.
Queen or show piece: The theory of computation has been and still is one of the core areas of computer science. It explores the fundamental
capabilities and limitations of models of computation. A model of computation is a mathematical abstraction of a
computing system. The most important model of sequential computation studied in computer science is the Turing machine,
first proposed by Alan Turing in 1936.
We can think of a Turing machine as a finite-state control attached to a tape head that can read and write symbols on the
squares of a semi-infinite tape. Initially, a finite string of length n representing the input is in the leftmost n squares of the
tape. An infinite sequence of blanks follows the input string. The tape head is reading the symbol in the leftmost square and
the finite control is in a predefined initial state.
The Turing machine then makes a sequence of moves. In a move it reads the symbol on the tape under the tape head and
consults a transition table in the finite-state control which specifies a symbol to be overprinted on the square under the tape
head, a direction the tape head is to move (one square to the left or right), and a state to enter next. If the Turing machine enters an accepting halting state (one with no next move), the string of nonblank symbols remaining on the input tape at that
point in time is its output.
Mathematically, a Turing machine consists of seven components: a finite set of states; a finite input alphabet (not
containing the blank); a finite tape alphabet (which includes the input alphabet and the blank); a transition function that maps
a state and a tape symbol into a state, tape symbol, and direction (left or right); a start state; an accept state from which there
are no further moves; and a reject state from which there are no further moves.
We can characterize the configuration of a Turing machine at a given moment in time by three quantities:
1. the state of the finite-state control,
2. the string of nonblank symbols on the tape, and
3. the location of the input head on the tape.
A computation of a Turing machine on an input w is a sequence of configurations the machine can go through starting from
the initial configuration with w on the tape and terminating (if the computation terminates) in a halting configuration. We say a function f from strings to strings is computable if there is some Turing machine M that given any input string w always
halts in the accepting state with just f (w) on its tape. We say that M computes f.
The Turing machine provides a precise definition for the term algorithm: an algorithm for a function f is just a Turing
machine that computes f.
There are scores of models of computation that are equivalent to Turing machines in the sense that these models compute
exactly the same set of functions that Turing machines can compute. Among these Turing-complete models of computation
are multitape Turing machines, lambda-calculus, random access machines, production systems, cellular automata,
and all general-purpose programming languages.
The reason there are so many different models of computation equivalent to Turing machines is that we rarely want to
implement an algorithm as a Turing machine program; we would like to use a computational notation such as a
programming language that is easy to write and easy to understand. But no matter what notation we choose, the famous Church-Turing thesis hypothesizes that any function that can be computed can be computed by a Turing machine.
Note that if there is one algorithm to compute a function f, then there is an infinite number. Much of computer science is
devoted to finding efficient algorithms to compute a given function.
For clarity, we should point out that we have defined a computation as a sequence of configurations a Turing machine can go
through on a given input. This sequence could be infinite if the machine does not halt or one of a number of possible
sequences in case the machine is nondeterministic.
The reason we went through this explanation is to point out how much detail is involved in precisely defining the term
computation for the Turing machine, one of the simplest models of computation. It is not surprising, then, as we move to
more complex models, the amount of effort needed to precisely formulate computation in terms of those models grows
substantially.
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Sublime synthesis not dismal anchorage:
Many real-world computational systems compute more than just a single functionβthe world has moved to interactive
computing [Goldin, Smolka, Wegner, 2006]. The term reactive system is used to describe a system that maintains an
ongoing interaction with its environment. Examples of reactive systems include operating systems and embedded systems. A distributed system is one that consists of autonomous computing systems that communicate with one another through
some kind of network using message passing. Examples of distributed systems include telecommunications systems, the
Internet, air-traffic control systems, and parallel computers. Many distributed systems are also reactive systems.
Perhaps the most intriguing examples of reactive distributed computing systems are biological systems such as cells and
organisms. We could even consider the human brain to be a biological computing system. Formulation of appropriate
models of computation for understanding biological processes is a formidable scientific challenge in the intersection of
biology and computer science.
Distributed systems can exhibit behaviors such as deadlock, live lock, race conditions, and the like that cannot be usefully
studied using a sequential model of computation. Moreover, solving problems such as determining the throughput, latency,
and performance of a distributed system cannot be productively formulated with a single-thread model of computation. For
these reasons, computer scientists have developed a number of models of concurrent computation which can be used to
study these phenomena and to architect tools and components for building distributed systems. Many authors have studied these aspects in wider detail (See for example Alfred V. Aho),
There are many theoretical models for concurrent computation. One is the message-passing Actor model, consisting of
computational entities called actors [Hewitt, Bishop, Steiger, 1973].
An actor can send and receive messages, make local decisions, create more actors, and fix the behavior to be used for the
next message it receives. These actions may be executed in parallel and in no fixed order. The Actor model was devised to
study the behavioral properties of parallel computing machines consisting of large numbers of independent processors
communicating by passing messages through a network. Other well-studied models of concurrent computation include Petri
nets and the process calculi such as pi-calculus and mu-calculus.
Many variants of computational models for distributed systems are being devised to study and understand the behaviors of
biological systems. For example, Dematte, Priami, and Romanel [2008] describe a language called BlenX that is based on a
process calculus called Beta-binders for modeling and simulating biological systems. We do not have the space to describe these concurrent models in any detail. However, it is still an open research area to find
practically useful concurrent models of computation that combine control and data for many areas of distributed computing.
Comprehensive envelope of expression not an identarian instance of semantic jugglery:
In addition to aiding education and understanding, there are many practical benefits to having appropriate models of
computation for the systems we are trying to build. In cloud computing, for example, there are still a host of poorly
understood concerns for systems of this scale. We need to better understand the architectural tradeoffs needed to achieve the
desired levels of reliability, performance, scalability and adaptivity in the services these systems are expected to provide. We
do not have appropriate abstractions to describe these properties in such a way that they can be automatically mapped from a
model of computation into an implementation (or the other way around).
In cloud computing, there are a host of research challenges for system developers and tool builders. As examples, we need
programming languages, compilers, verification tools, defect detection tools, and service management tools that can scale to
the huge number of clients and servers involved in the networks and data centers of the future. Cloud computing is one important area that can benefit from innovative computational thinking.
The Finale:
Mathematical abstractions called models of computation are at the heart of computation and computational thinking.
Computation is a process that is defined in terms of an underlying model of computation and computational thinking is the
thought processes involved in formulating problems so their solutions can be represented as computational steps and
algorithms. Useful models of computation for solving problems arising in sequential computation can range from
simple finite-state machines to Turing-complete models such as random access machines. Useful models of concurrent
computation for solving problems arising in the design and analysis of complex distributed systems are still a subject of
current research.
The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is
necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing
machine, introduced by Alan Turing in 1936 [Tur36]. Although the model was introduced before physical computers were
built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of
computable function.
Examples of Turing machines
3-state busy beaver
Formal definition
Hopcroft and Ullman (1979, p. 148) formally define a (one-tape) Turing machine as a 7-
tuple where
Is a finite, non-empty set of states
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Is a finite, non-empty set of the tape alphabet/symbols
is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step during the computation)
is the set of input symbols
is the initial state
is the set of final or accepting states.
is a partial function called the transition function, where L is left shift, R is right shift. (A relatively uncommon variant allows "no shift", say N, as a third element of the latter set.)
Anything that operates according to these specifications is a Turing machine.
The 7-tuple for the 3-state busy beaver looks like this (see more about this busy beaver at Turing machine examples):
("Blank")
(the initial state)
see state-table below
Initially all tape cells are marked with 0.
State table for 3 state, 2 symbol busy beaver
Tape symbol-Current state A-Current state B-Current state C -Write symbol-Move tape-Next state-Write symbol-Move tape-Next state-Write symbol-Move tape-Next state
0-1-R-B-1-L-A-1-L-B
1-1-L-C-1-R-B-1-R-HALT
In the words of van Emde Boas (1990), p. 6: "The set-theoretical object his formal seven-tuple description similar to the
above] provides only partial information on how the machine will behave and what its computations will look like."
For instance,
There will need to be some decision on what the symbols actually look like, and a failproof way of reading and writing
symbols indefinitely.
The shift left and shift right operations may shift the tape head across the tape, but when actually building a Turing machine
it is more practical to make the tape slide back and forth under the head instead.
The tape can be finite, and automatically extended with blanks as needed (which is closest to the mathematical definition), but it is more common to think of it as stretching infinitely at both ends and being pre-filled with blanks except on the
explicitly given finite fragment the tape head is on. (This is, of course, not implementable in practice.) The tape cannot be
fixed in length, since that would not correspond to the given definition and would seriously limit the range of computations
the machine can perform to those of alinear bounded automaton.
Contradictions and complementarities:
Definitions in literature sometimes differ slightly, to make arguments or proofs easier or clearer, but this is always done in
such a way that the resulting machine has the same computational power. For example, changing the set
to , where N ("None" or "No-operation") would allow the machine to stay on the same tape cell instead of moving left or right, does not increase the machine's computational power.
The most common convention represents each "Turing instruction" in a "Turing table" by one of nine 5-tuples, per the
convention of Turing/Davis (Turing (1936) in Undecidable, p. 126-127 and Davis (2000) p. 152):
(Definition 1): (qi, Sj, Sk/E/N, L/R/N, qm)
(Current state qi , symbol scanned Sj , print symbol Sk/erase E/none N , move_tape_one_square left L/right R/none N , new
state qm )
Other authors (Minsky (1967) p. 119, Hopcroft and Ullman (1979) p. 158, Stone (1972) p. 9) adopt a different convention,
with new state qm listed immediately after the scanned symbol Sj:
(Definition 2): (qi, Sj, qm, Sk/E/N, L/R/N)
(Current state qi , symbol scanned Sj , new state qm , print symbol Sk/erase E/none N , move_tape_one_square left L/right R/none N )
For the remainder of this article "definition 1" (the Turing/Davis convention) will be used.
Example: state table for the 3-state 2-symbol busy beaver reduced to 5-tuples
Current state-Scanned symbol--Print symbol-Move tape-Final (i.e. next) state-5-tuples
A-0--1-R-B-(A, 0, 1, R, B)
A-1--1-L-C-(A, 1, 1, L, C)
B-0--1-L-A-(B, 0, 1, L, A)
B-1--1-R-B-(B, 1, 1, R, B)
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C-0--1-L-B-(C, 0, 1, L, B)
C-1--1-N-H-(C, 1, 1, N, H)
In the following table, Turing's original model allowed only the first three lines that he called N1, N2, N3 (cf Turing
in Undecidable, p. 126). He allowed for erasure of the "scanned square" by naming a 0th symbol S0 = "erase" or "blank", etc. However, he did not allow for non-printing, so every instruction-line includes "print symbol Sk" or "erase" (cf footnote
12 in Post (1947), Undecidable p. 300). The abbreviations are Turing's (Undecidable p. 119). Subsequent to Turing's original
paper in 1936β1937, machine-models have allowed all nine possible types of five-tuples:
-Current m-configuration (Turing state)-Tape symbol-Print-operation-Tape-motion-Final m-configuration (Turing state)-5-
tuple-5-tuple comments-4-tuple
N1-qi-Sj-Print(Sk)-Left L-qm-(qi, Sj, Sk, L, qm)-"blank" = S0, 1=S1, etc.-
N2-qi-Sj-Print(Sk)-Right R-qm-(qi, Sj, Sk, R, qm)-"blank" = S0, 1=S1, etc.-
N3-qi-Sj-Print(Sk)-None N-qm-(qi, Sj, Sk, N, qm)-"blank" = S0, 1=S1, etc.-(qi, Sj, Sk, qm)
4-qi-Sj-None N-Left L-qm-(qi, Sj, N, L, qm)--(qi, Sj, L, qm)
5-qi-Sj-None N-Right R-qm-(qi, Sj, N, R, qm)--(qi, Sj, R, qm)
6-qi-Sj-None N-None N-qm-(qi, Sj, N, N, qm)-Direct "jump"-(qi, Sj, N, qm)
7-qi-Sj-Erase-Left L-qm-(qi, Sj, E, L, qm)-- 8-qi-Sj-Erase-Right R-qm-(qi, Sj, E, R, qm)--
9-qi-Sj-Erase-None N-qm-(qi, Sj, E, N, qm)--(qi, Sj, E, qm)
Any Turing table (list of instructions) can be constructed from the above nine 5-tuples. For technical reasons, the three non-
printing or "N" instructions (4, 5, 6) can usually be dispensed with. For examples see Turing machine examples.
Less frequently the use of 4-tuples is encountered: these represent a further atomization of the Turing instructions (cf Post
(1947), Boolos & Jeffrey (1974, 1999), Davis-Sigal-Weyuker (1994)); also see more at PostβTuring machine.
The "state"
The word "state" used in context of Turing machines can be a source of confusion, as it can mean two things. Most
commentators after Turing have used "state" to mean the name/designator of the current instruction to be performedβi.e. the
contents of the state register. But Turing (1936) made a strong distinction between a record of what he called the machine's
"m-configuration", (its internal state) and the machine's (or person's) "state of progress" through the computation - the current state of the total system. What Turing called "the state formula" includes both the current instruction and all the
symbols on the tape:
Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the
symbols on the tape. That is, the state of the system may be described by a single expression (sequence of symbols)
consisting of the symbols on the tape followed by Ξ (which we suppose not to appear elsewhere) and then by the note of
instructions. This expression is called the 'state formula'.
βUndecidable, p.139β140, emphasis added
Earlier in his paper Turing carried this even further: he gives an example where he places a symbol of the current "m-
configuration"βthe instruction's labelβbeneath the scanned square, together with all the symbols on the tape (Undecidable,
p. 121); this he calls "the complete configuration" (Undecidable, p. 118). To print the "complete configuration" on one line
he places the state-label/m-configuration to the left of the scanned symbol.
A variant of this is seen in Kleene (1952) where Kleene shows how to write the GΓΆdel number of a machine's "situation": he places the "m-configuration" symbol q4 over the scanned square in roughly the center of the 6 non-blank squares on the tape
(see the Turing-tape figure in this article) and puts it to the right of the scanned square. But Kleene refers to "q4" itself as
"the machine state" (Kleene, p. 374-375). Hopcroft and Ullman call this composite the "instantaneous description" and
follow the Turing convention of putting the "current state" (instruction-label, m-configuration) to the left of the scanned
symbol (p. 149).
Example: total state of 3-state 2-symbol busy beaver after 3 "moves" (taken from example "run" in the figure below):
1A1
This means: after three moves the tape has ... 000110000 ... on it, the head is scanning the right-most 1, and the state is A.
Blanks (in this case represented by "0"s) can be part of the total state as shown here: B01 ; the tape has a single 1 on it, but
the head is scanning the 0 ("blank") to its left and the state is B.
"State" in the context of Turing machines should be clarified as to which is being described: (i) the current instruction, or (ii) the list of symbols on the tape together with the current instruction, or (iii) the list of symbols on the tape together with the
current instruction placed to the left of the scanned symbol or to the right of the scanned symbol.
Turing's biographer Andrew Hodges (1983: 107) has noted and discussed this confusion.
Turing machine "state" diagrams
The table for the 3-state busy beaver ("P" = print/write a "1")
Tape symbol-Current state A-Current state B-Current state C
-Write symbol-Move tape-Next state-Write symbol-Move tape-Next state-Write symbol-Move tape-Next state
0-P-R-B-P-L-A-P-L-B
1-P-L-C-P-R-B-P-R-HALT
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The "3-state busy beaver" Turing machine in a finite state representation. Each circle represents a "state" of the
TABLEβan "m-configuration" or "instruction". "Direction" of a state transition is shown by an arrow. The label
(e.g. 0/P, R) near the outgoing state (at the "tail" of the arrow) specifies the scanned symbol that causes a particular
transition (e.g. 0) followed by a slash /, followed by the subsequent "behaviors" of the machine, e.g. "P Print" then
move tape "R Right". No general accepted format exists. The convention shown is after McClusky (1965), Booth
(1967), Hill and Peterson (1974).
To the right: the above TABLE as expressed as a "state transition" diagram.
Usually large TABLES are better left as tables (Booth, p. 74). They are more readily simulated by computer in tabular form (Booth, p. 74). However, certain conceptsβe.g. machines with "reset" states and machines with repeating patterns (cf Hill
and Peterson p. 244ff)βcan be more readily seen when viewed as a drawing.
Whether a drawing represents an improvement on its TABLE must be decided by the reader for the particular context.
See Finite state machine for more.
The evolution of the busy-beaver's computation starts at the top and proceeds to the bottom.
The reader should again be cautioned that such diagrams represent a snapshot of their TABLE frozen in time, not the course
("trajectory") of a computation through time and/or space. While every time the busy beaver machine "runs" it will always
follow the same state-trajectory, this is not true for the "copy" machine that can be provided with variable input "parameters".
The diagram "Progress of the computation" shows the 3-state busy beaver's "state" (instruction) progress through its
computation from start to finish. On the far right is the Turing "complete configuration" (Kleene "situation", Hopcroftβ
Ullman "instantaneous description") at each step. If the machine were to be stopped and cleared to blank both the "state
register" and entire tape, these "configurations" could be used to rekindle a computation anywhere in its progress (cf Turing
(1936) Undecidable pp. 139β140).
Register machine,
Machines that might be thought to have more computational capability than a simple universal Turing machine can be
shown to have no more power (Hopcroft and Ullman p. 159, cf Minsky (1967)). They might compute faster, perhaps, or use
less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical
functions). (Recall that the ChurchβTuring thesis hypothesizes this to be true for any kind of machine: that anything that can be "computed" can be computed by some Turing machine.)
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A Turing machine is equivalent to a pushdown automaton that has been made more flexible and concise by relaxing the last-
in-first-out requirement of its stack.
At the other extreme, some very simple models turn out to be Turing-equivalent, i.e. to have the same computational power
as the Turing machine model. Common equivalent models are the multi-tape Turing machine, multi-track Turing machine, machines with input and output,
and the non-deterministic Turing machine(NDTM) as opposed to the deterministic Turing machine (DTM) for which the
action table has at most one entry for each combination of symbol and state.
Read-only, right-moving Turing machines are equivalent to NDFAs (as well as DFAs by conversion using the NDFA to
DFA conversion algorithm).
For practical and didactical intentions the equivalent register machine can be used as a usual assembly programming
language.
Choice c-machines, Oracle o-machines
Early in his paper (1936) Turing makes a distinction between an "automatic machine"βits "motion ... completely
determined by the configuration" and a "choice machine":
...whose motion is only partially determined by the configuration ... When such a machine reaches one of these ambiguous configurations; it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if
we were using machines to deal with axiomatic systems.
βUndecidable, p. 118
Turing (1936) does not elaborate further except in a footnote in which he describes how to use an a-machine to "find all the
provable formulae of the [Hilbert] calculus" rather than use a choice machine. He "supposes[s] that the choices are always
between two possibilities 0 and 1. Each proof will then be determined by a sequence of choices i1, i2, ..., in (i1 = 0 or 1, i2 =
0 or 1, ..., in = 0 or 1), and hence the number 2n + i12n-1 + i22n-2 + ... +in completely determines the proof. The automatic
machine carries out successively proof 1, proof 2, proof 3, ..." (Footnote β‘, Undecidable, p. 138)
This is indeed the technique by which a deterministic (i.e. a-) Turing machine can be used to mimic the action of
a nondeterministic Turing machine; Turing solved the matter in a footnote and appears to dismiss it from further
consideration. An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its
calculation, it "awaits the decision" of "the oracle"βan unspecified entity "apart from saying that it cannot be a machine"
(Turing (1939), Undecidable p. 166β168). The concept is now actively used by mathematicians.
Universal Turing machines
As Turing wrote in Undecidable, p. 128 (italics added):
It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is
supplied with the tape on the beginning of which is written the string of quintuples separated by semicolons of some
computing machine M, then U will compute the same sequence as M.
This finding is now taken for granted, but at the time (1936) it was considered astonishing. The model of computation that
Turing called his "universal machine"β"U" for shortβis considered by some (cf Davis (2000)) to have been the
fundamental theoretical breakthrough that led to the notion of the Stored-program computer. Turing's paper ... contains, in essence, the invention of the modern computer and some of the programming techniques that
accompanied it.
βMinsky (1967), p. 104
In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor
compared to the machines it simulates. This result was obtained in 1966 by F. C. Hennie and R. E. Stearns. (Arora and
Barak, 2009, theorem 1.9)
Comparison with real machines
It is often said that Turing machines, unlike simpler automata, are as powerful as real machines, and are able to execute any
operation that a real program can. What is missed in this statement is that, because a real machine can only be in finitely
many configurations, in fact this "real machine" is nothing but a linear bounded automaton. On the other hand, Turing machines are equivalent to machines that have an unlimited amount of storage space for their computations. In fact, Turing
machines are not intended to model computers, but rather they are intended to model computation itself; historically,
computers, which compute only on their (fixed) internal storage, were developed only later.
There are a number of ways to explain why Turing machines are useful models of real computers:
Anything a real computer can compute, a Turing machine can also compute. For example: "A Turing machine can simulate
any type of subroutine found in programming languages, including recursive procedures and any of the known parameter-
passing mechanisms" (Hopcroft and Ullman p. 157). A large enough FSA can also model any real computer, disregarding
IO. Thus, a statement about the limitations of Turing machines will also apply to real computers.
The difference lies only with the ability of a Turing machine to manipulate an unbounded amount of data. However, given a
finite amount of time, a Turing machine (like a real machine) can only manipulate a finite amount of data.
Like a Turing machine, a real machine can have its storage space enlarged as needed, by acquiring more disks or other storage media. If the supply of these runs short, the Turing machine may become less useful as a model. But the fact is that
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neither Turing machines nor real machines need astronomical amounts of storage space in order to perform useful
computation. The processing time required is usually much more of a problem.
Descriptions of real machine programs using simpler abstract models are often much more complex than descriptions using
Turing machines. For example, a Turing machine describing an algorithm may have a few hundred states, while the equivalent deterministic finite automaton on a given real machine has quadrillions. This makes the DFA representation
infeasible to analyze.
Turing machines describe algorithms independent of how much memory they use. There is a limit to the memory possessed
by any current machine, but this limit can rise arbitrarily in time. Turing machines allow us to make statements about
algorithms which will (theoretically) hold forever, regardless of advances in conventional computing machine architecture.
Turing machines simplify the statement of algorithms. Algorithms running on Turing-equivalent abstract machines are
usually more general than their counterparts running on real machines, because they have arbitrary-precision data types
available and never have to deal with unexpected conditions (including, but not limited to, running out of memory).
One way in which Turing machines are a poor model for programs is that many real programs, such as operating
systems and word processors, are written to receive unbounded input over time, and therefore do not halt. Turing machines
do not model such ongoing computation well (but can still model portions of it, such as individual procedures).
Computational complexity theory
A limitation of Turing machines is that they do not model the strengths of a particular arrangement well. For instance,
modern stored-program computers are actually instances of a more specific form of abstract machine known as the random
access stored program machine or RASP machine model. Like the Universal Turing machine the RASP stores its "program"
in "memory" external to its finite-state machine's "instructions". Unlike the universal Turing machine, the RASP has an
infinite number of distinguishable, numbered but unbounded "registers"βmemory "cells" that can contain any integer (cf.
Elgot and Robinson (1964), Hartmanis (1971), and in particular Cook-Rechow (1973); references at random access
machine). The RASP's finite-state machine is equipped with the capability for indirect addressing (e.g. the contents of one
register can be used as an address to specify another register); thus the RASP's "program" can address any register in the
register-sequence. The upshot of this distinction is that there are computational optimizations that can be performed based on
the memory indices, which are not possible in a general Turing machine; thus when Turing machines are used as the basis for bounding running times, a 'false lower bound' can be proven on certain algorithms' running times (due to the false
simplifying assumption of a Turing machine). An example of this is binary search, an algorithm that can be shown to
perform more quickly when using the RASP model of computation rather than the Turing machine model.
Concurrency
Another limitation of Turing machines is that they do not model concurrency well. For example, there is a bound on the size
of integer that can be computed by an always-halting nondeterministic Turing machine starting on a blank tape. (See article
on unbounded nondeterminism.) By contrast, there are always-halting concurrent systems with no inputs that can compute an
integer of unbounded size. (A process can be created with local storage that is initialized with a count of 0 that concurrently
sends itself both a stop and a go message. When it receives a go message, it increments its count by 1 and sends itself a go
message. When it receives a stop message, it stops with an unbounded number in its local storage.)
βAβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED ONE
NOTATION :
πΊ13 : CATEGORY ONE OFβAβ
πΊ14 : CATEGORY TWO OFβAβ
πΊ15 : CATEGORY THREE OF βAβ
π13 : CATEGORY ONE OF βBβ
π14 : CATEGORY TWO OF βBβ
π15 :CATEGORY THREE OF βBβ
βBβ AND βAβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED TWO
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO.
=================================================================
πΊ16 : CATEGORY ONE OF βBβ (NOTE THAT THEY REPRESENT CONFIGURATIONS,INSTRUCTIONS OR
STATES)
πΊ17 : CATEGORY TWO OF βBβ
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πΊ18 : CATEGORY THREE OF βBβ
π16 :CATEGORY ONE OF βAβ
π17 : CATEGORY TWO OF βAβ π18 : CATEGORY THREE OFβAβ
=============================================================================
Aβ AND βCβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED THREE
=======================================================================
πΊ20 : CATEGORY ONE OFβAβ
πΊ21 :CATEGORY TWO OFβAβ
πΊ22 : CATEGORY THREE OFβAβ
π20 : CATEGORY ONE OF βCβ
π21 :CATEGORY TWO OF βCβ
π22 : CATEGORY THREE OF βCβ
βCβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED FOUR
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO :
============================================================================
πΊ24 : CATEGORY ONE OF βCβ(EVALUATIVE PARAMETRICIZATION OF SITUATIONAL ORIENTATIONS AND
ESSENTIAL COGNITIVE ORIENTATION AND CHOICE VARIABLES OF THE SYSTEM TO WHICH
CONFIGURATION IS APPLICABLE)
πΊ25 : CATEGORY TWO OF βCβ
πΊ26 : CATEGORY THREE OF βCβ
π24 :CATEGORY ONE OF βBβ
π25 :CATEGORY TWO OF βBβ(SYSTEMIC INSTRUMENTAL CHARACTERISATIONS AND ACTION
ORIENTATIONS AND FUNCTIONAL IMPERATIVES OF CHANGE MANIFESTED THEREIN ) π26 : CATEGORY THREE OFβBβ
βCβ AND βHβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED FIVE
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO:
=============================================================================
πΊ28 : CATEGORY ONE OF βCβ
πΊ29 : CATEGORY TWO OFβCβ
πΊ30 :CATEGORY THREE OF βCβ
π28 :CATEGORY ONE OF βHβ
π29 :CATEGORY TWO OF βHβ
π30 :CATEGORY THREE OF βHβ
=========================================================================
βBβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
THE SYSTEM HERE IS ONE OF SELF TRANSFORMATIONAL,SYSTEM CHANGING,STRUCTURALLY
MUTATIONAL,SYLLOGISTICALLY CHANGEABLE AND CONFIGURATIONALLY ALTERABLE(VERY
VERY IMPORTANT SYSTEM IN ALMOST ALL SUBJECTS BE IT IN QUANTUM SYSTEMS OR
DISSIPATIVE STRUCTURES
MODULE NUMBERED SIX
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
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EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO:
=============================================================================
πΊ32 : CATEGORY ONE OF βBβ
πΊ33 : CATEGORY TWO OF βBβ
πΊ34 : CATEGORY THREE OFβBβ
INTERACTS WITH:ITSELF:
π32 : CATEGORY ONE OF βBβ
π33 : CATEGORY TWO OFβBβ π34 : CATEGORY THREE OF βBβ
βINPUTβ AND βAβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR
STATE: THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED SEVEN
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO:
==========================================================================
πΊ36 : CATEGORY ONE OF βINPUTβ
πΊ37 : CATEGORY TWO OF βINPUTβ
πΊ38 : CATEGORY THREE OFβINPUTβ (INPUT FEEDING AND CONCOMITANT GENERATION OF ENERGY DIFFERENTIAL-TIME LAG OR INSTANTANEOUSNESSMIGHT EXISTS WHEREBY ACCENTUATION AND
ATTRITIONS MODEL MAY ASSUME ZERO POSITIONS)
π36 : CATEGORY ONE OF βAβ
π37 : CATEGORY TWO OF "A" π38 : CATEGORY THREE OFβAβ
===============================================================================
π13 1 , π14
1 , π15 1 , π13
1 , π14 1 , π15
1 π16 2 , π17
2 , π18 2 π16
2 , π17 2 , π18
2 :
π20 3 , π21
3 , π22 3 , π20
3 , π21 3 , π22
3
π24 4 , π25
4 , π26 4 , π24
4 , π25 4 , π26
4 , π28 5 , π29
5 , π30 5 , π28
5 , π29 5 , π30
5 , π32
6 , π33 6 , π34
6 , π32 6 , π33
6 , π34 6
are Accentuation coefficients
π13β² 1 , π14
β² 1 , π15β²
1 , π13
β² 1 , π14β² 1 , π15
β² 1
, π16β² 2 , π17
β² 2 , π18β² 2 , π16
β² 2 , π17β² 2 , π18
β² 2
, π20β² 3 , π21
β² 3 , π22β² 3 , π20
β² 3 , π21β² 3 , π22
β² 3
π24β² 4 , π25
β² 4
, π26β² 4 , π24
β² 4 , π25β²
4 , π26
β² 4 , π28β² 5 , π29
β² 5 , π30β² 5 π28
β² 5 , π29β² 5 , π30
β² 5 ,
π32β² 6 , π33
β² 6 , π34β² 6 , π32
β² 6 , π33β² 6 , π34
β² 6
are Dissipation coefficients-
βAβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED ONE
The differential system of this model is now (Module Numbered one)-1 ππΊ13
ππ‘= π13
1 πΊ14 β π13β² 1 + π13
β²β² 1 π14 , π‘ πΊ13 -2 ππΊ14
ππ‘= π14
1 πΊ13 β π14β² 1 + π14
β²β² 1 π14 , π‘ πΊ14 -3
ππΊ15
ππ‘= π15
1 πΊ14 β π15β²
1 + π15
β²β² 1
π14 , π‘ πΊ15 -4
ππ13
ππ‘= π13
1 π14 β π13β² 1 β π13
β²β² 1 πΊ, π‘ π13 -5 ππ14
ππ‘= π14
1 π13 β π14β² 1 β π14
β²β² 1 πΊ, π‘ π14 -6
ππ15
ππ‘= π15
1 π14 β π15β²
1 β π15
β²β² 1
πΊ, π‘ π15 -7
+ π13β²β² 1 π14 , π‘ = First augmentation factor -8
β π13β²β² 1 πΊ, π‘ = First detritions factor-
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βBβ AND βAβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED TWO
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO.
The differential system of this model is now ( Module numbered two)-9 ππΊ16
ππ‘= π16
2 πΊ17 β π16β² 2 + π16
β²β² 2 π17 , π‘ πΊ16 -10 ππΊ17
ππ‘= π17
2 πΊ16 β π17β² 2 + π17
β²β² 2 π17 , π‘ πΊ17 -11 ππΊ18
ππ‘= π18
2 πΊ17 β π18β² 2 + π18
β²β² 2 π17 , π‘ πΊ18 -12 ππ16
ππ‘= π16
2 π17 β π16β² 2 β π16
β²β² 2 πΊ19 , π‘ π16 -13 ππ17
ππ‘= π17
2 π16 β π17β² 2 β π17
β²β² 2 πΊ19 , π‘ π17 -14 ππ18
ππ‘= π18
2 π17 β π18β² 2 β π18
β²β² 2 πΊ19 , π‘ π18 -15
+ π16β²β² 2 π17 , π‘ = First augmentation factor -16
β π16β²β² 2 πΊ19 , π‘ = First detritions factor -17
:
============================================================================= Aβ AND
βCβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE: THE
CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED THREE
The differential system of this model is now (Module numbered three)-18 ππΊ20
ππ‘= π20
3 πΊ21 β π20β² 3 + π20
β²β² 3 π21 , π‘ πΊ20 -19 ππΊ21
ππ‘= π21
3 πΊ20 β π21β² 3 + π21
β²β² 3 π21 , π‘ πΊ21 -20 ππΊ22
ππ‘= π22
3 πΊ21 β π22β² 3 + π22
β²β² 3 π21 , π‘ πΊ22 -21 ππ20
ππ‘= π20
3 π21 β π20β² 3 β π20
β²β² 3 πΊ23 , π‘ π20 -22 ππ21
ππ‘= π21
3 π20 β π21β² 3 β π21
β²β² 3 πΊ23 , π‘ π21 -23 ππ22
ππ‘= π22
3 π21 β π22β² 3 β π22
β²β² 3 πΊ23 , π‘ π22 -24
+ π20β²β² 3 π21 , π‘ = First augmentation factor-
β π20β²β² 3 πΊ23 , π‘ = First detritions factor -25
βCβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED FOUR
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO
The differential system of this model is now (Module numbered Four)-26 ππΊ24
ππ‘= π24
4 πΊ25 β π24β² 4 + π24
β²β² 4 π25 , π‘ πΊ24 -27
ππΊ25
ππ‘= π25
4 πΊ24 β π25β²
4 + π25
β²β² 4
π25 , π‘ πΊ25 -28 ππΊ26
ππ‘= π26
4 πΊ25 β π26β² 4 + π26
β²β² 4 π25 , π‘ πΊ26 -29 ππ24
ππ‘= π24
4 π25 β π24β² 4 β π24
β²β² 4 πΊ27 , π‘ π24 -30
ππ25
ππ‘= π25
4 π24 β π25β²
4 β π25
β²β² 4
πΊ27 , π‘ π25 -31
ππ26
ππ‘= π26
4 π25 β π26β² 4 β π26
β²β² 4 πΊ27 , π‘ π26 -32
+ π24β²β² 4 π25 , π‘ = First augmentation factor-33
β π24β²β² 4 πΊ27 , π‘ = First detritions factor -34
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βCβ AND βHβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED FIVE
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO
The differential system of this model is now (Module number five)-35 ππΊ28
ππ‘= π28
5 πΊ29 β π28β² 5 + π28
β²β² 5 π29 , π‘ πΊ28 -36 ππΊ29
ππ‘= π29
5 πΊ28 β π29β² 5 + π29
β²β² 5 π29 , π‘ πΊ29 -37 ππΊ30
ππ‘= π30
5 πΊ29 β π30β² 5 + π30
β²β² 5 π29 , π‘ πΊ30 -38 ππ28
ππ‘= π28
5 π29 β π28β² 5 β π28
β²β² 5 πΊ31 , π‘ π28 -39 ππ29
ππ‘= π29
5 π28 β π29β² 5 β π29
β²β² 5 πΊ31 , π‘ π29 -40 ππ30
ππ‘= π30
5 π29 β π30β² 5 β π30
β²β² 5 πΊ31 , π‘ π30 -41
+ π28β²β² 5 π29 , π‘ = First augmentation factor -42
β π28β²β² 5 πΊ31 , π‘ = First detritions factor -43
Bβ AND βBβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR STATE:
THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
THE SYSTEM HERE IS ONE OF SELF TRANSFORMATIONAL,SYSTEM CHANGING,STRUCTURALLY
MUTATIONAL,SYLLOGISTICALLY CHANGEABLE AND CONFIGURATIONALLY ALTERABLE(VERY
VERY IMPORTANT SYSTEM IN ALMOST ALL SUBJECTS BE IT IN QUANTUM SYSTEMS OR
DISSIPATIVE STRUCTURES
MODULE NUMBERED SIX
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO
:
The differential system of this model is now (Module numbered Six)-44
45 ππΊ32
ππ‘= π32
6 πΊ33 β π32β² 6 + π32
β²β² 6 π33 , π‘ πΊ32 -46 ππΊ33
ππ‘= π33
6 πΊ32 β π33β² 6 + π33
β²β² 6 π33 , π‘ πΊ33 -47 ππΊ34
ππ‘= π34
6 πΊ33 β π34β² 6 + π34
β²β² 6 π33 , π‘ πΊ34 -48 ππ32
ππ‘= π32
6 π33 β π32β² 6 β π32
β²β² 6 πΊ35 , π‘ π32 -49 ππ33
ππ‘= π33
6 π32 β π33β² 6 β π33
β²β² 6 πΊ35 , π‘ π33 -50 ππ34
ππ‘= π34
6 π33 β π34β² 6 β π34
β²β² 6 πΊ35 , π‘ π34 -51
+ π32β²β² 6 π33 , π‘ = First augmentation factor-52
βINPUTβ AND βAβ(SEE FIGURE REPRESENTS AN βM CONFIGURATIONβ OR βINSTRUCTIONS) OR
STATE: THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.
MODULE NUMBERED SEVEN
NOTE: THE ACCENTUATION COEFFICIENT AND DISSIPATION COEFFICIENT NEED NOT BE THE SAME
,IT MAY BE ZERO, OR MIGHT BE SAME,I ALL THE THREE CASES THE MODEL COULD BE CHANGED
EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF
ZERO
=============================================================================
: The differential system of this model is now (SEVENTH MODULE)
-53 ππΊ36
ππ‘= π36
7 πΊ37 β π36β² 7 + π36
β²β² 7 π37 , π‘ πΊ36 -54 ππΊ37
ππ‘= π37
7 πΊ36 β π37β² 7 + π37
β²β² 7 π37 , π‘ πΊ37 -55
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ππΊ38
ππ‘= π38
7 πΊ37 β π38β² 7 + π38
β²β² 7 π37 , π‘ πΊ38 -56 ππ36
ππ‘= π36
7 π37 β π36β² 7 β π36
β²β² 7 πΊ39 , π‘ π36 -57 ππ37
ππ‘= π37
7 π36 β π37β² 7 β π37
β²β² 7 πΊ39 , π‘ π37 -58
59 ππ38
ππ‘= π38
7 π37 β π38β² 7 β π38
β²β² 7 πΊ39 , π‘ π38 -60
+ π36β²β² 7 π37 , π‘ = First augmentation factor -61
β π36β²β² 7 πΊ39 , π‘ = First detritions factor
FIRST MODULE CONCATENATION:
ππΊ13
ππ‘= π13
1 πΊ14 β
π13
β² 1 + π13β²β² 1 π14 , π‘ + π16
β²β² 2,2, π17 , π‘ + π20β²β² 3,3, π21 , π‘
+ π24β²β² 4,4,4,4, π25 , π‘ + π28
β²β² 5,5,5,5, π29 , π‘ + π32β²β² 6,6,6,6, π33 , π‘
+ π36β²β² 7 π37 , π‘
πΊ13
ππΊ14
ππ‘= π14
1 πΊ13 β
π14β² 1 + π14
β²β² 1 π14 , π‘ + π17β²β² 2,2, π17 , π‘ + π21
β²β² 3,3, π21 , π‘
+ π25β²β²
4,4,4,4, π25 , π‘ + π29
β²β² 5,5,5,5, π29 , π‘ + π33β²β² 6,6,6,6, π33 , π‘
+ π37β²β² 7 π37 , π‘
πΊ14
ππΊ15
ππ‘= π15
1 πΊ14 β
π15
β² 1
+ π15β²β²
1 π14 , π‘ + π18
β²β² 2,2, π17 , π‘ + π22β²β² 3,3, π21 , π‘
+ π26β²β² 4,4,4,4, π25 , π‘ + π30
β²β² 5,5,5,5, π29 , π‘ + π34β²β² 6,6,6,6, π33 , π‘
+ π38β²β² 7 π37 , π‘
πΊ15
Where π13β²β² 1 π14 , π‘ , π14
β²β² 1 π14 , π‘ , π15β²β²
1 π14 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π16β²β² 2,2, π17 , π‘ , + π17
β²β² 2,2, π17 , π‘ , + π18β²β² 2,2, π17 , π‘ are second augmentation coefficient for category 1, 2 and
3
+ π20β²β² 3,3, π21 , π‘ , + π21
β²β² 3,3, π21 , π‘ , + π22β²β² 3,3, π21 , π‘ are third augmentation coefficient for category 1, 2 and 3
+ π24β²β² 4,4,4,4, π25 , π‘ , + π25
β²β² 4,4,4,4,
π25 , π‘ , + π26β²β² 4,4,4,4, π25 , π‘ are fourth augmentation coefficient for category 1,
2 and 3
+ π28β²β² 5,5,5,5, π29 , π‘ , + π29
β²β² 5,5,5,5, π29 , π‘ , + π30β²β² 5,5,5,5, π29 , π‘ are fifth augmentation coefficient for category 1, 2
and 3
+ π32β²β² 6,6,6,6, π33 , π‘ , + π33
β²β² 6,6,6,6, π33 , π‘ , + π34β²β² 6,6,6,6, π33 , π‘ are sixth augmentation coefficient for category 1, 2
and 3
+ π36β²β² 7 π37 , π‘ + π37
β²β² 7 π37 , π‘ + π38β²β² 7 π37 , π‘ ARESEVENTHAUGMENTATION COEFFICIENTS
ππ13
ππ‘= π13
1 π14 β
π13β² 1 β π16
β²β² 1 πΊ, π‘ β π36β²β² 7, πΊ39 , π‘ β π20
β²β² 3,3, πΊ23 , π‘
β π24β²β² 4,4,4,4, πΊ27 , π‘ β π28
β²β² 5,5,5,5, πΊ31 , π‘ β π32β²β² 6,6,6,6, πΊ35 , π‘
β π36β²β² 7, πΊ39 , π‘
π13
ππ14
ππ‘= π14
1 π13 β
π14
β² 1 β π14β²β² 1 πΊ, π‘ β π17
β²β² 2,2, πΊ19 , π‘ β π21β²β² 3,3, πΊ23 , π‘
β π25β²β²
4,4,4,4, πΊ27 , π‘ β π29
β²β² 5,5,5,5, πΊ31 , π‘ β π33β²β² 6,6,6,6, πΊ35 , π‘
β π37β²β² 7, πΊ39 , π‘
π14
ππ15
ππ‘= π15
1 π14 β
π15
β² 1
β π15β²β²
1 πΊ, π‘ β π18
β²β² 2,2, πΊ19 , π‘ β π22β²β² 3,3, πΊ23 , π‘
β π26β²β² 4,4,4,4, πΊ27 , π‘ β π30
β²β² 5,5,5,5, πΊ31 , π‘ β π34β²β² 6,6,6,6, πΊ35 , π‘
β π38β²β² 7, πΊ39 , π‘
π15
Where β π13β²β² 1 πΊ, π‘ , β π14
β²β² 1 πΊ, π‘ , β π15β²β²
1 πΊ, π‘ are first detritions coefficients for category 1, 2 and 3
β π16β²β² 2,2, πΊ19 , π‘ , β π17
β²β² 2,2, πΊ19 , π‘ , β π18β²β² 2,2, πΊ19 , π‘ are second detritions coefficients for category 1, 2 and 3
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β π20β²β² 3,3, πΊ23 , π‘ , β π21
β²β² 3,3, πΊ23 , π‘ , β π22β²β² 3,3, πΊ23 , π‘ are third detritions coefficients for category 1, 2 and 3
β π24β²β² 4,4,4,4, πΊ27 , π‘ , β π25
β²β² 4,4,4,4,
πΊ27 , π‘ , β π26β²β² 4,4,4,4, πΊ27 , π‘ are fourth detritions coefficients for category 1, 2
and 3
β π28β²β² 5,5,5,5, πΊ31 , π‘ , β π29
β²β² 5,5,5,5, πΊ31 , π‘ , β π30β²β² 5,5,5,5, πΊ31 , π‘ are fifth detritions coefficients for category 1, 2
and 3
β π32β²β² 6,6,6,6, πΊ35 , π‘ , β π33
β²β² 6,6,6,6, πΊ35 , π‘ , β π34β²β² 6,6,6,6, πΊ35 , π‘ are sixth detritions coefficients for category 1, 2
and 3
β π36β²β² 7, πΊ39 , π‘ β π36
β²β² 7, πΊ39 , π‘ β π36β²β² 7, πΊ39 , π‘ ARE SEVENTH DETRITION COEFFICIENTS
-62
ππ15
ππ‘= π15
1 π14 β π15
β² 1
β π15β²β²
1 πΊ, π‘ β π18
β²β² 2,2, πΊ19 , π‘ β π22β²β² 3,3, πΊ23 , π‘
β π26β²β² 4,4,4,4, πΊ27 , π‘ β π30
β²β² 5,5,5,5, πΊ31 , π‘ β π34β²β² 6,6,6,6, πΊ35 , π‘
π15 -63
Where β π13β²β² 1 πΊ, π‘ , β π14
β²β² 1 πΊ, π‘ , β π15β²β²
1 πΊ, π‘ are first detrition coefficients for category 1, 2 and 3
β π16β²β² 2,2, πΊ19 , π‘ , β π17
β²β² 2,2, πΊ19 , π‘ , β π18β²β² 2,2, πΊ19 , π‘ are second detritions coefficients for category 1, 2 and 3
β π20β²β² 3,3, πΊ23 , π‘ , β π21
β²β² 3,3, πΊ23 , π‘ , β π22β²β² 3,3, πΊ23 , π‘ are third detritions coefficients for category 1, 2 and 3
β π24β²β² 4,4,4,4, πΊ27 , π‘ , β π25
β²β² 4,4,4,4,
πΊ27 , π‘ , β π26β²β² 4,4,4,4, πΊ27 , π‘ are fourth detritions coefficients for category 1, 2
and 3
β π28β²β² 5,5,5,5, πΊ31 , π‘ , β π29
β²β² 5,5,5,5, πΊ31 , π‘ , β π30β²β² 5,5,5,5, πΊ31 , π‘ are fifth detritions coefficients for category 1, 2
and 3
β π32β²β² 6,6,6,6, πΊ35 , π‘ , β π33
β²β² 6,6,6,6, πΊ35 , π‘ , β π34β²β² 6,6,6,6, πΊ35 , π‘ are sixth detritions coefficients for category 1, 2
and 3 -64
SECOND MODULE CONCATENATION:-
65ππΊ16
ππ‘= π16
2 πΊ17 β
π16β² 2 + π16
β²β² 2 π17 , π‘ + π13β²β² 1,1, π14 , π‘ + π20
β²β² 3,3,3 π21 , π‘
+ π24β²β² 4,4,4,4,4 π25 , π‘ + π28
β²β² 5,5,5,5,5 π29 , π‘ + π32β²β² 6,6,6,6,6 π33 , π‘
+ π36β²β² 7,7, π37 , π‘
πΊ16 -66
ππΊ17
ππ‘= π17
2 πΊ16 β
π17β² 2 + π17
β²β² 2 π17 , π‘ + π14β²β² 1,1, π14 , π‘ + π21
β²β² 3,3,3 π21 , π‘
+ π25β²β²
4,4,4,4,4 π25 , π‘ + π29
β²β² 5,5,5,5,5 π29 , π‘ + π33β²β² 6,6,6,6,6 π33 , π‘
+ π37β²β² 7,7, π37 , π‘
πΊ17 -67
ππΊ18
ππ‘= π18
2 πΊ17 β
π18β² 2 + π18
β²β² 2 π17 , π‘ + π15β²β²
1,1, π14 , π‘ + π22
β²β² 3,3,3 π21 , π‘
+ π26β²β² 4,4,4,4,4 π25 , π‘ + π30
β²β² 5,5,5,5,5 π29 , π‘ + π34β²β² 6,6,6,6,6 π33 , π‘
+ π38β²β² 7,7, π37 , π‘
πΊ18 -68
Where + π16β²β² 2 π17 , π‘ , + π17
β²β² 2 π17 , π‘ , + π18β²β² 2 π17 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π13β²β² 1,1, π14 , π‘ , + π14
β²β² 1,1, π14 , π‘ , + π15β²β²
1,1, π14 , π‘ are second augmentation coefficient for category 1, 2 and 3
+ π20β²β² 3,3,3 π21 , π‘ , + π21
β²β² 3,3,3 π21 , π‘ , + π22β²β² 3,3,3 π21 , π‘ are third augmentation coefficient for category 1, 2 and
3
+ π24β²β² 4,4,4,4,4 π25 , π‘ , + π25
β²β² 4,4,4,4,4
π25 , π‘ , + π26β²β² 4,4,4,4,4 π25 , π‘ are fourth augmentation coefficient for category
1, 2 and 3
+ π28β²β² 5,5,5,5,5 π29 , π‘ , + π29
β²β² 5,5,5,5,5 π29 , π‘ , + π30β²β² 5,5,5,5,5 π29 , π‘ are fifth augmentation coefficient for category
1, 2 and 3
+ π32β²β² 6,6,6,6,6 π33 , π‘ , + π33
β²β² 6,6,6,6,6 π33 , π‘ , + π34β²β² 6,6,6,6,6 π33 , π‘ are sixth augmentation coefficient for category
1, 2 and 3 -69
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70
+ π36β²β² 7,7, π37 , π‘ + π37
β²β² 7,7, π37 , π‘ + π38β²β² 7,7, π37 , π‘ ARE SEVENTH DETRITION COEFFICIENTS-71
ππ16
ππ‘= π16
2 π17 β
π16β² 2 β π16
β²β² 2 πΊ19 , π‘ β π13β²β² 1,1, πΊ, π‘ β π20
β²β² 3,3,3, πΊ23 , π‘
β π24β²β² 4,4,4,4,4 πΊ27 , π‘ β π28
β²β² 5,5,5,5,5 πΊ31 , π‘ β π32β²β² 6,6,6,6,6 πΊ35 , π‘
β π36β²β² 7,7 πΊ39 , π‘
π16 -72
ππ17
ππ‘= π17
2 π16 β
π17β² 2 β π17
β²β² 2 πΊ19 , π‘ β π14β²β² 1,1, πΊ, π‘ β π21
β²β² 3,3,3, πΊ23 , π‘
β π25β²β²
4,4,4,4,4 πΊ27 , π‘ β π29
β²β² 5,5,5,5,5 πΊ31 , π‘ β π33β²β² 6,6,6,6,6 πΊ35 , π‘
β π37β²β² 7,7 πΊ39 , π‘
π17 -73
ππ18
ππ‘= π18
2 π17 β
π18β² 2 β π18
β²β² 2 πΊ19 , π‘ β π15β²β²
1,1, πΊ, π‘ β π22
β²β² 3,3,3, πΊ23 , π‘
β π26β²β² 4,4,4,4,4 πΊ27 , π‘ β π30
β²β² 5,5,5,5,5 πΊ31 , π‘ β π34β²β² 6,6,6,6,6 πΊ35 , π‘
β π38β²β² 7,7 πΊ39 , π‘
π18 -74
where β b16β²β² 2 G19 , t , β b17
β²β² 2 G19 , t , β b18β²β² 2 G19 , t are first detrition coefficients for category 1, 2 and 3
β π13β²β² 1,1, πΊ, π‘ , β π14
β²β² 1,1, πΊ, π‘ , β π15β²β²
1,1, πΊ, π‘ are second detrition coefficients for category 1,2 and 3
β π20β²β² 3,3,3, πΊ23 , π‘ , β π21
β²β² 3,3,3, πΊ23 , π‘ , β π22β²β² 3,3,3, πΊ23 , π‘ are third detrition coefficients for category 1,2 and 3
β π24β²β² 4,4,4,4,4 πΊ27 , π‘ , β π25
β²β² 4,4,4,4,4
πΊ27 , π‘ , β π26β²β² 4,4,4,4,4 πΊ27 , π‘ are fourth detritions coefficients for category 1,2
and 3
β π28β²β² 5,5,5,5,5 πΊ31 , π‘ , β π29
β²β² 5,5,5,5,5 πΊ31 , π‘ , β π30β²β² 5,5,5,5,5 πΊ31 , π‘ are fifth detritions coefficients for category 1,2
and 3
β π32β²β² 6,6,6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6,6,6 πΊ35 , π‘ are sixth detritions coefficients for category 1,2
and 3
β π36β²β² 7,7 πΊ39 , π‘ β π36
β²β² 7,7 πΊ39 , π‘ β π36β²β² 7,7 πΊ39 , π‘ πππ π ππ£πππ‘π πππ‘πππ‘πππ πππππππππππ‘π
THIRD MODULE CONCATENATION:-75
ππΊ20
ππ‘= π20
3 πΊ21 β
π20β² 3 + π20
β²β² 3 π21 , π‘ + π16β²β² 2,2,2 π17 , π‘ + π13
β²β² 1,1,1, π14 , π‘
+ π24β²β² 4,4,4,4,4,4 π25 , π‘ + π28
β²β² 5,5,5,5,5,5 π29 , π‘ + π32β²β² 6,6,6,6,6,6 π33 , π‘
+ π36β²β² 7.7.7. π37 , π‘
πΊ20 -76
ππΊ21
ππ‘= π21
3 πΊ20 β
π21
β² 3 + π21β²β² 3 π21 , π‘ + π17
β²β² 2,2,2 π17 , π‘ + π14β²β² 1,1,1, π14 , π‘
+ π25β²β²
4,4,4,4,4,4 π25 , π‘ + π29
β²β² 5,5,5,5,5,5 π29 , π‘ + π33β²β² 6,6,6,6,6,6 π33 , π‘
+ π37β²β² 7.7.7. π37 , π‘
πΊ21 -77
ππΊ22
ππ‘= π22
3 πΊ21 β
π22
β² 3 + π22β²β² 3 π21 , π‘ + π18
β²β² 2,2,2 π17 , π‘ + π15β²β²
1,1,1, π14 , π‘
+ π26β²β² 4,4,4,4,4,4 π25 , π‘ + π30
β²β² 5,5,5,5,5,5 π29 , π‘ + π34β²β² 6,6,6,6,6,6 π33 , π‘
+ π38β²β² 7.7.7. π37 , π‘
πΊ22 -78
+ π20β²β² 3 π21 , π‘ , + π21
β²β² 3 π21 , π‘ , + π22β²β² 3 π21 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π16β²β² 2,2,2 π17 , π‘ , + π17
β²β² 2,2,2 π17 , π‘ , + π18β²β² 2,2,2 π17 , π‘ are second augmentation coefficients for category 1, 2
and 3
+ π13β²β² 1,1,1, π14 , π‘ , + π14
β²β² 1,1,1, π14 , π‘ , + π15β²β²
1,1,1, π14 , π‘ are third augmentation coefficients for category 1, 2
and 3
+ π24β²β² 4,4,4,4,4,4 π25 , π‘ , + π25
β²β² 4,4,4,4,4,4
π25 , π‘ , + π26β²β² 4,4,4,4,4,4 π25 , π‘ are fourth augmentation coefficients for
category 1, 2 and 3
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+ π28β²β² 5,5,5,5,5,5 π29 , π‘ , + π29
β²β² 5,5,5,5,5,5 π29 , π‘ , + π30β²β² 5,5,5,5,5,5 π29 , π‘ are fifth augmentation coefficients for
category 1, 2 and 3
+ π32β²β² 6,6,6,6,6,6 π33 , π‘ , + π33
β²β² 6,6,6,6,6,6 π33 , π‘ , + π34β²β² 6,6,6,6,6,6 π33 , π‘ are sixth augmentation coefficients for
category 1, 2 and 3 -79
80
+ π36β²β² 7.7.7. π37 , π‘ + π37
β²β² 7.7.7. π37 , π‘ + π38β²β² 7.7.7. π37 , π‘ are seventh augmentation coefficient-81
ππ20
ππ‘= π20
3 π21 β
π20β² 3 β π20
β²β² 3 πΊ23 , π‘ β π36β²β² 7,7,7 πΊ19 , π‘ β π13
β²β² 1,1,1, πΊ, π‘
β π24β²β² 4,4,4,4,4,4 πΊ27 , π‘ β π28
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π32β²β² 6,6,6,6,6,6 πΊ35 , π‘
β π36β²β² 7,7,7 πΊ39 , π‘
π20 -82
ππ21
ππ‘= π21
3 π20 β
π21
β² 3 β π21β²β² 3 πΊ23 , π‘ β π17
β²β² 2,2,2 πΊ19 , π‘ β π14β²β² 1,1,1, πΊ, π‘
β π25β²β²
4,4,4,4,4,4 πΊ27 , π‘ β π29
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π33β²β² 6,6,6,6,6,6 πΊ35 , π‘
β π37β²β² 7,7,7 πΊ39 , π‘
π21 -83
ππ22
ππ‘= π22
3 π21 β
π22β² 3 β π22
β²β² 3 πΊ23 , π‘ β π18β²β² 2,2,2 πΊ19 , π‘ β π15
β²β² 1,1,1,
πΊ, π‘
β π26β²β² 4,4,4,4,4,4 πΊ27 , π‘ β π30
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π34β²β² 6,6,6,6,6,6 πΊ35 , π‘
β π38β²β² 7,7,7 πΊ39 , π‘
π22 -84
β π20β²β² 3 πΊ23 , π‘ , β π21
β²β² 3 πΊ23 , π‘ , β π22β²β² 3 πΊ23 , π‘ are first detritions coefficients for category 1, 2 and 3
β π16β²β² 2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2 πΊ19 , π‘ , β π18β²β² 2,2,2 πΊ19 , π‘ are second detritions coefficients for category 1, 2 and
3
β π13β²β² 1,1,1, πΊ, π‘ , β π14
β²β² 1,1,1, πΊ, π‘ , β π15β²β²
1,1,1, πΊ, π‘ are third detrition coefficients for category 1,2 and 3
β π24β²β² 4,4,4,4,4,4 πΊ27 , π‘ , β π25
β²β² 4,4,4,4,4,4
πΊ27 , π‘ , β π26β²β² 4,4,4,4,4,4 πΊ27 , π‘ are fourth detritions coefficients for
category 1, 2 and 3
β π28β²β² 5,5,5,5,5,5 πΊ31 , π‘ , β π29
β²β² 5,5,5,5,5,5 πΊ31 , π‘ , β π30β²β² 5,5,5,5,5,5 πΊ31 , π‘ are fifth detritions coefficients for
category 1, 2 and 3
β π32β²β² 6,6,6,6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6,6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6,6,6,6 πΊ35 , π‘ are sixth detritions coefficients for category
1, 2 and 3 -85
β π36β²β² 7,7,7 πΊ39 , π‘ β π37
β²β² 7,7,7 πΊ39 , π‘ β π38β²β² 7,7,7 πΊ39 , π‘ are seventh detritions coefficients
====================================================================================
FOURTH MODULE CONCATENATION:-86
ππΊ24
ππ‘= π24
4 πΊ25 β
π24
β² 4 + π24β²β² 4 π25 , π‘ + π28
β²β² 5,5, π29 , π‘ + π32β²β² 6,6, π33 , π‘
+ π13β²β² 1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3 π21 , π‘
+ π36β²β² 7,7,7,7, π37 , π‘
πΊ24 -87
ππΊ25
ππ‘= π25
4 πΊ24 β
π25
β² 4
+ π25β²β²
4 π25 , π‘ + π29
β²β² 5,5, π29 , π‘ + π33β²β² 6,6 π33 , π‘
+ π14β²β² 1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3 π21 , π‘
+ π37β²β² 7,7,7,7, π37 , π‘
πΊ25 -88
ππΊ26
ππ‘= π26
4 πΊ25 β
π26
β² 4 + π26β²β² 4 π25 , π‘ + π30
β²β² 5,5, π29 , π‘ + π34β²β² 6,6, π33 , π‘
+ π15β²β²
1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3 π21 , π‘
+ π38β²β² 7,7,7,7, π37 , π‘
πΊ26 -89
πππππ π24β²β² 4 π25 , π‘ , π25
β²β² 4
π25 , π‘ , π26β²β² 4 π25 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π28β²β² 5,5, π29 , π‘ , + π29
β²β² 5,5, π29 , π‘ , + π30β²β² 5,5, π29 , π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
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+ π32β²β² 6,6, π33 , π‘ , + π33
β²β² 6,6, π33 , π‘ , + π34β²β² 6,6, π33 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1 π14 , π‘ , + π15β²β²
1,1,1,1 π14 , π‘ are fourth augmentation coefficients for category 1,
2,and 3
+ π16β²β² 2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2 π17 , π‘ are fifth augmentation coefficients for category 1,
2,and 3
+ π20β²β² 3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3 π21 , π‘ are sixth augmentation coefficients for category 1,
2,and 3
+ π36β²β² 7,7,7,7, π37 , π‘ + π36
β²β² 7,7,7,7, π37 , π‘ + π36β²β² 7,7,7,7, π37 , π‘ ARE SEVENTH augmentation coefficients-90
91
-92
ππ24
ππ‘= π24
4 π25 β
π24
β² 4 β π24β²β² 4 πΊ27 , π‘ β π28
β²β² 5,5, πΊ31 , π‘ β π32β²β² 6,6, πΊ35 , π‘
β π13β²β² 1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3 πΊ23 , π‘
β π36β²β² 7,7,7,7,,, πΊ39 , π‘
π24 -93
ππ25
ππ‘= π25
4 π24 β
π25
β² 4
β π25β²β²
4 πΊ27 , π‘ β π29
β²β² 5,5, πΊ31 , π‘ β π33β²β² 6,6, πΊ35 , π‘
β π14β²β² 1,1,1,1 πΊ, π‘ β π17
β²β² 2,2,2,2 πΊ19 , π‘ β π21β²β² 3,3,3,3 πΊ23 , π‘
β π37β²β² 7,7,7,77,, πΊ39 , π‘
π25 -94
ππ26
ππ‘= π26
4 π25 β
π26
β² 4 β π26β²β² 4 πΊ27 , π‘ β π30
β²β² 5,5, πΊ31 , π‘ β π34β²β² 6,6, πΊ35 , π‘
β π15β²β²
1,1,1,1 πΊ, π‘ β π18
β²β² 2,2,2,2 πΊ19 , π‘ β π22β²β² 3,3,3,3 πΊ23 , π‘
β π38β²β² 7,7,7,,7,, πΊ39 , π‘
π26 -95
πππππ β π24β²β² 4 πΊ27 , π‘ , β π25
β²β² 4
πΊ27 , π‘ , β π26β²β² 4 πΊ27 , π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π28β²β² 5,5, πΊ31 , π‘ , β π29
β²β² 5,5, πΊ31 , π‘ , β π30β²β² 5,5, πΊ31 , π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π32β²β² 6,6, πΊ35 , π‘ , β π33
β²β² 6,6, πΊ35 , π‘ , β π34β²β² 6,6, πΊ35 , π‘ πππ π‘ππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π13β²β² 1,1,1,1 πΊ, π‘ , β π14
β²β² 1,1,1,1 πΊ, π‘ , β π15β²β²
1,1,1,1 πΊ, π‘
πππ πππ’ππ‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π16β²β² 2,2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2,2 πΊ19 , π‘ , β π18β²β² 2,2,2,2 πΊ19 , π‘
πππ ππππ‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π20β²β² 3,3,3,3 πΊ23 , π‘ , β π21
β²β² 3,3,3,3 πΊ23 , π‘ , β π22β²β² 3,3,3,3 πΊ23 , π‘
πππ π ππ₯π‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π36β²β² 7,7,7,7,7,, πΊ39 , π‘ β π37
β²β² 7,7,7,7,7,, πΊ39 , π‘ β π38β²β² 7,7,7,7,7,, πΊ39 , π‘ π΄π πΈ SEVENTH DETRITION
COEFFICIENTS-96
-97
FIFTH MODULE CONCATENATION:-
98ππΊ28
ππ‘= π28
5 πΊ29 β
π28
β² 5 + π28β²β² 5 π29 , π‘ + π24
β²β² 4,4, π25 , π‘ + π32β²β² 6,6,6 π33 , π‘
+ π13β²β² 1,1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3,3 π21 , π‘
+ π36β²β² 7,7,,7,,7,7 π37 , π‘
πΊ28 -99
ππΊ29
ππ‘= π29
5 πΊ28 β
π29
β² 5 + π29β²β² 5 π29 , π‘ + π25
β²β² 4,4,
π25 , π‘ + π33β²β² 6,6,6 π33 , π‘
+ π14β²β² 1,1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3,3 π21 , π‘
+ π37β²β² 7,7,,,7,,7,7 π37 , π‘
πΊ29 -100
ππΊ30
ππ‘= π30
5 πΊ29 β
π30β² 5 + π30
β²β² 5 π29 , π‘ + π26β²β² 4,4, π25 , π‘ + π34
β²β² 6,6,6 π33 , π‘
+ π15β²β²
1,1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3,3 π21 , π‘
+ π38β²β² 7,7,,7,,7,7 π37 , π‘
πΊ30 -101
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πππππ + π28β²β² 5 π29 , π‘ , + π29
β²β² 5 π29 , π‘ , + π30β²β² 5 π29 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
π΄ππ + π24β²β² 4,4, π25 , π‘ , + π25
β²β² 4,4,
π25 , π‘ , + π26β²β² 4,4, π25 , π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π32β²β² 6,6,6 π33 , π‘ , + π33
β²β² 6,6,6 π33 , π‘ , + π34β²β² 6,6,6 π33 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1,1 π14 , π‘ , + π15β²β²
1,1,1,1,1 π14 , π‘ are fourth augmentation coefficients for category
1,2, and 3
+ π16β²β² 2,2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2,2 π17 , π‘ are fifth augmentation coefficients for category
1,2,and 3
+ π20β²β² 3,3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3,3 π21 , π‘ are sixth augmentation coefficients for category
1,2, 3 -102
-103
ππ28
ππ‘= π28
5 π29 β
π28
β² 5 β π28β²β² 5 πΊ31 , π‘ β π24
β²β² 4,4, πΊ23 , π‘ β π32β²β² 6,6,6 πΊ35 , π‘
β π13β²β² 1,1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3,3 πΊ23 , π‘
β π36β²β² 7,7,,7,7,7, πΊ38 , π‘
π28 -104
ππ29
ππ‘= π29
5 π28 β
π29
β² 5 β π29β²β² 5 πΊ31 , π‘ β π25
β²β² 4,4,
πΊ27 , π‘ β π33β²β² 6,6,6 πΊ35 , π‘
β π14β²β² 1,1,1,1,1 πΊ, π‘ β π17
β²β² 2,2,2,2,2 πΊ19 , π‘ β π21β²β² 3,3,3,3,3 πΊ23 , π‘
β π37β²β² 7,7,7,7,7, πΊ38 , π‘
π29 -105
ππ30
ππ‘= π30
5 π29 β
π30
β² 5 β π30β²β² 5 πΊ31 , π‘ β π26
β²β² 4,4, πΊ27 , π‘ β π34β²β² 6,6,6 πΊ35 , π‘
β π15β²β²
1,1,1,1,1, πΊ, π‘ β π18
β²β² 2,2,2,2,2 πΊ19 , π‘ β π22β²β² 3,3,3,3,3 πΊ23 , π‘
β π38β²β² 7,7,7,7,7, πΊ38 , π‘
π30 -106
π€ππππ β π28β²β² 5 πΊ31 , π‘ , β π29
β²β² 5 πΊ31 , π‘ , β π30β²β² 5 πΊ31 , π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘π
πππ πππ‘πππππ¦ 1, 2 πππ 3
β π24β²β² 4,4, πΊ27 , π‘ , β π25
β²β² 4,4,
πΊ27 , π‘ , β π26β²β² 4,4, πΊ27 , π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π32β²β² 6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6 πΊ35 , π‘ πππ π‘ππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π13β²β² 1,1,1,1,1 πΊ, π‘ , β π14
β²β² 1,1,1,1,1 πΊ, π‘ , β π15β²β²
1,1,1,1,1, πΊ, π‘ are fourth detrition coefficients for category 1,2, and
3
β π16β²β² 2,2,2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2,2,2 πΊ19 , π‘ , β π18β²β² 2,2,2,2,2 πΊ19 , π‘ are fifth detrition coefficients for category 1,2,
and 3
β π20β²β² 3,3,3,3,3 πΊ23 , π‘ , β π21
β²β² 3,3,3,3,3 πΊ23 , π‘ , β π22β²β² 3,3,3,3,3 πΊ23 , π‘ are sixth detrition coefficients for category 1,2,
and 3-107
SIXTH MODULE CONCATENATION-108
ππΊ32
ππ‘= π32
6 πΊ33 β
π32
β² 6 + π32β²β² 6 π33 , π‘ + π28
β²β² 5,5,5 π29 , π‘ + π24β²β² 4,4,4, π25 , π‘
+ π13β²β² 1,1,1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3,3,3 π21 , π‘
+ π36β²β² 7,7,7,7,7,7, π37 , π‘
πΊ32 -109
ππΊ33
ππ‘= π33
6 πΊ32 β
π33
β² 6 + π33β²β² 6 π33 , π‘ + π29
β²β² 5,5,5 π29 , π‘ + π25β²β²
4,4,4, π25 , π‘
+ π14β²β² 1,1,1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3,3,3 π21 , π‘
+ π37β²β² 7,7,7,,7,7,7, π37 , π‘
πΊ33 -110
ππΊ34
ππ‘= π34
6 πΊ33 β
π34
β² 6 + π34β²β² 6 π33 , π‘ + π30
β²β² 5,5,5 π29 , π‘ + π26β²β² 4,4,4, π25 , π‘
+ π15β²β²
1,1,1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3,3,3 π21 , π‘
+ π38β²β² 7,7,7,7,7,7, π37 , π‘
πΊ34 -111
+ π32β²β² 6 π33 , π‘ , + π33
β²β² 6 π33 , π‘ , + π34β²β² 6 π33 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π28β²β² 5,5,5 π29 , π‘ , + π29
β²β² 5,5,5 π29 , π‘ , + π30β²β² 5,5,5 π29 , π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
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+ π24β²β² 4,4,4, π25 , π‘ , + π25
β²β² 4,4,4,
π25 , π‘ , + π26β²β² 4,4,4, π25 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1,1,1 π14 , π‘ , + π15β²β²
1,1,1,1,1,1 π14 , π‘ - are fourth augmentation coefficients
+ π16β²β² 2,2,2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2,2,2 π17 , π‘ - fifth augmentation coefficients
+ π20β²β² 3,3,3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3,3,3 π21 , π‘ sixth augmentation coefficients
+ π36β²β² 7,7,7,7,7,7, π37 , π‘ + π36
β²β² 7,7,,77,7,,7, π37 , π‘ + π36β²β² 7,7,7,,7,7,7, π37 , π‘ ARE SVENTH AUGMENTATION
COEFFICIENTS-112
-113
ππ32
ππ‘= π32
6 π33 β
π32
β² 6 β π32β²β² 6 πΊ35 , π‘ β π28
β²β² 5,5,5 πΊ31 , π‘ β π24β²β² 4,4,4, πΊ27 , π‘
β π13β²β² 1,1,1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3,3,3 πΊ23 , π‘
β π36β²β² 7,7,7,,7,7,7 πΊ39 , π‘
π32 -114
π π 33
ππ‘= π 33
6 π 32 β
π 33
β² 6
β π 33β²β²
6 πΊ 35, π‘ β π 29
β²β² 5,5,5
πΊ 31, π‘ β π 25β²β²
4,4,4, πΊ 27, π‘
β π 14β²β²
1,1,1,1,1,1 πΊ , π‘ β π 17
β²β² 2,2,2,2,2,2
πΊ 19, π‘ β π 21β²β²
3,3,3,3,3,3 πΊ 23, π‘
β π 37β²β²
7,7,7,,7,7,7 πΊ 39, π‘
π 33 -115
π π 34
ππ‘= π 34
6 π 33 β
π 34
β² 6
β π 34β²β²
6 πΊ 35, π‘ β π 30
β²β² 5,5,5
πΊ 31, π‘ β π 26β²β²
4,4,4, πΊ 27, π‘
β π 15β²β²
1,1,1,1,1,1 πΊ , π‘ β π 18
β²β² 2,2,2,2,2,2
πΊ 19, π‘ β π 22β²β²
3,3,3,3,3,3 πΊ 23, π‘
β π 38β²β²
7,7,7,,7,7,7 πΊ 39, π‘
π 34 -116
β π 32β²β²
6 πΊ 35, π‘ , β π 33
β²β² 6
πΊ 35, π‘ , β π 34β²β²
6 πΊ 35, π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘ π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π 28β²β²
5,5,5 πΊ 31, π‘ , β π 29
β²β² 5,5,5
πΊ 31, π‘ , β π 30β²β²
5,5,5 πΊ 31, π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π 24β²β²
4,4,4, πΊ 27, π‘ , β π 25
β²β² 4,4,4,
πΊ 27, π‘ , β π 26β²β²
4,4,4, πΊ 27, π‘ πππ π‘ π πππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π 13β²β²
1,1,1,1,1,1 πΊ , π‘ , β π 14
β²β² 1,1,1,1,1,1
πΊ , π‘ , β π 15β²β²
1,1,1,1,1,1 πΊ , π‘ are fourth detrition coefficients for category 1,
2, and 3
β π 16β²β²
2,2,2,2,2,2 πΊ 19, π‘ , β π 17
β²β² 2,2,2,2,2,2
πΊ 19, π‘ , β π 18β²β²
2,2,2,2,2,2 πΊ 19, π‘ are fifth detrition coefficients for
category 1, 2, and 3
β π 20β²β²
3,3,3,3,3,3 πΊ 23, π‘ , β π 21
β²β² 3,3,3,3,3,3
πΊ 23, π‘ , β π 22β²β²
3,3,3,3,3,3 πΊ 23, π‘ are sixth detrition coefficients for category
1, 2, and 3
β π 36β²β²
7,7,7,7,7,7 πΊ 39, π‘ β π 36
β²β² 7,7,7,7,7,7
πΊ 39, π‘ β π 36β²β²
7,7,7,7,7,7 πΊ 39, π‘ ARE SEVENTH DETRITION
COEFFICIENTS-117
-118
SEVENTH MODULE CONCATENATION:-119 π πΊ 36
ππ‘=
π 36 7 πΊ 37 β π 36
β² 7
+ π 36β²β²
7 π 37, π‘
+ π 16β²β²
7 π 17, π‘ + π 20
β²β² 7
π 21, π‘ + π 24β²β²
7 π 23, π‘ πΊ 36 +
οΏ½28β²β²7οΏ½29,οΏ½ + οΏ½32β²β²7οΏ½33,οΏ½ +οΏ½13β²β²7οΏ½14,οΏ½ οΏ½36-120
121 π πΊ 37
ππ‘=
π 37 7 πΊ 36 β π 37
β² 7
+ π 37β²β²
7 π 37, π‘ + π 14
β²β² 7
π 14, π‘ + π 21β²β²
7 π 21, π‘ + π 17
β²β² 7
π 17, π‘ +
οΏ½25β²β²7οΏ½25,οΏ½ +οΏ½33β²β²7οΏ½33,οΏ½ + οΏ½29β²β²7οΏ½29,οΏ½ οΏ½37
-122
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π πΊ 38
ππ‘=
π 38 7 πΊ 37 β π 38
β² 7
+ π 38β²β²
7 π 37, π‘ + π 15
β²β² 7
π 14, π‘
+ π 22
β²β² 7
π 21, π‘ + + π 18β²β²
7 π 17, π‘ +
οΏ½26β²β²7οΏ½25,οΏ½ + οΏ½34β²β²7οΏ½33,οΏ½ + οΏ½30β²β²7οΏ½29,οΏ½ οΏ½38
-123
124
125
π π 36
ππ‘= π 36
7 π 37 β π 36β²
7 β π 36
β²β² 7
πΊ 39 , π‘ β π 16β²β²
7 πΊ 19 , π‘ β π 13
β²β² 7
πΊ 14 , π‘ β
οΏ½20β²β²7οΏ½231,οΏ½ β οΏ½24β²β²7οΏ½27,οΏ½ βοΏ½28β²β²7οΏ½31,οΏ½ βοΏ½32β²β²7οΏ½35,οΏ½ οΏ½36
-126
π π 37
ππ‘= π 37
7 π 36 β π 36β²
7 β π 37
β²β² 7
πΊ 39 , π‘ β π 17β²β²
7 πΊ 19 , π‘ β π 19
β²β² 7
πΊ 14 , π‘ β
οΏ½21β²β²7οΏ½231,οΏ½ β οΏ½25β²β²7οΏ½27,οΏ½ βοΏ½29β²β²7οΏ½31,οΏ½ βοΏ½33β²β²7οΏ½35,οΏ½
οΏ½37
-127
Where we suppose
(A) ππ 1 , ππ
β² 1 , ππβ²β² 1 , ππ
1 , ππβ² 1 , ππ
β²β² 1 > 0, π, π = 13,14,15
(B) The functions ππβ²β² 1 , ππ
β²β² 1 are positive continuous increasing and bounded.
Definition of (ππ) 1 , (ππ)
1 :
ππβ²β² 1 (π14 , π‘) β€ (ππ)
1 β€ ( π΄ 13 )(1)
ππβ²β² 1 (πΊ, π‘) β€ (ππ)
1 β€ (ππβ²) 1 β€ ( π΅ 13 )(1)
(C) ππππ2ββ ππβ²β² 1 π14 , π‘ = (ππ)
1
limGββ ππβ²β² 1 πΊ , π‘ = (ππ)
1
Definition of ( π΄ 13 )(1), ( π΅ 13 )(1) :
Where ( π΄ 13 )(1), ( π΅ 13 )(1), (ππ) 1 , (ππ)
1 are positive constants
and π = 13,14,15
They satisfy Lipschitz condition:
|(ππβ²β² ) 1 π14
β² , π‘ β (ππβ² β² ) 1 π14 , π‘ | β€ ( π 13 )(1)|π14 β π14
β² |πβ( π 13 )(1)π‘
|(ππβ²β² ) 1 πΊ β² , π‘ β (ππ
β²β² ) 1 πΊ,π | < ( π 13 )(1)||πΊ β πΊ β² ||πβ( π 13 )(1)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 1 π14
β² , π‘ and(ππβ²β² ) 1 π14 , π‘ . π14
β² , π‘ and π14 , π‘ are points belonging to the interval ( π 13 )(1), ( π 13 )(1) . It is to be noted that (ππ
β²β² ) 1 π14 , π‘ is uniformly continuous.
In the eventuality of the fact, that if ( π 13 )(1) = 1 then the function (ππβ²β² ) 1 π14 , π‘ , the first augmentation coefficient
attributable to terrestrial organisms, would be absolutely continuous.
Definition of ( π 13 )(1), ( π 13 )(1) :
(D) ( π 13 )(1), ( π 13 )(1), are positive constants
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(ππ) 1
( π 13 )(1) ,(ππ) 1
( π 13 )(1) < 1
Definition of ( π 13 )(1), ( π 13 )(1) :
(E) There exists two constants ( π 13 )(1) and ( π 13 )(1) which together with ( π 13 )(1), ( π 13 )(1), (π΄ 13 )(1)πππ ( π΅ 13 )(1) and
the constants (ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ)
1 , π = 13,14,15, satisfy the inequalities
1
( π 13 )(1)[ (ππ)
1 + (ππβ²) 1 + ( π΄ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
1
( π 13 )(1)[ (ππ)
1 + (ππβ²) 1 + ( π΅ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
ππ38
ππ‘= π38
7 π37 β π38β² 7 β π38
β²β² 7 πΊ39 , π‘ β π18β²β² 7 πΊ19 , π‘ β π20
β²β² 7 πΊ14 , π‘ β
π22β²β²7πΊ23,π‘ β π26β²β²7πΊ27,π‘ βπ30β²β²7πΊ31,π‘ βπ34β²β²7πΊ35,π‘
π38
128
129
130
131
132
+ π36β²β² 7 π37 , π‘ = First augmentation factor 134
(1) ππ 2 , ππ
β² 2 , ππβ²β² 2 , ππ
2 , ππβ² 2 , ππ
β²β² 2 > 0, π, π = 16,17,18 135
(F) (2) The functions ππβ²β² 2 , ππ
β²β² 2 are positive continuous increasing and bounded. 136
Definition of (pi) 2 , (ri)
2 : 137
ππβ²β² 2 π17 , π‘ β€ (ππ)
2 β€ π΄ 16 2
138
ππβ²β² 2 (πΊ19 , π‘) β€ (ππ)
2 β€ (ππβ²) 2 β€ ( π΅ 16 )(2) 139
(G) (3) limπ2ββ ππβ²β² 2 π17 , π‘ = (ππ)
2 140
limπΊββ ππβ²β² 2 πΊ19 , π‘ = (ππ)
2 141
Definition of ( π΄ 16 )(2), ( π΅ 16 )(2) :
Where ( π΄ 16 )(2), ( π΅ 16 )(2), (ππ) 2 , (ππ)
2 are positive constants and π = 16,17,18
142
They satisfy Lipschitz condition: 143
|(ππβ²β² ) 2 π17
β² , π‘ β (ππβ²β² ) 2 π17 , π‘ | β€ ( π 16 )(2)|π17 β π17
β² |πβ( π 16 )(2)π‘ 144
|(ππβ²β² ) 2 πΊ19
β² , π‘ β (ππβ²β² ) 2 πΊ19 , π‘ | < ( π 16 )(2)|| πΊ19 β πΊ19
β² ||πβ( π 16 )(2)π‘ 145
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 2 π17
β² , π‘ and(ππ
β²β² ) 2 π17 , π‘ . π17β² , π‘ And π17 , π‘ are points belonging to the interval ( π 16 )(2), ( π 16 )(2) . It is to be
noted that (ππβ²β² ) 2 π17 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 16 )(2) = 1 then the
function (ππβ²β² ) 2 π17 , π‘ , the SECOND augmentation coefficient would be absolutely continuous.
146
Definition of ( π 16 )(2), ( π 16 )(2) : 147
(H) (4) ( π 16 )(2), ( π 16 )(2), are positive constants 148
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(ππ) 2
( π 16 )(2) ,(ππ) 2
( π 16 )(2) < 1
Definition of ( π 13 )(2), ( π 13 )(2) :
There exists two constants ( π 16 )(2) and ( π 16 )(2) which together
with ( π 16 )(2), ( π 16 )(2), (π΄ 16 )(2)πππ ( π΅ 16 )(2) and the constants (ππ)
2 , (ππβ² ) 2 , (ππ)
2 , (ππβ²) 2 , (ππ)
2 , (ππ) 2 , π = 16,17,18,
satisfy the inequalities
149
1
( M 16 )(2) [ (ai) 2 + (ai
β²) 2 + ( A 16 )(2) + ( P 16 )(2) ( k 16 )(2)] < 1 150
1
( π 16 )(2) [ (ππ) 2 + (ππ
β²) 2 + ( π΅ 16 )(2) + ( π 16 )(2) ( π 16 )(2)] < 1 151
Where we suppose 152
(I) (5) ππ 3 , ππ
β² 3 , ππβ²β² 3 , ππ
3 , ππβ² 3 , ππ
β²β² 3 > 0, π, π = 20,21,22
The functions ππβ²β² 3 , ππ
β²β² 3 are positive continuous increasing and bounded.
Definition of (ππ) 3 , (ri )
3 :
ππβ²β² 3 (π21 , π‘) β€ (ππ)
3 β€ ( π΄ 20 )(3)
ππβ²β² 3 (πΊ23 , π‘) β€ (ππ)
3 β€ (ππβ²) 3 β€ ( π΅ 20 )(3)
153
ππππ2ββ ππβ²β² 3 π21 , π‘ = (ππ)
3
limGββ ππβ²β² 3 πΊ23 , π‘ = (ππ)
3
Definition of ( π΄ 20 )(3), ( π΅ 20 )(3) :
Where ( π΄ 20 )(3), ( π΅ 20 )(3), (ππ) 3 , (ππ)
3 are positive constants and π = 20,21,22
154
155
156
They satisfy Lipschitz condition:
|(ππβ²β² ) 3 π21
β² , π‘ β (ππβ²β² ) 3 π21 , π‘ | β€ ( π 20 )(3)|π21 β π21
β² |πβ( π 20 )(3)π‘
|(ππβ²β² ) 3 πΊ23
β² , π‘ β (ππβ²β² ) 3 πΊ23 , π‘ | < ( π 20 )(3)||πΊ23 β πΊ23
β² ||πβ( π 20 )(3)π‘
157
158
159
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 3 π21
β² , π‘
and(ππβ²β² ) 3 π21 , π‘ . π21
β² , π‘ And π21 , π‘ are points belonging to the interval ( π 20 )(3), ( π 20 )(3) . It is to be
noted that (ππβ²β² ) 3 π21 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 20 )(3) = 1 then the
function (ππβ²β² ) 3 π21 , π‘ , the THIRD augmentation coefficient, would be absolutely continuous.
160
Definition of ( π 20 )(3), ( π 20 )(3) :
(J) (6) ( π 20 )(3), ( π 20 )(3), are positive constants
(ππ) 3
( π 20 )(3) ,(ππ) 3
( π 20 )(3) < 1
161
There exists two constants There exists two constants ( π 20 )(3) and ( π 20 )(3) which together with
( π 20 )(3), ( π 20 )(3), (π΄ 20 )(3)πππ ( π΅ 20 )(3) and the constants (ππ) 3 , (ππ
β² ) 3 , (ππ) 3 , (ππ
β² ) 3 , (ππ) 3 , (ππ)
3 , π =20,21,22, satisfy the inequalities
1
( π 20 )(3) [ (ππ) 3 + (ππ
β² ) 3 + ( π΄ 20 )(3) + ( π 20 )(3) ( π 20 )(3)] < 1
162
163
164
165
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1
( π 20 )(3) [ (ππ) 3 + (ππ
β²) 3 + ( π΅ 20 )(3) + ( π 20 )(3) ( π 20 )(3)] < 1 166
167
Where we suppose 168
(K) ππ 4 , ππ
β² 4 , ππβ²β² 4 , ππ
4 , ππβ² 4 , ππ
β²β² 4 > 0, π, π = 24,25,26
(L) (7) The functions ππβ²β² 4 , ππ
β²β² 4 are positive continuous increasing and bounded.
Definition of (ππ) 4 , (ππ)
4 :
ππβ²β² 4 (π25 , π‘) β€ (ππ)
4 β€ ( π΄ 24 )(4)
ππβ²β² 4 πΊ27 , π‘ β€ (ππ)
4 β€ (ππβ²) 4 β€ ( π΅ 24 )(4)
169
(M) (8) ππππ2ββ ππβ²β² 4 π25 , π‘ = (ππ)
4
limGββ ππβ²β² 4 πΊ27 , π‘ = (ππ)
4
Definition of ( π΄ 24 )(4), ( π΅ 24 )(4) :
Where ( π΄ 24 )(4), ( π΅ 24 )(4), (ππ) 4 , (ππ)
4 are positive constants and π = 24,25,26
170
They satisfy Lipschitz condition:
|(ππβ²β² ) 4 π25
β² , π‘ β (ππβ²β² ) 4 π25 , π‘ | β€ ( π 24 )(4)|π25 β π25
β² |πβ( π 24 )(4)π‘
|(ππβ²β² ) 4 πΊ27
β² , π‘ β (ππβ²β² ) 4 πΊ27 , π‘ | < ( π 24 )(4)|| πΊ27 β πΊ27
β² ||πβ( π 24 )(4)π‘
171
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 4 π25
β² , π‘
and(ππβ²β² ) 4 π25 , π‘ . π25
β² , π‘ And π25 , π‘ are points belonging to the interval ( π 24 )(4), ( π 24 )(4) . It is to be
noted that (ππβ²β² ) 4 π25 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 24 )(4) = 4 then the
function (ππβ²β² ) 4 π25 , π‘ , the FOURTH augmentation coefficient WOULD be absolutely continuous.
172
173
Definition of ( π 24 )(4), ( π 24 )(4) :
(N) ( π 24 )176175(4), ( π 24 )(4), are positive constants (O)
(ππ) 4
( π 24 )(4) ,(ππ) 4
( π 24 )(4) < 1
174
Definition of ( π 24 )(4), ( π 24 )(4) :
(P) (9) There exists two constants ( π 24 )(4) and ( π 24 )(4) which together with
( π 24 )(4), ( π 24 )(4), (π΄ 24 )(4)πππ ( π΅ 24 )(4) and the constants
(ππ) 4 , (ππ
β² ) 4 , (ππ) 4 , (ππ
β²) 4 , (ππ) 4 , (ππ)
4 , π = 24,25,26, satisfy the inequalities
1
( π 24 )(4) [ (ππ) 4 + (ππ
β² ) 4 + ( π΄ 24 )(4) + ( π 24 )(4) ( π 24 )(4)] < 1
1
( π 24 )(4) [ (ππ) 4 + (ππ
β²) 4 + ( π΅ 24 )(4) + ( π 24 )(4) ( π 24 )(4)] < 1
175
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Where we suppose 176
(Q) ππ 5 , ππ
β² 5 , ππβ²β² 5 , ππ
5 , ππβ² 5 , ππ
β²β² 5 > 0, π, π = 28,29,30
(R) (10) The functions ππβ²β² 5 , ππ
β²β² 5 are positive continuous increasing and bounded.
Definition of (ππ) 5 , (ππ)
5 :
ππβ²β² 5 (π29 , π‘) β€ (ππ)
5 β€ ( π΄ 28 )(5)
ππβ²β² 5 πΊ31 , π‘ β€ (ππ)
5 β€ (ππβ²) 5 β€ ( π΅ 28 )(5)
177
(S) (11) ππππ2ββ ππβ²β² 5 π29 , π‘ = (ππ)
5
limGββ ππβ²β² 5 πΊ31 , π‘ = (ππ)
5
Definition of ( π΄ 28 )(5), ( π΅ 28 )(5) :
Where ( π΄ 28 )(5), ( π΅ 28 )(5), (ππ) 5 , (ππ)
5 are positive constants and π = 28,29,30
178
They satisfy Lipschitz condition:
|(ππβ²β² ) 5 π29
β² , π‘ β (ππβ²β² ) 5 π29 , π‘ | β€ ( π 28 )(5)|π29 β π29
β² |πβ( π 28 )(5)π‘
|(ππβ²β² ) 5 πΊ31
β² , π‘ β (ππβ²β² ) 5 πΊ31 , π‘ | < ( π 28 )(5)|| πΊ31 β πΊ31
β² ||πβ( π 28 )(5)π‘
179
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 5 π29
β² , π‘
and(ππβ²β² ) 5 π29 , π‘ . π29
β² , π‘ and π29 , π‘ are points belonging to the interval ( π 28 )(5), ( π 28 )(5) . It is to be
noted that (ππβ²β² ) 5 π29 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 28 )(5) = 5 then the
function (ππβ²β² ) 5 π29 , π‘ , theFIFTH augmentation coefficient attributable would be absolutely continuous.
180
Definition of ( π 28 )(5), ( π 28 )(5) :
(T) ( π 28 )(5), ( π 28 )(5), are positive constants
(ππ) 5
( π 28 )(5) ,(ππ) 5
( π 28 )(5) < 1
181
Definition of ( π 28 )(5), ( π 28 )(5) :
(U) There exists two constants ( π 28 )(5) and ( π 28 )(5) which together with
( π 28 )(5), ( π 28 )(5), (π΄ 28 )(5)πππ ( π΅ 28 )(5) and the constants
(ππ) 5 , (ππ
β² ) 5 , (ππ) 5 , (ππ
β²) 5 , (ππ) 5 , (ππ)
5 , π = 28,29,30, satisfy the inequalities
1
( π 28 )(5) [ (ππ) 5 + (ππ
β² ) 5 + ( π΄ 28 )(5) + ( π 28 )(5) ( π 28 )(5)] < 1
1
( π 28 )(5) [ (ππ) 5 + (ππ
β²) 5 + ( π΅ 28 )(5) + ( π 28 )(5) ( π 28 )(5)] < 1
182
Where we suppose 183
ππ 6 , ππ
β² 6 , ππβ²β² 6 , ππ
6 , ππβ² 6 , ππ
β²β² 6 > 0, π, π = 32,33,34
(12) The functions ππβ²β² 6 , ππ
β²β² 6 are positive continuous increasing and bounded.
Definition of (ππ) 6 , (ππ)
6 :
ππβ²β² 6 (π33 , π‘) β€ (ππ)
6 β€ ( π΄ 32 )(6)
ππβ²β² 6 ( πΊ35 , π‘) β€ (ππ)
6 β€ (ππβ²) 6 β€ ( π΅ 32 )(6)
184
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(13) ππππ2ββ ππβ²β² 6 π33 , π‘ = (ππ)
6
limGββ ππβ²β² 6 πΊ35 , π‘ = (ππ)
6
Definition of ( π΄ 32 )(6), ( π΅ 32 )(6) :
Where ( π΄ 32 )(6), ( π΅ 32 )(6), (ππ) 6 , (ππ)
6 are positive constants and π = 32,33,34
185
They satisfy Lipschitz condition:
|(ππβ²β² ) 6 π33
β² , π‘ β (ππβ²β² ) 6 π33 , π‘ | β€ ( π 32 )(6)|π33 β π33
β² |πβ( π 32 )(6)π‘
|(ππβ²β² ) 6 πΊ35
β² , π‘ β (ππβ²β² ) 6 πΊ35 , π‘ | < ( π 32 )(6)|| πΊ35 β πΊ35
β² ||πβ( π 32 )(6)π‘
186
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 6 π33
β² , π‘
and(ππβ²β² ) 6 π33 , π‘ . π33
β² , π‘ and π33 , π‘ are points belonging to the interval ( π 32 )(6), ( π 32 )(6) . It is to be
noted that (ππβ²β² ) 6 π33 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 32 )(6) = 6 then the
function (ππβ²β² ) 6 π33 , π‘ , the SIXTH augmentation coefficient would be absolutely continuous.
187
Definition of ( π 32 )(6), ( π 32 )(6) :
( π 32 )(6), ( π 32 )(6), are positive constants
(ππ) 6
( π 32 )(6) ,(ππ) 6
( π 32 )(6) < 1
188
Definition of ( π 32 )(6), ( π 32 )(6) :
There exists two constants ( π 32 )(6) and ( π 32 )(6) which together with
( π 32 )(6), ( π 32 )(6), (π΄ 32 )(6)πππ ( π΅ 32 )(6) and the constants (ππ) 6 , (ππ
β²) 6 , (ππ) 6 , (ππ
β²) 6 , (ππ) 6 , (ππ)
6 , π =32,33,34, satisfy the inequalities
1
( π 32 )(6) [ (ππ) 6 + (ππ
β² ) 6 + ( π΄ 32 )(6) + ( π 32 )(6) ( π 32 )(6)] < 1
1
( π 32 )(6) [ (ππ) 6 + (ππ
β²) 6 + ( π΅ 32 )(6) + ( π 32 )(6) ( π 32 )(6)] < 1
189
Where we suppose
190
(V) ππ 7 , ππ
β² 7
, ππβ²β²
7 , ππ
7 , ππβ²
7 , ππ
β²β² 7
> 0,
π, π = 36,37,38
(W) The functions ππβ²β²
7 , ππ
β²β² 7
are positive continuous increasing and bounded.
Definition of (ππ) 7 , (ππ)
7 :
ππβ²β²
7 (π37 , π‘) β€ (ππ)
7 β€ ( π΄ 36 )(7)
ππβ²β²
7 (πΊ, π‘) β€ (ππ)
7 β€ (ππβ² ) 7 β€ ( π΅ 36 )(7)
191
limπ2ββ ππβ²β²
7 π37 , π‘ = (ππ)
7
limGββ ππβ²β²
7 πΊ39 , π‘ = (ππ)
7
192
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Definition of ( π΄ 36 )(7), ( π΅ 36 )(7) :
Where ( π΄ 36 )(7), ( π΅ 36 )(7), (ππ) 7 , (ππ)
7 are positive constants
and π = 36,37,38
They satisfy Lipschitz condition:
|(ππβ²β²) 7 π37
β² , π‘ β (ππβ²β²) 7 π37 , π‘ | β€ ( π 36 )(7)|π37 β π37
β² |πβ( π 36 )(7)π‘
|(ππβ²β²) 7 πΊ39
β², π‘ β (ππβ²β²) 7 πΊ39 , π39 | < ( π 36 )(7)|| πΊ39 β πΊ39
β²||πβ( π 36 )(7)π‘
193
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β²) 7 π37
β² , π‘
and(ππβ²β²) 7 π37 , π‘ . π37
β² , π‘ and π37 , π‘ are points belonging to the interval ( π 36 )(7), ( π 36 )(7) . It is to be
noted that (ππβ²β²) 7 π37 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 36 )(7) = 7 then the
function (ππβ²β²) 7 π37 , π‘ , the first augmentation coefficient attributable to terrestrial organisms, would be
absolutely continuous.
194
Definition of ( π 36 )(7), ( π 36 )(7) :
(X) ( π 36 )(7), ( π 36 )(7), are positive constants
(ππ) 7
( π 36 )(7) ,(ππ) 7
( π 36 )(7) < 1
195
Definition of ( π 36 )(7), ( π 36 )(7) :
(Y) There exists two constants ( π 36 )(7) and ( π 36 )(7) which together with
( π 36 )(7), ( π 36 )(7), (π΄ 36 )(7)πππ ( π΅ 36 )(7) and the constants
(ππ) 7 , (ππ
β² ) 7 , (ππ) 7 , (ππ
β² ) 7 , (ππ) 7 , (ππ)
7 , π = 36,37,38, satisfy the inequalities
1
( π 36 )(7)[ (ππ)
7 + (ππβ² ) 7 + ( π΄ 36 )(7) + ( π 36 )(7) ( π 36 )(7)] < 1
1
( π 36 )(7)[ (ππ)
7 + (ππβ² ) 7 + ( π΅ 36 )(7) + ( π 36 )(7) ( π 36 )(7)] < 1
196
Definition of πΊπ 0 ,ππ 0 :
πΊπ π‘ β€ π 28 5
π π 28 5 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 28 )(5)π( π 28 )(5)π‘ , ππ 0 = ππ0 > 0
197
198
Definition of πΊπ 0 ,ππ 0 :
πΊπ π‘ β€ π 32 6
π π 32 6 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 32 )(6)π( π 32 )(6)π‘ , ππ 0 = ππ0 > 0
===================================================================================
Definition of πΊπ 0 ,ππ 0 :
πΊπ π‘ β€ π 36 7
π π 36 7 π‘ , πΊπ 0 = πΊπ0 > 0
199
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ππ(π‘) β€ ( π 36 )(7)π( π 36 )(7)π‘ , ππ 0 = ππ0 > 0
Proof: Consider operator π(1) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which satisfy
200
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 13 )(1) ,ππ
0 β€ ( π 13 )(1), 201
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 13 )(1)π( π 13 )(1)π‘ 202
0 β€ ππ π‘ β ππ0 β€ ( π 13 )(1)π( π 13 )(1)π‘ 203
By
πΊ 13 π‘ = πΊ130 + (π13 ) 1 πΊ14 π 13 β (π13
β² ) 1 + π13β²β² ) 1 π14 π 13 , π 13 πΊ13 π 13 ππ 13
π‘
0
204
πΊ 14 π‘ = πΊ140 + (π14 ) 1 πΊ13 π 13 β (π14
β² ) 1 + (π14β²β² ) 1 π14 π 13 , π 13 πΊ14 π 13 ππ 13
π‘
0 205
πΊ 15 π‘ = πΊ150 + (π15 ) 1 πΊ14 π 13 β (π15
β² ) 1 + (π15β²β² ) 1 π14 π 13 , π 13 πΊ15 π 13 ππ 13
π‘
0 206
π 13 π‘ = π130 + (π13 ) 1 π14 π 13 β (π13
β² ) 1 β (π13β²β² ) 1 πΊ π 13 , π 13 π13 π 13 ππ 13
π‘
0 207
π 14 π‘ = π140 + (π14 ) 1 π13 π 13 β (π14
β² ) 1 β (π14β²β² ) 1 πΊ π 13 , π 13 π14 π 13 ππ 13
π‘
0 208
T 15 t = T150 + (π15 ) 1 π14 π 13 β (π15
β² ) 1 β (π15β²β² ) 1 πΊ π 13 , π 13 π15 π 13 ππ 13
π‘
0
Where π 13 is the integrand that is integrated over an interval 0, π‘
209
210
if the conditions IN THE FOREGOING above are fulfilled, there exists a solution satisfying the conditions Definition of πΊπ 0 ,ππ 0 :
πΊπ π‘ β€ π 36 7
π π 36 7 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 36 )(7)π( π 36 )(7)π‘ , ππ 0 = ππ0 > 0
Consider operator π(7) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which satisfy πΊπ 0 = πΊπ
0 , ππ 0 = ππ0 , πΊπ
0 β€ ( π 36 )(7) , ππ0 β€ ( π 36 )(7),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 36 )(7)π( π 36 )(7)π‘
0 β€ ππ π‘ β ππ0 β€ ( π 36 )(7)π( π 36 )(7)π‘
By
πΊ 36 π‘ = πΊ360 + (π36 ) 7 πΊ37 π 36 β (π36
β² ) 7 + π36β²β² ) 7 π37 π 36 , π 36 πΊ36 π 36 ππ 36
π‘
0
πΊ 37 π‘ = πΊ37
0 +
(π37 ) 7 πΊ36 π 36 β (π37β² ) 7 + (π37
β²β² ) 7 π37 π 36 , π 36 πΊ37 π 36 ππ 36 π‘
0
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πΊ 38 π‘ = πΊ380 +
(π38 ) 7 πΊ37 π 36 β (π38β² ) 7 + (π38
β²β² ) 7 π37 π 36 , π 36 πΊ38 π 36 ππ 36 π‘
0
π 36 π‘ = π360 + (π36 ) 7 π37 π 36 β (π36
β² ) 7 β (π36β²β² ) 7 πΊ π 36 , π 36 π36 π 36 ππ 36
π‘
0
π 37 π‘ = π370 + (π37 ) 7 π36 π 36 β (π37
β² ) 7 β (π37β²β² ) 7 πΊ π 36 , π 36 π37 π 36 ππ 36
π‘
0
T 38 t = T380 +
(π38) 7 π37 π 36 β (π38β² ) 7 β (π38
β²β² ) 7 πΊ π 36 , π 36 π38 π 36 ππ 36 π‘
0
Where π 36 is the integrand that is integrated over an interval 0, π‘
Consider operator π(2) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
211
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 16 )(2) ,ππ
0 β€ ( π 16 )(2), 212
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 16 )(2)π( π 16 )(2)π‘ 213
0 β€ ππ π‘ β ππ0 β€ ( π 16 )(2)π( π 16 )(2)π‘ 214
By
πΊ 16 π‘ = πΊ160 + (π16 ) 2 πΊ17 π 16 β (π16
β² ) 2 + π16β²β² ) 2 π17 π 16 , π 16 πΊ16 π 16 ππ 16
π‘
0
215
πΊ 17 π‘ = πΊ170 + (π17 ) 2 πΊ16 π 16 β (π17
β² ) 2 + (π17β²β² ) 2 π17 π 16 , π 17 πΊ17 π 16 ππ 16
π‘
0 216
πΊ 18 π‘ = πΊ180 + (π18 ) 2 πΊ17 π 16 β (π18
β² ) 2 + (π18β² β² ) 2 π17 π 16 , π 16 πΊ18 π 16 ππ 16
π‘
0 217
π 16 π‘ = π160 + (π16 ) 2 π17 π 16 β (π16
β² ) 2 β (π16β²β² ) 2 πΊ π 16 , π 16 π16 π 16 ππ 16
π‘
0 218
π 17 π‘ = π170 + (π17 ) 2 π16 π 16 β (π17
β² ) 2 β (π17β²β² ) 2 πΊ π 16 , π 16 π17 π 16 ππ 16
π‘
0 219
π 18 π‘ = π180 + (π18 ) 2 π17 π 16 β (π18
β² ) 2 β (π18β²β² ) 2 πΊ π 16 , π 16 π18 π 16 ππ 16
π‘
0
Where π 16 is the integrand that is integrated over an interval 0, π‘
220
Consider operator π(3) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
221
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 20 )(3) , ππ
0 β€ ( π 20 )(3), 222
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 20 )(3)π( π 20 )(3)π‘ 223
0 β€ ππ π‘ β ππ0 β€ ( π 20 )(3)π( π 20 )(3)π‘ 224
By 225
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πΊ 20 π‘ = πΊ200 + (π20 ) 3 πΊ21 π 20 β (π20
β² ) 3 + π20β²β² ) 3 π21 π 20 , π 20 πΊ20 π 20 ππ 20
π‘
0
πΊ 21 π‘ = πΊ210 + (π21 ) 3 πΊ20 π 20 β (π21
β² ) 3 + (π21β²β² ) 3 π21 π 20 , π 20 πΊ21 π 20 ππ 20
π‘
0 226
πΊ 22 π‘ = πΊ220 + (π22 ) 3 πΊ21 π 20 β (π22
β² ) 3 + (π22β²β² ) 3 π21 π 20 , π 20 πΊ22 π 20 ππ 20
π‘
0 227
π 20 π‘ = π200 + (π20 ) 3 π21 π 20 β (π20
β² ) 3 β (π20β²β² ) 3 πΊ π 20 , π 20 π20 π 20 ππ 20
π‘
0 228
π 21 π‘ = π210 + (π21 ) 3 π20 π 20 β (π21
β² ) 3 β (π21β²β² ) 3 πΊ π 20 , π 20 π21 π 20 ππ 20
π‘
0 229
T 22 t = T220 + (π22) 3 π21 π 20 β (π22
β² ) 3 β (π22β²β² ) 3 πΊ π 20 , π 20 π22 π 20 ππ 20
π‘
0
Where π 20 is the integrand that is integrated over an interval 0, π‘
230
Consider operator π(4) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
231
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 24 )(4) , ππ
0 β€ ( π 24 )(4), 232
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 24 )(4)π( π 24 )(4)π‘ 233
0 β€ ππ π‘ β ππ0 β€ ( π 24 )(4)π( π 24 )(4)π‘ 234
By
πΊ 24 π‘ = πΊ240 + (π24 ) 4 πΊ25 π 24 β (π24
β² ) 4 + π24β²β² ) 4 π25 π 24 , π 24 πΊ24 π 24 ππ 24
π‘
0
235
πΊ 25 π‘ = πΊ250 + (π25 ) 4 πΊ24 π 24 β (π25
β² ) 4 + (π25β²β² ) 4 π25 π 24 , π 24 πΊ25 π 24 ππ 24
π‘
0 236
πΊ 26 π‘ = πΊ260 + (π26 ) 4 πΊ25 π 24 β (π26
β² ) 4 + (π26β²β² ) 4 π25 π 24 ,π 24 πΊ26 π 24 ππ 24
π‘
0 237
π 24 π‘ = π240 + (π24 ) 4 π25 π 24 β (π24
β² ) 4 β (π24β²β² ) 4 πΊ π 24 , π 24 π24 π 24 ππ 24
π‘
0 238
π 25 π‘ = π250 + (π25 ) 4 π24 π 24 β (π25
β² ) 4 β (π25β²β² ) 4 πΊ π 24 , π 24 π25 π 24 ππ 24
π‘
0 239
T 26 t = T260 + (π26) 4 π25 π 24 β (π26
β² ) 4 β (π26β²β² ) 4 πΊ π 24 , π 24 π26 π 24 ππ 24
π‘
0
Where π 24 is the integrand that is integrated over an interval 0, π‘
240
Consider operator π(5) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
241
242
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 28 )(5) , ππ
0 β€ ( π 28 )(5), 243
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 28 )(5)π( π 28 )(5)π‘ 244
0 β€ ππ π‘ β ππ0 β€ ( π 28 )(5)π( π 28 )(5)π‘ 245
By
πΊ 28 π‘ = πΊ280 + (π28 ) 5 πΊ29 π 28 β (π28
β² ) 5 + π28β²β² ) 5 π29 π 28 , π 28 πΊ28 π 28 ππ 28
π‘
0
246
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πΊ 29 π‘ = πΊ290 + (π29) 5 πΊ28 π 28 β (π29
β² ) 5 + (π29β²β² ) 5 π29 π 28 , π 28 πΊ29 π 28 ππ 28
π‘
0 247
πΊ 30 π‘ = πΊ300 + (π30 ) 5 πΊ29 π 28 β (π30
β² ) 5 + (π30β²β² ) 5 π29 π 28 , π 28 πΊ30 π 28 ππ 28
π‘
0 248
π 28 π‘ = π280 + (π28 ) 5 π29 π 28 β (π28
β² ) 5 β (π28β²β² ) 5 πΊ π 28 , π 28 π28 π 28 ππ 28
π‘
0 249
π 29 π‘ = π290 + (π29) 5 π28 π 28 β (π29
β² ) 5 β (π29β²β² ) 5 πΊ π 28 , π 28 π29 π 28 ππ 28
π‘
0 250
T 30 t = T300 + (π30) 5 π29 π 28 β (π30
β² ) 5 β (π30β²β² ) 5 πΊ π 28 , π 28 π30 π 28 ππ 28
π‘
0
Where π 28 is the integrand that is integrated over an interval 0, π‘
251
Consider operator π(6) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
252
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 32 )(6) , ππ
0 β€ ( π 32 )(6), 253
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 32 )(6)π( π 32 )(6)π‘ 254
0 β€ ππ π‘ β ππ0 β€ ( π 32 )(6)π( π 32 )(6)π‘ 255
By
πΊ 32 π‘ = πΊ320 + (π32 ) 6 πΊ33 π 32 β (π32
β² ) 6 + π32β²β² ) 6 π33 π 32 , π 32 πΊ32 π 32 ππ 32
π‘
0
256
πΊ 33 π‘ = πΊ330 + (π33 ) 6 πΊ32 π 32 β (π33
β² ) 6 + (π33β²β² ) 6 π33 π 32 , π 32 πΊ33 π 32 ππ 32
π‘
0 257
πΊ 34 π‘ = πΊ340 + (π34 ) 6 πΊ33 π 32 β (π34
β² ) 6 + (π34β²β² ) 6 π33 π 32 ,π 32 πΊ34 π 32 ππ 32
π‘
0 258
π 32 π‘ = π320 + (π32 ) 6 π33 π 32 β (π32
β² ) 6 β (π32β²β² ) 6 πΊ π 32 , π 32 π32 π 32 ππ 32
π‘
0 259
π 33 π‘ = π330 + (π33 ) 6 π32 π 32 β (π33
β² ) 6 β (π33β²β² ) 6 πΊ π 32 , π 32 π33 π 32 ππ 32
π‘
0 260
T 34 t = T340 + (π34) 6 π33 π 32 β (π34
β² ) 6 β (π34β²β² ) 6 πΊ π 32 , π 32 π34 π 32 ππ 32
π‘
0
Where π 32 is the integrand that is integrated over an interval 0, π‘
261
: if the conditions IN THE FOREGOING are fulfilled, there exists a solution satisfying the conditions
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 36 7
π π 36 7 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 36 )(7)π( π 36 )(7)π‘ , ππ 0 = ππ0 > 0
Proof:
Consider operator π(7) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
262
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πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 36 )(7) , ππ
0 β€ ( π 36 )(7),
263
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 36 )(7)π( π 36 )(7)π‘
264
0 β€ ππ π‘ β ππ0 β€ ( π 36 )(7)π( π 36 )(7)π‘ 265
By
πΊ 36 π‘ = πΊ360 + (π36 ) 7 πΊ37 π 36 β (π36
β² ) 7 + π36β²β² ) 7 π37 π 36 , π 36 πΊ36 π 36 ππ 36
π‘
0
266
πΊ 37 π‘ = πΊ370 +
(π37 ) 7 πΊ36 π 36 β (π37β² ) 7 + (π37
β²β² ) 7 π37 π 36 , π 36 πΊ37 π 36 ππ 36 π‘
0
267
πΊ 38 π‘ = πΊ380 +
(π38 ) 7 πΊ37 π 36 β (π38β² ) 7 + (π38
β²β² ) 7 π37 π 36 , π 36 πΊ38 π 36 ππ 36 π‘
0
268
π 36 π‘ = π360 + (π36 ) 7 π37 π 36 β (π36
β² ) 7 β (π36β²β² ) 7 πΊ π 36 , π 36 π36 π 36 ππ 36
π‘
0
269
π 37 π‘ = π370 + (π37 ) 7 π36 π 36 β (π37
β² ) 7 β (π37β²β² ) 7 πΊ π 36 , π 36 π37 π 36 ππ 36
π‘
0
270
T 38 t = T380 +
(π38) 7 π37 π 36 β (π38β² ) 7 β (π38
β²β² ) 7 πΊ π 36 , π 36 π38 π 36 ππ 36 π‘
0
Where π 36 is the integrand that is integrated over an interval 0, π‘
271
Analogous inequalities hold also for πΊ21 ,πΊ22 ,π20 ,π21 , π22 272
(a) The operator π(4) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
273
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πΊ24 π‘ β€ πΊ240 + (π24 ) 4 πΊ25
0 +( π 24 )(4)π( π 24 )(4)π 24 π‘
0ππ 24 =
1 + (π24 ) 4 π‘ πΊ250 +
(π24 ) 4 ( π 24 )(4)
( π 24 )(4) π( π 24 )(4)π‘ β 1
From which it follows that
πΊ24 π‘ β πΊ240 πβ( π 24 )(4)π‘ β€
(π24 ) 4
( π 24 )(4) ( π 24 )(4) + πΊ250 π
β ( π 24 )(4)+πΊ25
0
πΊ250
+ ( π 24 )(4)
πΊπ0 is as defined in the statement of theorem 1
274
(b) The operator π(5) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
πΊ28 π‘ β€ πΊ280 + (π28 ) 5 πΊ29
0 +( π 28 )(5)π( π 28 )(5)π 28 π‘
0ππ 28 =
1 + (π28 ) 5 π‘ πΊ290 +
(π28 ) 5 ( π 28 )(5)
( π 28 )(5) π( π 28 )(5)π‘ β 1
275
From which it follows that
πΊ28 π‘ β πΊ280 πβ( π 28 )(5)π‘ β€
(π28 ) 5
( π 28 )(5) ( π 28 )(5) + πΊ290 π
β ( π 28 )(5)+πΊ29
0
πΊ290
+ ( π 28 )(5)
πΊπ0 is as defined in the statement of theorem 1
276
(c) The operator π(6) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
πΊ32 π‘ β€ πΊ320 + (π32 ) 6 πΊ33
0 +( π 32 )(6)π( π 32 )(6)π 32 π‘
0ππ 32 =
1 + (π32 ) 6 π‘ πΊ330 +
(π32 ) 6 ( π 32 )(6)
( π 32 )(6) π( π 32 )(6)π‘ β 1
277
From which it follows that
πΊ32 π‘ β πΊ320 πβ( π 32 )(6)π‘ β€
(π32 ) 6
( π 32 )(6) ( π 32 )(6) + πΊ330 π
β ( π 32 )(6)+πΊ33
0
πΊ330
+ ( π 32 )(6)
πΊπ0 is as defined in the statement of theorem1
Analogous inequalities hold also for πΊ25 ,πΊ26 , π24 ,π25 ,π26
278
(d) The operator π(7) maps the space of functions satisfying 37,35,36 into itself .Indeed it is obvious that
πΊ36 π‘ β€ πΊ360 + (π36 ) 7 πΊ37
0 +( π 36 )(7)π( π 36 )(7)π 36 π‘
0ππ 36 =
1 + (π36 ) 7 π‘ πΊ370 +
(π36 ) 7 ( π 36 )(7)
( π 36 )(7) π( π 36 )(7)π‘ β 1
279
280
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From which it follows that
πΊ36 π‘ β πΊ360 πβ( π 36 )(7)π‘ β€
(π36 ) 7
( π 36 )(7) ( π 36 )(7) + πΊ370 π
β ( π 36 )(7)+πΊ37
0
πΊ370
+ ( π 36 )(7)
πΊπ0 is as defined in the statement of theorem 7
It is now sufficient to take (ππ) 1
( π 13 )(1) ,(ππ) 1
( π 13 )(1) < 1 and to choose
( P 13 )(1) and ( Q 13 )(1) large to have
281
282
(ππ) 1
(π 13 ) 1 ( π 13 ) 1 + ( π 13 )(1) + πΊπ0 π
β ( π 13 )(1)+πΊπ
0
πΊπ0
β€ ( π 13 )(1)
283
(ππ) 1
(π 13 ) 1 ( π 13 )(1) + ππ0 π
β ( π 13 )(1)+ππ
0
ππ0
+ ( π 13 )(1) β€ ( π 13 )(1)
284
In order that the operator π(1) transforms the space of sextuples of functions πΊπ ,ππ satisfying GLOBAL
EQUATIONS into itself
285
The operator π(1) is a contraction with respect to the metric
π πΊ 1 ,π 1 , πΊ 2 ,π 2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 13 ) 1 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 13 ) 1 π‘}
286
Indeed if we denote
Definition of πΊ ,π :
πΊ ,π = π(1)(πΊ,π)
It results
πΊ 13 1
β πΊ π 2
β€ (π13 ) 1 π‘
0 πΊ14
1 β πΊ14
2 πβ( π 13 ) 1 π 13 π( π 13 ) 1 π 13 ππ 13 +
{(π13β² ) 1 πΊ13
1 β πΊ13
2 πβ( π 13 ) 1 π 13 πβ( π 13 ) 1 π 13
π‘
0+
(π13β²β² ) 1 π14
1 , π 13 πΊ13
1 β πΊ13
2 πβ( π 13 ) 1 π 13 π( π 13 ) 1 π 13 +
πΊ13 2
|(π13β²β² ) 1 π14
1 , π 13 β (π13
β²β² ) 1 π14 2
, π 13 | πβ( π 13 ) 1 π 13 π( π 13 ) 1 π 13 }ππ 13
Where π 13 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
287
πΊ 1 β πΊ 2 πβ( π 13 ) 1 π‘ β€1
( π 13 ) 1 (π13 ) 1 + (π13β² ) 1 + ( π΄ 13 ) 1 + ( π 13 ) 1 ( π 13 ) 1 π πΊ 1 ,π 1 ; πΊ 2 ,π 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
288
Remark 1: The fact that we supposed (π13β²β² ) 1 and (π13
β²β² ) 1 depending also on t can be considered as not 289
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conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 13 ) 1 π( π 13 ) 1 π‘ πππ ( π 13 ) 1 π( π 13 ) 1 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices to
consider that (ππβ²β² ) 1 and (ππ
β²β² ) 1 , π = 13,14,15 depend only on T14 and respectively on πΊ(πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From 19 to 24 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 1 β(ππβ²β² ) 1 π14 π 13 ,π 13 ππ 13
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 1 π‘ > 0 for t > 0
290
291
Definition of ( π 13 ) 1 1, πππ ( π 13 ) 1
3 :
Remark 3: if πΊ13 is bounded, the same property have also πΊ14 πππ πΊ15 . indeed if
πΊ13 < ( π 13 ) 1 it follows ππΊ14
ππ‘β€ ( π 13 ) 1
1β (π14
β² ) 1 πΊ14 and by integrating
πΊ14 β€ ( π 13 ) 1 2
= πΊ140 + 2(π14 ) 1 ( π 13 ) 1
1/(π14
β² ) 1
In the same way , one can obtain
πΊ15 β€ ( π 13 ) 1 3
= πΊ150 + 2(π15 ) 1 ( π 13 ) 1
2/(π15
β² ) 1
If πΊ14 ππ πΊ15 is bounded, the same property follows for πΊ13 , πΊ15 and πΊ13 , πΊ14 respectively.
292
Remark 4: If πΊ13 ππ bounded, from below, the same property holds for πΊ14 πππ πΊ15 . The proof is analogous
with the preceding one. An analogous property is true if πΊ14 is bounded from below.
293
Remark 5: If T13 is bounded from below and limπ‘ββ((ππβ²β² ) 1 (πΊ π‘ , π‘)) = (π14
β² ) 1 then π14 β β.
Definition of π 1 and π1 :
Indeed let π‘1 be so that for π‘ > π‘1
(π14 ) 1 β (ππβ²β² ) 1 (πΊ π‘ , π‘) < π1 , π13 (π‘) > π 1
294
Then ππ14
ππ‘β₯ (π14 ) 1 π 1 β π1π14 which leads to
π14 β₯ (π14 ) 1 π 1
π1 1 β πβπ1π‘ + π14
0 πβπ1π‘ If we take t such that πβπ1π‘ = 1
2 it results
π14 β₯ (π14 ) 1 π 1
2 , π‘ = πππ
2
π1 By taking now π1 sufficiently small one sees that T14 is unbounded. The
same property holds for π15 if limπ‘ββ(π15β²β² ) 1 πΊ π‘ , π‘ = (π15
β² ) 1
We now state a more precise theorem about the behaviors at infinity of the solutions
295
296
It is now sufficient to take (ππ) 2
( π 16 )(2) ,(ππ) 2
( π 16 )(2) < 1 and to choose
( π 16 )(2) πππ ( π 16 )(2) large to have
297
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(ππ) 2
(π 16 ) 2 ( π 16 ) 2 + ( π 16 )(2) + πΊπ0 π
β ( π 16 )(2)+πΊπ
0
πΊπ0
β€ ( π 16 )(2)
298
(ππ) 2
(π 16 ) 2 ( π 16 )(2) + ππ0 π
β ( π 16 )(2)+ππ
0
ππ0
+ ( π 16 )(2) β€ ( π 16 )(2)
299
In order that the operator π(2) transforms the space of sextuples of functions πΊπ ,ππ satisfying 300
The operator π(2) is a contraction with respect to the metric
π πΊ19 1 , π19
1 , πΊ19 2 , π19
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1
π‘ β πΊπ 2
π‘ πβ(π 16 ) 2 π‘ ,πππ₯π‘ββ+
ππ 1
π‘ β ππ 2
π‘ πβ(π 16 ) 2 π‘}
301
Indeed if we denote
Definition of πΊ19 ,π19
: πΊ19 , π19
= π(2)(πΊ19 ,π19 )
302
It results
πΊ 16 1
β πΊ π 2
β€ (π16 ) 2 π‘
0 πΊ17
1 β πΊ17
2 πβ( π 16 ) 2 π 16 π( π 16 ) 2 π 16 ππ 16 +
{(π16β² ) 2 πΊ16
1 β πΊ16
2 πβ( π 16 ) 2 π 16 πβ( π 16 ) 2 π 16
π‘
0+
(π16β²β² ) 2 π17
1 , π 16 πΊ16
1 β πΊ16
2 πβ( π 16 ) 2 π 16 π( π 16 ) 2 π 16 +
πΊ16 2
|(π16β²β² ) 2 π17
1 , π 16 β (π16
β²β² ) 2 π17 2
, π 16 | πβ( π 16 ) 2 π 16 π( π 16 ) 2 π 16 }ππ 16
303
Where π 16 represents integrand that is integrated over the interval 0, π‘
From the hypotheses it follows
304
πΊ19 1 β πΊ19
2 eβ( M 16 ) 2 t β€1
( M 16 ) 2 (π16 ) 2 + (π16β² ) 2 + ( A 16 ) 2 + ( P 16 ) 2 ( π 16 ) 2 d πΊ19
1 , π19 1 ; πΊ19
2 , π19 2
305
And analogous inequalities for Gπ and Tπ. Taking into account the hypothesis the result follows 306
Remark 1: The fact that we supposed (π16β²β² ) 2 and (π16
β²β² ) 2 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( P 16 ) 2 e( M 16 ) 2 t and ( Q 16 ) 2 e( M 16 ) 2 t
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices to
consider that (ππβ²β² ) 2 and (ππ
β²β² ) 2 , π = 16,17,18 depend only on T17 and respectively on πΊ19 (and not on t) and hypothesis can replaced by a usual Lipschitz condition.
307
Remark 2: There does not exist any t where Gπ t = 0 and Tπ t = 0
From 19 to 24 it results
Gπ t β₯ Gπ0e β (ππ
β² ) 2 β(ππβ²β² ) 2 T17 π 16 ,π 16 dπ 16
t0 β₯ 0
Tπ t β₯ Tπ0e β(ππ
β² ) 2 t > 0 for t > 0
308
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Definition of ( M 16 ) 2 1
, ( M 16 ) 2 2
and ( M 16 ) 2 3 :
Remark 3: if G16 is bounded, the same property have also G17 and G18 . indeed if
G16 < ( M 16 ) 2 it follows dG17
dtβ€ ( M 16 ) 2
1β (π17
β² ) 2 G17 and by integrating
G17 β€ ( M 16 ) 2 2
= G170 + 2(π17 ) 2 ( M 16 ) 2
1/(π17
β² ) 2
In the same way , one can obtain
G18 β€ ( M 16 ) 2 3
= G180 + 2(π18 ) 2 ( M 16 ) 2
2/(π18
β² ) 2
If G17 or G18 is bounded, the same property follows for G16 , G18 and G16 , G17 respectively.
309
310
Remark 4: If G16 is bounded, from below, the same property holds for G17 and G18 . The proof is analogous
with the preceding one. An analogous property is true if G17 is bounded from below.
311
Remark 5: If T16 is bounded from below and limtββ((ππβ²β² ) 2 ( πΊ19 t , t)) = (π17
β² ) 2 then T17 β β.
Definition of π 2 and Ξ΅2 :
Indeed let t2 be so that for t > t2
(π17 ) 2 β (ππβ²β² ) 2 ( πΊ19 t , t) < Ξ΅2 , T16 (t) > π 2
312
Then dT17
dtβ₯ (π17 ) 2 π 2 β Ξ΅2T17 which leads to
T17 β₯ (π17 ) 2 π 2
Ξ΅2 1 β eβΞ΅2t + T17
0 eβΞ΅2t If we take t such that eβΞ΅2t = 1
2 it results
313
T17 β₯ (π17 ) 2 π 2
2 , π‘ = log
2
Ξ΅2 By taking now Ξ΅2 sufficiently small one sees that T17 is unbounded. The
same property holds for T18 if limπ‘ββ(π18β²β² ) 2 πΊ19 t , t = (π18
β² ) 2
We now state a more precise theorem about the behaviors at infinity of the solutions
314
315
It is now sufficient to take (ππ) 3
( π 20 )(3) ,(ππ) 3
( π 20 )(3) < 1 and to choose
( P 20 )(3) and ( Q 20 )(3) large to have
316
(ππ) 3
(π 20 ) 3 ( π 20 ) 3 + ( π 20 )(3) + πΊπ0 π
β ( π 20 )(3)+πΊπ
0
πΊπ0
β€ ( π 20 )(3)
317
(ππ) 3
(π 20 ) 3 ( π 20 )(3) + ππ0 π
β ( π 20 )(3)+ππ
0
ππ0
+ ( π 20 )(3) β€ ( π 20 )(3)
318
In order that the operator π(3) transforms the space of sextuples of functions πΊπ ,ππ into itself 319
The operator π(3) is a contraction with respect to the metric
π πΊ23 1 , π23
1 , πΊ23 2 , π23
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 20 ) 3 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 20 ) 3 π‘}
320
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Indeed if we denote
Definition of πΊ23 , π23
: πΊ23 , π23 = π(3) πΊ23 , π23
321
It results
πΊ 20 1
β πΊ π 2
β€ (π20 ) 3 π‘
0 πΊ21
1 β πΊ21
2 πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 ππ 20 +
{(π20β² ) 3 πΊ20
1 β πΊ20
2 πβ( π 20 ) 3 π 20 πβ( π 20 ) 3 π 20
π‘
0+
(π20β²β² ) 3 π21
1 , π 20 πΊ20
1 β πΊ20
2 πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 +
πΊ20 2
|(π20β²β² ) 3 π21
1 , π 20 β (π20
β²β² ) 3 π21 2
,π 20 | πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 }ππ 20
Where π 20 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
322
323
πΊ 1 β πΊ 2 πβ( π 20 ) 3 π‘ β€1
( π 20 ) 3 (π20 ) 3 + (π20β² ) 3 + ( π΄ 20 ) 3 + ( π 20 ) 3 ( π 20 ) 3 π πΊ23
1 , π23 1 ; πΊ23
2 , π23 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
324
Remark 1: The fact that we supposed (π20β²β² ) 3 and (π20
β²β² ) 3 depending also on t can be considered as not conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 20 ) 3 π( π 20 ) 3 π‘ πππ ( π 20) 3 π( π 20 ) 3 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices to
consider that (ππβ²β² ) 3 and (ππ
β²β² ) 3 , π = 20,21,22 depend only on T21 and respectively on πΊ23 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
325
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From 19 to 24 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 3 β(ππβ²β² ) 3 π21 π 20 ,π 20 ππ 20
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 3 π‘ > 0 for t > 0
326
Definition of ( π 20 ) 3 1, ( π 20) 3
2 πππ ( π 20) 3
3 :
Remark 3: if πΊ20 is bounded, the same property have also πΊ21 πππ πΊ22 . indeed if
πΊ20 < ( π 20 ) 3 it follows ππΊ21
ππ‘β€ ( π 20) 3
1β (π21
β² ) 3 πΊ21 and by integrating
πΊ21 β€ ( π 20 ) 3 2
= πΊ210 + 2(π21 ) 3 ( π 20) 3
1/(π21
β² ) 3
In the same way , one can obtain
πΊ22 β€ ( π 20 ) 3 3
= πΊ220 + 2(π22 ) 3 ( π 20) 3
2/(π22
β² ) 3
If πΊ21 ππ πΊ22 is bounded, the same property follows for πΊ20 , πΊ22 and πΊ20 , πΊ21 respectively.
327
Remark 4: If πΊ20 ππ bounded, from below, the same property holds for πΊ21 πππ πΊ22 . The proof is analogous
with the preceding one. An analogous property is true if πΊ21 is bounded from below.
328
Remark 5: If T20 is bounded from below and limπ‘ββ((ππβ²β² ) 3 πΊ23 π‘ , π‘) = (π21
β² ) 3 then π21 β β. 329
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Definition of π 3 and π3 :
Indeed let π‘3 be so that for π‘ > π‘3
(π21 ) 3 β (ππβ²β² ) 3 πΊ23 π‘ , π‘ < π3 , π20 (π‘) > π 3
330
Then ππ21
ππ‘β₯ (π21 ) 3 π 3 β π3π21 which leads to
π21 β₯ (π21 ) 3 π 3
π3 1 β πβπ3π‘ + π21
0 πβπ3π‘ If we take t such that πβπ3π‘ = 1
2 it results
π21 β₯ (π21 ) 3 π 3
2 , π‘ = πππ
2
π3 By taking now π3 sufficiently small one sees that T21 is unbounded. The
same property holds for π22 if limπ‘ββ(π22β²β² ) 3 πΊ23 π‘ , π‘ = (π22
β² ) 3
We now state a more precise theorem about the behaviors at infinity of the solutions
331
332
It is now sufficient to take (ππ)
4
( π 24 )(4) ,(ππ) 4
( π 24 )(4) < 1 and to choose
( P 24 )(4) and ( Q 24 )(4) large to have
333
(ππ) 4
(π 24 ) 4 ( π 24 ) 4 + ( π 24 )(4) + πΊπ0 π
β ( π 24 )(4)+πΊπ
0
πΊπ0
β€ ( π 24 )(4)
334
(ππ) 4
(π 24 ) 4 ( π 24 )(4) + ππ0 π
β ( π 24 )(4)+ππ
0
ππ0
+ ( π 24 )(4) β€ ( π 24 )(4)
335
In order that the operator π(4) transforms the space of sextuples of functions πΊπ , ππ satisfying IN to itself 336
The operator π(4) is a contraction with respect to the metric
π πΊ27 1 , π27
1 , πΊ27 2 , π27
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 24 ) 4 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 24 ) 4 π‘}
Indeed if we denote
Definition of πΊ27 , π27 : πΊ27 , π27 = π(4)( πΊ27 , π27 )
It results
πΊ 24 1
β πΊ π 2
β€ (π24 ) 4 π‘
0 πΊ25
1 β πΊ25
2 πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 ππ 24 +
{(π24β² ) 4 πΊ24
1 β πΊ24
2 πβ( π 24 ) 4 π 24 πβ( π 24 ) 4 π 24
π‘
0+
(π24β²β² ) 4 π25
1 , π 24 πΊ24
1 β πΊ24
2 πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 +
πΊ24 2
|(π24β²β² ) 4 π25
1 , π 24 β (π24
β²β² ) 4 π25 2
,π 24 | πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 }ππ 24
Where π 24 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
337
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338
πΊ27 1 β πΊ27
2 πβ( π 24 ) 4 π‘ β€1
( π 24 ) 4 (π24 ) 4 + (π24β² ) 4 + ( π΄ 24 ) 4 + ( π 24 ) 4 ( π 24 ) 4 π πΊ27
1 , π27 1 ; πΊ27
2 , π27 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
339
Remark 1: The fact that we supposed (π24β²β² ) 4 and (π24
β²β² ) 4 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 24 ) 4 π( π 24 ) 4 π‘ πππ ( π 24 ) 4 π( π 24 ) 4 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices
to consider that (ππβ²β² ) 4 and (ππ
β²β² ) 4 , π = 24,25,26 depend only on T25 and respectively on
πΊ27 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
340
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From GLOBAL EQUATIONS it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 4 β(ππβ²β² ) 4 π25 π 24 ,π 24 ππ 24
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 4 π‘ > 0 for t > 0
341
Definition of ( π 24) 4 1
, ( π 24) 4 2
πππ ( π 24) 4 3 :
Remark 3: if πΊ24 is bounded, the same property have also πΊ25 πππ πΊ26 . indeed if
πΊ24 < ( π 24 ) 4 it follows ππΊ25
ππ‘β€ ( π 24 ) 4
1β (π25
β² ) 4 πΊ25 and by integrating
πΊ25 β€ ( π 24 ) 4 2
= πΊ250 + 2(π25 ) 4 ( π 24) 4
1/(π25
β² ) 4
In the same way , one can obtain
πΊ26 β€ ( π 24 ) 4 3
= πΊ260 + 2(π26 ) 4 ( π 24) 4
2/(π26
β² ) 4
If πΊ25 ππ πΊ26 is bounded, the same property follows for πΊ24 , πΊ26 and πΊ24 , πΊ25 respectively.
342
Remark 4: If πΊ24 ππ bounded, from below, the same property holds for πΊ25 πππ πΊ26 . The proof is analogous
with the preceding one. An analogous property is true if πΊ25 is bounded from below.
343
Remark 5: If T24 is bounded from below and limπ‘ββ((ππβ²β² ) 4 ( πΊ27 π‘ , π‘)) = (π25
β² ) 4 then π25 β β.
Definition of π 4 and π4 :
Indeed let π‘4 be so that for π‘ > π‘4
(π25 ) 4 β (ππβ²β² ) 4 ( πΊ27 π‘ , π‘) < π4 , π24 (π‘) > π 4
344
Then ππ25
ππ‘β₯ (π25 ) 4 π 4 β π4π25 which leads to
π25 β₯ (π25 ) 4 π 4
π4 1 β πβπ4π‘ + π25
0 πβπ4π‘ If we take t such that πβπ4π‘ = 1
2 it results
π25 β₯ (π25 ) 4 π 4
2 , π‘ = πππ
2
π4 By taking now π4 sufficiently small one sees that T25 is unbounded. The
345
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same property holds for π26 if limπ‘ββ(π26β²β² ) 4 πΊ27 π‘ , π‘ = (π26
β² ) 4
We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS inequalities
hold also for πΊ29 ,πΊ30 ,π28 , π29 ,π30
346
It is now sufficient to take (ππ) 5
( π 28 )(5) ,(ππ) 5
( π 28 )(5) < 1 and to choose
( P 28 )(5) and ( Q 28 )(5) large to have
347
(ππ) 5
(π 28 ) 5 ( π 28 ) 5 + ( π 28 )(5) + πΊπ0 π
β ( π 28 )(5)+πΊπ
0
πΊπ0
β€ ( π 28 )(5)
348
(ππ) 5
(π 28 ) 5 ( π 28 )(5) + ππ0 π
β ( π 28 )(5)+ππ
0
ππ0
+ ( π 28 )(5) β€ ( π 28 )(5)
349
In order that the operator π(5) transforms the space of sextuples of functions πΊπ , ππ into itself 350
The operator π(5) is a contraction with respect to the metric
π πΊ31 1 , π31
1 , πΊ31 2 , π31
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 28 ) 5 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 28 ) 5 π‘}
Indeed if we denote
Definition of πΊ31 , π31 : πΊ31 , π31 = π(5) πΊ31 , π31
It results
πΊ 28 1
β πΊ π 2
β€ (π28 ) 5 π‘
0 πΊ29
1 β πΊ29
2 πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 ππ 28 +
{(π28β² ) 5 πΊ28
1 β πΊ28
2 πβ( π 28 ) 5 π 28 πβ( π 28 ) 5 π 28
π‘
0+
(π28β²β² ) 5 π29
1 , π 28 πΊ28
1 β πΊ28
2 πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 +
πΊ28 2
|(π28β²β² ) 5 π29
1 , π 28 β (π28
β²β² ) 5 π29 2
,π 28 | πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 }ππ 28
Where π 28 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
351
352
πΊ31 1 β πΊ31
2 πβ( π 28 ) 5 π‘ β€1
( π 28 ) 5 (π28 ) 5 + (π28β² ) 5 + ( π΄ 28 ) 5 + ( π 28 ) 5 ( π 28 ) 5 π πΊ31
1 , π31 1 ; πΊ31
2 , π31 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis (35,35,36) the result follows
353
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Remark 1: The fact that we supposed (π28β²β² ) 5 and (π28
β²β² ) 5 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 28 ) 5 π( π 28 ) 5 π‘ πππ ( π 28 ) 5 π( π 28 ) 5 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices
to consider that (ππβ²β² ) 5 and (ππ
β²β² ) 5 , π = 28,29,30 depend only on T29 and respectively on
πΊ31 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
354
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From GLOBAL EQUATIONS it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 5 β(ππβ²β² ) 5 π29 π 28 ,π 28 ππ 28
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 5 π‘ > 0 for t > 0
355
Definition of ( π 28) 5 1
, ( π 28) 5 2
πππ ( π 28) 5 3 :
Remark 3: if πΊ28 is bounded, the same property have also πΊ29 πππ πΊ30 . indeed if
πΊ28 < ( π 28 ) 5 it follows ππΊ29
ππ‘β€ ( π 28 ) 5
1β (π29
β² ) 5 πΊ29 and by integrating
πΊ29 β€ ( π 28 ) 5 2
= πΊ290 + 2(π29) 5 ( π 28 ) 5
1/(π29
β² ) 5
In the same way , one can obtain
πΊ30 β€ ( π 28 ) 5 3
= πΊ300 + 2(π30 ) 5 ( π 28) 5
2/(π30
β² ) 5
If πΊ29 ππ πΊ30 is bounded, the same property follows for πΊ28 , πΊ30 and πΊ28 , πΊ29 respectively.
356
Remark 4: If πΊ28 ππ bounded, from below, the same property holds for πΊ29 πππ πΊ30 . The proof is analogous
with the preceding one. An analogous property is true if πΊ29 is bounded from below.
357
Remark 5: If T28 is bounded from below and limπ‘ββ((ππβ²β² ) 5 ( πΊ31 π‘ , π‘)) = (π29
β² ) 5 then π29 β β.
Definition of π 5 and π5 :
Indeed let π‘5 be so that for π‘ > π‘5
(π29) 5 β (ππβ²β² ) 5 ( πΊ31 π‘ , π‘) < π5 , π28 (π‘) > π 5
358
359
Then ππ29
ππ‘β₯ (π29) 5 π 5 β π5π29 which leads to
π29 β₯ (π29 ) 5 π 5
π5 1 β πβπ5π‘ + π29
0 πβπ5π‘ If we take t such that πβπ5π‘ = 1
2 it results
π29 β₯ (π29 ) 5 π 5
2 , π‘ = πππ
2
π5 By taking now π5 sufficiently small one sees that T29 is unbounded. The
same property holds for π30 if limπ‘ββ(π30β²β² ) 5 πΊ31 π‘ , π‘ = (π30
β² ) 5
We now state a more precise theorem about the behaviors at infinity of the solutions
360
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Analogous inequalities hold also for πΊ33 , πΊ34 , π32 ,π33 ,π34
361
It is now sufficient to take (ππ)
6
( π 32 )(6) ,(ππ) 6
( π 32 )(6) < 1 and to choose
( P 32 )(6) and ( Q 32 )(6) large to have
362
(ππ) 6
(π 32 ) 6 ( π 32 ) 6 + ( π 32 )(6) + πΊπ0 π
β ( π 32 )(6)+πΊπ
0
πΊπ0
β€ ( π 32 )(6)
363
(ππ) 6
(π 32 ) 6 ( π 32 )(6) + ππ0 π
β ( π 32 )(6)+ππ
0
ππ0
+ ( π 32 )(6) β€ ( π 32 )(6)
364
In order that the operator π(6) transforms the space of sextuples of functions πΊπ , ππ into itself 365
The operator π(6) is a contraction with respect to the metric
π πΊ35 1 , π35
1 , πΊ35 2 , π35
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 32 ) 6 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 32 ) 6 π‘}
Indeed if we denote
Definition of πΊ35 , π35 : πΊ35 , π35 = π(6) πΊ35 , π35
It results
πΊ 32 1
β πΊ π 2
β€ (π32 ) 6 π‘
0 πΊ33
1 β πΊ33
2 πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 ππ 32 +
{(π32β² ) 6 πΊ32
1 β πΊ32
2 πβ( π 32 ) 6 π 32 πβ( π 32 ) 6 π 32
π‘
0+
(π32β²β² ) 6 π33
1 , π 32 πΊ32
1 β πΊ32
2 πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 +
πΊ32 2
|(π32β²β² ) 6 π33
1 , π 32 β (π32
β²β² ) 6 π33 2
,π 32 | πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 }ππ 32
Where π 32 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
366
367
(1) ππβ² 1 , ππ
β²β² 1 , ππ 1 , ππ
β² 1 , ππβ²β² 1 > 0,
π, π = 13,14,15
(2)The functions ππβ²β² 1 , ππ
β²β² 1 are positive continuous increasing and bounded.
Definition of (ππ) 1 , (ππ)
1 :
ππβ²β² 1 (π14 , π‘) β€ (ππ)
1 β€ ( π΄ 13 )(1)
ππβ²β² 1 (πΊ, π‘) β€ (ππ)
1 β€ (ππβ²) 1 β€ ( π΅ 13 )(1)
(3) ππππ2ββ ππβ²β² 1 π14 , π‘ = (ππ)
1
limGββ ππβ²β² 1 πΊ , π‘ = (ππ)
1
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Definition of ( π΄ 13 )(1), ( π΅ 13 )(1) :
Where ( π΄ 13 )(1), ( π΅ 13 )(1), (ππ) 1 , (ππ)
1 are positive constants
and π = 13,14,15
They satisfy Lipschitz condition:
|(ππβ²β² ) 1 π14
β² , π‘ β (ππβ²β² ) 1 π14 , π‘ | β€ ( π 13 )(1)|π14 β π14
β² |πβ( π 13 )(1)π‘
|(ππβ²β² ) 1 πΊ β² , π‘ β (ππ
β²β² ) 1 πΊ,π | < ( π 13 )(1)||πΊ β πΊ β² ||πβ( π 13 )(1)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 1 π14
β² , π‘ and(ππβ²β² ) 1 π14 , π‘ . π14
β² , π‘ and π14 , π‘ are points belonging to the interval ( π 13 )(1), ( π 13 )(1) . It is to be noted that (ππ
β²β² ) 1 π14 , π‘ is uniformly continuous. In
the eventuality of the fact, that if ( π 13 )(1) = 1 then the function (ππβ²β² ) 1 π14 , π‘ , the first augmentation coefficient attributable
to terrestrial organisms, would be absolutely continuous.
Definition of ( π 13 )(1), ( π 13 )(1) :
(Z) ( π 13 )(1), ( π 13 )(1), are positive constants
(ππ) 1
( π 13 )(1) ,(ππ) 1
( π 13 )(1) < 1
Definition of ( π 13 )(1), ( π 13 )(1) :
(AA) There exists two constants ( π 13 )(1) and ( π 13 )(1) which together with ( π 13 )(1), ( π 13 )(1), (π΄ 13 )(1)πππ ( π΅ 13 )(1) and the
constants (ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ)
1 , π = 13,14,15, satisfy the inequalities
1
( π 13 )(1)[ (ππ)
1 + (ππβ² ) 1 + ( π΄ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
1
( π 13 )(1)[ (ππ)
1 + (ππβ²) 1 + ( π΅ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
Analogous inequalities hold also for πΊ37 ,πΊ38 ,π36 ,π37 , π38
It is now sufficient to take (ππ) 7
( π 36 )(7) ,(ππ) 7
( π 36 )(7) < 7 and to choose
( P 36 )(7) and ( Q 36 )(7) large to have
368
(ππ) 7
(π 36 ) 7 ( π 36 ) 7 + ( π 36 )(7) + πΊπ0 π
β ( π 36 )(7)+πΊπ
0
πΊπ0
β€ ( π 36 )(7)
369
(ππ) 7
(π 36 ) 7 ( π 36 )(7) + ππ0 π
β ( π 36 )(7)+ππ
0
ππ0
+ ( π 36 )(7) β€ ( π 36 )(7)
370
371
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The operator π(7) is a contraction with respect to the metric
π πΊ39 1 , π39
1 , πΊ39 2 , π39
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 36 ) 7 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 36 ) 7 π‘}
Indeed if we denote
Definition of πΊ39 , π39 :
πΊ39 , π39
= π(7)( πΊ39 , π39 )
It results
πΊ 36 1
β πΊ π 2
β€ (π36 ) 7 π‘
0 πΊ37
1 β πΊ37
2 πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 ππ 36 +
{(π36β² ) 7 πΊ36
1 β πΊ36
2 πβ( π 36 ) 7 π 36 πβ( π 36 ) 7 π 36
π‘
0+
(π36β²β² ) 7 π37
1 , π 36 πΊ36
1 β πΊ36
2 πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 +
πΊ36 2
|(π36β²β² ) 7 π37
1 , π 36 β (π36
β²β² ) 7 π37 2
,π 36 | πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 }ππ 36
Where π 36 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
372
πΊ39 1 β πΊ39
2 πβ( π 36 ) 7 π‘ β€1
( π 36 ) 7 (π36 ) 7 + (π36β² ) 7 + ( π΄ 36 ) 7 + ( π 36 ) 7 ( π 36 ) 7 π πΊ39
1 , π39 1 ; πΊ39
2 , π39 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis (37,35,36) the result follows
373
374
Remark 1: The fact that we supposed (π36β²β² ) 7 and (π36
β²β² ) 7 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 36 ) 7 π( π 36 ) 7 π‘ πππ ( π 36) 7 π( π 36 ) 7 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices to
consider that (ππβ²β²) 7 and (ππ
β²β²) 7 , π = 36,37,38 depend only on T37 and respectively on πΊ39 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
375
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From 79 to 36 it results
πΊπ π‘ β₯ πΊπ0π
β (ππβ² ) 7 β(ππ
β²β²) 7 π37 π 36 ,π 36 ππ 36 π‘
0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 7 π‘ > 0 for t > 0
376
Definition of ( π 36 ) 7 1, ( π 36) 7
2 πππ ( π 36) 7
3 :
Remark 3: if πΊ36 is bounded, the same property have also πΊ37 πππ πΊ38 . indeed if
πΊ36 < ( π 36 ) 7 it follows ππΊ37
ππ‘β€ ( π 36) 7
1β (π37
β² ) 7 πΊ37 and by integrating
πΊ37 β€ ( π 36 ) 7 2
= πΊ370 + 2(π37 ) 7 ( π 36) 7
1/(π37
β² ) 7
In the same way , one can obtain
377
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πΊ38 β€ ( π 36 ) 7 3
= πΊ380 + 2(π38 ) 7 ( π 36) 7
2/(π38
β² ) 7
If πΊ37 ππ πΊ38 is bounded, the same property follows for πΊ36 , πΊ38 and πΊ36 , πΊ37 respectively.
Remark 7: If πΊ36 ππ bounded, from below, the same property holds for πΊ37 πππ πΊ38 . The proof is analogous
with the preceding one. An analogous property is true if πΊ37 is bounded from below.
378
Remark 5: If T36 is bounded from below and limπ‘ββ((ππβ²β²) 7 ( πΊ39 π‘ , π‘)) = (π37
β² ) 7 then π37 β β. Definition of π 7 and π7 :
Indeed let π‘7 be so that for π‘ > π‘7
(π37 ) 7 β (ππβ²β²) 7 ( πΊ39 π‘ , π‘) < π7 , π36 (π‘) > π 7
379
Then ππ37
ππ‘β₯ (π37 ) 7 π 7 β π7π37 which leads to
π37 β₯ (π37 ) 7 π 7
π7 1 β πβπ7π‘ + π37
0 πβπ7π‘ If we take t such that πβπ7π‘ = 1
2 it results
π37 β₯ (π37 ) 7 π 7
2 , π‘ = πππ
2
π7 By taking now π7 sufficiently small one sees that T37 is unbounded. The
same property holds for π38 if limπ‘ββ(π38β²β² ) 7 πΊ39 π‘ , π‘ = (π38
β² ) 7
We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 72
380
In order that the operator π(7) transforms the space of sextuples of functions πΊπ , ππ satisfying GLOBAL
EQUATIONS AND ITS CONCOMITANT CONDITIONALITIES into itself
381
382
The operator π(7) is a contraction with respect to the metric
π πΊ39 1 , π39
1 , πΊ39 2 , π39
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 36 ) 7 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 36 ) 7 π‘}
Indeed if we denote
Definition of πΊ39 , π39 :
πΊ39 , π39
= π(7)( πΊ39 , π39 )
It results
πΊ 36 1
β πΊ π 2
β€ (π36 ) 7 π‘
0 πΊ37
1 β πΊ37
2 πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 ππ 36 +
{(π36β² ) 7 πΊ36
1 β πΊ36
2 πβ( π 36 ) 7 π 36 πβ( π 36 ) 7 π 36
π‘
0+
(π36β²β² ) 7 π37
1 , π 36 πΊ36
1 β πΊ36
2 πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 +
πΊ36 2
|(π36β²β² ) 7 π37
1 , π 36 β (π36
β²β² ) 7 π37 2
,π 36 | πβ( π 36 ) 7 π 36 π( π 36 ) 7 π 36 }ππ 36
Where π 36 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
383
384
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πΊ39 1 β πΊ39
2 πβ( π 36 ) 7 π‘ β€1
( π 36 ) 7 (π36 ) 7 + (π36β² ) 7 + ( π΄ 36 ) 7 + ( π 36 ) 7 ( π 36 ) 7 π πΊ39
1 , π39 1 ; πΊ39
2 , π39 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π36β²β² ) 7 and (π36
β²β² ) 7 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 36 ) 7 π( π 36 ) 7 π‘ πππ ( π 36 ) 7 π( π 36 ) 7 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it suffices
to consider that (ππβ²β² ) 7 and (ππ
β²β² ) 7 , π = 36,37,38 depend only on T37 and respectively on
πΊ39 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
385
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From CONCATENATED GLOBAL EQUATIONS it results
πΊπ π‘ β₯ πΊπ0π
β (ππβ² ) 7 β(ππ
β²β² ) 7 π37 π 36 ,π 36 ππ 36 π‘
0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 7 π‘ > 0 for t > 0
386
Definition of ( π 36) 7 1
, ( π 36) 7 2
πππ ( π 36) 7 3 :
Remark 3: if πΊ36 is bounded, the same property have also πΊ37 πππ πΊ38 . indeed if
πΊ36 < ( π 36 ) 7 it follows ππΊ37
ππ‘β€ ( π 36 ) 7
1β (π37
β² ) 7 πΊ37 and by integrating
πΊ37 β€ ( π 36 ) 7 2
= πΊ370 + 2(π37 ) 7 ( π 36) 7
1/(π37
β² ) 7
In the same way , one can obtain
πΊ38 β€ ( π 36 ) 7 3
= πΊ380 + 2(π38 ) 7 ( π 36) 7
2/(π38
β² ) 7
If πΊ37 ππ πΊ38 is bounded, the same property follows for πΊ36 , πΊ38 and πΊ36 , πΊ37 respectively.
387
Remark 7: If πΊ36 ππ bounded, from below, the same property holds for πΊ37 πππ πΊ38 . The proof is analogous
with the preceding one. An analogous property is true if πΊ37 is bounded from below.
388
Remark 5: If T36 is bounded from below and limπ‘ββ((ππβ²β² ) 7 ( πΊ39 π‘ , π‘)) = (π37
β² ) 7 then π37 β β.
Definition of π 7 and π7 :
Indeed let π‘7 be so that for π‘ > π‘7
(π37 ) 7 β (ππβ²β² ) 7 ( πΊ39 π‘ , π‘) < π7 , π36 (π‘) > π 7
389
Then ππ37
ππ‘β₯ (π37 ) 7 π 7 β π7π37 which leads to
π37 β₯ (π37 ) 7 π 7
π7 1 β πβπ7π‘ + π37
0 πβπ7π‘ If we take t such that πβπ7π‘ = 1
2 it results
π37 β₯ (π37 ) 7 π 7
2 , π‘ = πππ
2
π7 By taking now π7 sufficiently small one sees that T37 is unbounded. The
390
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same property holds for π38 if limπ‘ββ(π38β²β² ) 7 πΊ39 π‘ , π‘ = (π38
β² ) 7
We now state a more precise theorem about the behaviors at infinity of the solutions
β(Ο2) 2 β€ β(π16β² ) 2 + (π17
β² ) 2 β (π16β²β² ) 2 T17 , π‘ + (π17
β²β² ) 2 T17 , π‘ β€ β(Ο1) 2 391
β(Ο2) 2 β€ β(π16β² ) 2 + (π17
β² ) 2 β (π16β²β² ) 2 πΊ19 , π‘ β (π17
β²β² ) 2 πΊ19 , π‘ β€ β(Ο1) 2 392
Definition of (π1 ) 2 , (Ξ½2) 2 , (π’1) 2 , (π’2) 2 : 393
By (π1) 2 > 0 , (Ξ½2) 2 < 0 and respectively (π’1) 2 > 0 , (π’2) 2 < 0 the roots 394
(a) of the equations (π17 ) 2 π 2 2
+ (Ο1) 2 π 2 β (π16 ) 2 = 0 395
and (π14 ) 2 π’ 2 2
+ (Ο1) 2 π’ 2 β (π16 ) 2 = 0 and 396
Definition of (π 1 ) 2 , , (π 2 ) 2 , (π’ 1) 2 , (π’ 2) 2 : 397
By (π 1) 2 > 0 , (Ξ½ 2) 2 < 0 and respectively (π’ 1) 2 > 0 , (π’ 2) 2 < 0 the 398
roots of the equations (π17 ) 2 π 2 2
+ (Ο2) 2 π 2 β (π16 ) 2 = 0 399
and (π17 ) 2 π’ 2 2
+ (Ο2) 2 π’ 2 β (π16 ) 2 = 0 400
Definition of (π1) 2 , (π2) 2 , (π1) 2 , (π2) 2 :- 401
(b) If we define (π1) 2 , (π2) 2 , (π1) 2 , (π2) 2 by 402
(π2) 2 = (π0 ) 2 , (π1) 2 = (π1 ) 2 , ππ (π0 ) 2 < (π1) 2 403
(π2) 2 = (π1 ) 2 , (π1) 2 = (π 1 ) 2 , ππ (π1) 2 < (π0 ) 2 < (π 1 ) 2 ,
and (π0 ) 2 =G16
0
G170
404
( π2) 2 = (π1 ) 2 , (π1) 2 = (π0 ) 2 , ππ (π 1) 2 < (π0 ) 2 405
and analogously
(π2) 2 = (π’0) 2 , (π1) 2 = (π’1) 2 , ππ (π’0) 2 < (π’1) 2
(π2) 2 = (π’1) 2 , (π1) 2 = (π’ 1) 2 , ππ (π’1) 2 < (π’0) 2 < (π’ 1) 2 ,
and (π’0) 2 =T16
0
T170
406
( π2) 2 = (π’1) 2 , (π1) 2 = (π’0) 2 , ππ (π’ 1) 2 < (π’0) 2 407
Then the solution satisfies the inequalities
G160 e (S1) 2 β(π16 ) 2 t β€ πΊ16 π‘ β€ G16
0 e(S1) 2 t
408
(ππ) 2 is defined 409
1
(π1) 2 G160 e (S1) 2 β(π16 ) 2 t β€ πΊ17 (π‘) β€
1
(π2 ) 2 G160 e(S1) 2 t 410
( (π18 ) 2 G16
0
(π1) 2 (S1) 2 β(π16 ) 2 β(S2 ) 2 e (S1) 2 β(π16 ) 2 t β eβ(S2 ) 2 t + G18
0 eβ(S2) 2 t β€ G18 (π‘) β€411
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(π18 ) 2 G160
(π2) 2 (S1) 2 β(π18β² ) 2
[e(S1 ) 2 t β eβ(π18β² ) 2 t] + G18
0 eβ(π18β² ) 2 t)
T160 e(R1) 2 π‘ β€ π16 (π‘) β€ T16
0 e (R1) 2 +(π16 ) 2 π‘ 412
1
(π1) 2 T160 e(R1) 2 π‘ β€ π16 (π‘) β€
1
(π2) 2 T160 e (R1) 2 +(π16 ) 2 π‘ 413
(π18 ) 2 T160
(π1) 2 (R1) 2 β(π18β² ) 2
e(R1 ) 2 π‘ β eβ(π18β² ) 2 π‘ + T18
0 eβ(π18β² ) 2 π‘ β€ π18 (π‘) β€
(π18 ) 2 T160
(π2) 2 (R1) 2 +(π16 ) 2 +(R2 ) 2 e (R1) 2 +(π16 ) 2 π‘ β eβ(R2) 2 π‘ + T18
0 eβ(R2) 2 π‘
414
Definition of (S1) 2 , (S2) 2 , (R1) 2 , (R2) 2 :- 415
Where (S1) 2 = (π16 ) 2 (π2) 2 β (π16β² ) 2
(S2) 2 = (π18 ) 2 β (π18 ) 2
416
(π 1) 2 = (π16 ) 2 (π2) 1 β (π16β² ) 2
(R2) 2 = (π18β² ) 2 β (π18 ) 2
417
418
Behavior of the solutions
If we denote and define
Definition of (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 :
(a) π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 four constants satisfying
β(π2) 3 β€ β(π20β² ) 3 + (π21
β² ) 3 β (π20β²β² ) 3 π21 , π‘ + (π21
β²β² ) 3 π21 , π‘ β€ β(π1) 3
β(π2) 3 β€ β(π20β² ) 3 + (π21
β² ) 3 β (π20β²β² ) 3 πΊ, π‘ β (π21
β²β² ) 3 πΊ23 , π‘ β€ β(π1) 3
419
Definition of (π1 ) 3 , (π2) 3 , (π’1) 3 , (π’2) 3 :
(b) By (π1) 3 > 0 , (π2 ) 3 < 0 and respectively (π’1) 3 > 0 , (π’2) 3 < 0 the roots of the equations
(π21 ) 3 π 3 2
+ (π1) 3 π 3 β (π20 ) 3 = 0
and (π21 ) 3 π’ 3 2
+ (π1) 3 π’ 3 β (π20 ) 3 = 0 and
By (π 1) 3 > 0 , (π 2 ) 3 < 0 and respectively (π’ 1) 3 > 0 , (π’ 2) 3 < 0 the
roots of the equations (π21 ) 3 π 3 2
+ (π2) 3 π 3 β (π20 ) 3 = 0
and (π21 ) 3 π’ 3 2
+ (π2) 3 π’ 3 β (π20 ) 3 = 0
420
Definition of (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 :-
(c) If we define (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 by
(π2) 3 = (π0 ) 3 , (π1) 3 = (π1) 3 , ππ (π0) 3 < (π1 ) 3
(π2) 3 = (π1) 3 , (π1) 3 = (π 1 ) 3 , ππ (π1 ) 3 < (π0 ) 3 < (π 1 ) 3 ,
and (π0 ) 3 =πΊ20
0
πΊ210
421
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( π2) 3 = (π1) 3 , (π1) 3 = (π0 ) 3 , ππ (π 1 ) 3 < (π0) 3
and analogously
(π2) 3 = (π’0) 3 , (π1) 3 = (π’1) 3 , ππ (π’0) 3 < (π’1) 3
(π2) 3 = (π’1) 3 , (π1) 3 = (π’ 1) 3 , ππ (π’1) 3 < (π’0) 3 < (π’ 1) 3 , and (π’0) 3 =π20
0
π210
( π2) 3 = (π’1) 3 , (π1) 3 = (π’0) 3 , ππ (π’ 1) 3 < (π’0) 3
Then the solution satisfies the inequalities
πΊ200 π (π1) 3 β(π20 ) 3 π‘ β€ πΊ20 (π‘) β€ πΊ20
0 π(π1) 3 π‘
(ππ) 3 is defined
422
423
1
(π1) 3 πΊ200 π (π1) 3 β(π20 ) 3 π‘ β€ πΊ21 (π‘) β€
1
(π2) 3 πΊ200 π(π1) 3 π‘ 424
( (π22 ) 3 πΊ20
0
(π1) 3 (π1) 3 β(π20 ) 3 β(π2) 3 π (π1) 3 β(π20 ) 3 π‘ β πβ(π2) 3 π‘ + πΊ22
0 πβ(π2) 3 π‘ β€ πΊ22(π‘) β€
(π22 ) 3 πΊ200
(π2) 3 (π1) 3 β(π22β² ) 3
[π(π1) 3 π‘ β πβ(π22β² ) 3 π‘ ] + πΊ22
0 πβ(π22β² ) 3 π‘)
425
π200 π(π 1 ) 3 π‘ β€ π20 (π‘) β€ π20
0 π (π 1) 3 +(π20 ) 3 π‘ 426
1
(π1) 3 π200 π(π 1) 3 π‘ β€ π20 (π‘) β€
1
(π2) 3 π200 π (π 1 ) 3 +(π20 ) 3 π‘ 427
(π22 ) 3 π200
(π1) 3 (π 1 ) 3 β(π22β² ) 3
π(π 1) 3 π‘ β πβ(π22β² ) 3 π‘ + π22
0 πβ(π22β² ) 3 π‘ β€ π22 (π‘) β€
(π22 ) 3 π200
(π2) 3 (π 1 ) 3 +(π20 ) 3 +(π 2 ) 3 π (π 1 ) 3 +(π20 ) 3 π‘ β πβ(π 2) 3 π‘ + π22
0 πβ(π 2) 3 π‘
428
Definition of (π1) 3 , (π2) 3 , (π 1) 3 , (π 2) 3 :-
Where (π1) 3 = (π20 ) 3 (π2) 3 β (π20β² ) 3
(π2) 3 = (π22 ) 3 β (π22 ) 3
(π 1) 3 = (π20 ) 3 (π2) 3 β (π20β² ) 3
(π 2) 3 = (π22β² ) 3 β (π22 ) 3
429
430
431
If we denote and define
Definition of (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 :
(d) (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 four constants satisfying
β(π2) 4 β€ β(π24β² ) 4 + (π25
β² ) 4 β (π24β²β² ) 4 π25 , π‘ + (π25
β²β² ) 4 π25 , π‘ β€ β(π1) 4
β(π2) 4 β€ β(π24β² ) 4 + (π25
β² ) 4 β (π24β²β² ) 4 πΊ27 , π‘ β (π25
β²β² ) 4 πΊ27 , π‘ β€ β(π1) 4
432
Definition of (π1 ) 4 , (π2 ) 4 , (π’1) 4 , (π’2) 4 ,π 4 ,π’ 4 :
(e) By (π1) 4 > 0 , (π2 ) 4 < 0 and respectively (π’1) 4 > 0 , (π’2) 4 < 0 the roots of the equations
433
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(π25 ) 4 π 4 2
+ (π1) 4 π 4 β (π24 ) 4 = 0
and (π25) 4 π’ 4 2
+ (π1) 4 π’ 4 β (π24 ) 4 = 0 and
Definition of (π 1 ) 4 , , (π 2) 4 , (π’ 1) 4 , (π’ 2) 4 :
By (π 1) 4 > 0 , (π 2 ) 4 < 0 and respectively (π’ 1) 4 > 0 , (π’ 2) 4 < 0 the
roots of the equations (π25 ) 4 π 4 2
+ (π2) 4 π 4 β (π24 ) 4 = 0
and (π25 ) 4 π’ 4 2
+ (π2) 4 π’ 4 β (π24 ) 4 = 0
Definition of (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 , (π0 ) 4 :-
(f) If we define (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 by
(π2) 4 = (π0 ) 4 , (π1) 4 = (π1) 4 , ππ (π0) 4 < (π1 ) 4
(π2) 4 = (π1 ) 4 , (π1) 4 = (π 1 ) 4 , ππ (π4) 4 < (π0 ) 4 < (π 1 ) 4 ,
and (π0 ) 4 =πΊ24
0
πΊ250
( π2) 4 = (π4) 4 , (π1) 4 = (π0 ) 4 , ππ (π 4) 4 < (π0) 4
434 435 436
and analogously
(π2) 4 = (π’0) 4 , (π1) 4 = (π’1) 4 , ππ (π’0) 4 < (π’1) 4
(π2) 4 = (π’1) 4 , (π1) 4 = (π’ 1) 4 , ππ (π’1) 4 < (π’0) 4 < (π’ 1) 4 ,
and (π’0) 4 =π24
0
π250
( π2) 4 = (π’1) 4 , (π1) 4 = (π’0) 4 , ππ (π’ 1) 4 < (π’0) 4 where (π’1) 4 , (π’ 1) 4 are defined respectively
437 438
Then the solution satisfies the inequalities
πΊ240 π (π1) 4 β(π24 ) 4 π‘ β€ πΊ24 π‘ β€ πΊ24
0 π(π1) 4 π‘
where (ππ) 4 is defined
439 440 441 442 443 444 445
1
(π1) 4 πΊ240 π (π1) 4 β(π24 ) 4 π‘ β€ πΊ25 π‘ β€
1
(π2) 4 πΊ240 π(π1) 4 π‘
446 447
(π26 ) 4 πΊ24
0
(π1) 4 (π1) 4 β(π24 ) 4 β(π2) 4 π (π1) 4 β(π24 ) 4 π‘ β πβ(π2) 4 π‘ + πΊ26
0 πβ(π2) 4 π‘ β€ πΊ26 π‘ β€
(π26)4πΊ240(π2)4(π1)4β(π26β²)4π(π1)4π‘βπβ(π26β²)4π‘+ πΊ260πβ(π26β²)4π‘
448
π240 π(π 1) 4 π‘ β€ π24 π‘ β€ π24
0 π (π 1 ) 4 +(π24 ) 4 π‘
449
1
(π1) 4 π240 π(π 1) 4 π‘ β€ π24 (π‘) β€
1
(π2) 4 π240 π (π 1 ) 4 +(π24 ) 4 π‘
450
(π26 ) 4 π240
(π1) 4 (π 1 ) 4 β(π26β² ) 4
π(π 1) 4 π‘ β πβ(π26β² ) 4 π‘ + π26
0 πβ(π26β² ) 4 π‘ β€ π26 (π‘) β€
(π26 ) 4 π24
0
(π2) 4 (π 1 ) 4 +(π24 ) 4 +(π 2 ) 4 π (π 1 ) 4 +(π24 ) 4 π‘ β πβ(π 2) 4 π‘ + π26
0 πβ(π 2) 4 π‘
451
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Definition of (π1) 4 , (π2) 4 , (π 1) 4 , (π 2) 4 :-
Where (π1) 4 = (π24 ) 4 (π2) 4 β (π24β² ) 4
(π2) 4 = (π26 ) 4 β (π26 ) 4
(π 1) 4 = (π24) 4 (π2) 4 β (π24β² ) 4
(π 2) 4 = (π26β² ) 4 β (π26 ) 4
452 453
Behavior of the solutions If we denote and define
Definition of (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 :
(g) (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 four constants satisfying
β(π2) 5 β€ β(π28β² ) 5 + (π29
β² ) 5 β (π28β²β² ) 5 π29 , π‘ + (π29
β²β² ) 5 π29 , π‘ β€ β(π1) 5
β(π2) 5 β€ β(π28β² ) 5 + (π29
β² ) 5 β (π28β²β² ) 5 πΊ31 , π‘ β (π29
β²β² ) 5 πΊ31 , π‘ β€ β(π1) 5
454
Definition of (π1 ) 5 , (π2 ) 5 , (π’1) 5 , (π’2) 5 ,π 5 ,π’ 5 :
(h) By (π1) 5 > 0 , (π2 ) 5 < 0 and respectively (π’1) 5 > 0 , (π’2) 5 < 0 the roots of the equations
(π29) 5 π 5 2
+ (π1) 5 π 5 β (π28 ) 5 = 0
and (π29) 5 π’ 5 2
+ (π1) 5 π’ 5 β (π28 ) 5 = 0 and
455
Definition of (π 1 ) 5 , , (π 2) 5 , (π’ 1) 5 , (π’ 2) 5 :
By (π 1) 5 > 0 , (π 2 ) 5 < 0 and respectively (π’ 1) 5 > 0 , (π’ 2) 5 < 0 the
roots of the equations (π29) 5 π 5 2
+ (π2) 5 π 5 β (π28 ) 5 = 0
and (π29) 5 π’ 5 2
+ (π2) 5 π’ 5 β (π28 ) 5 = 0
Definition of (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 , (π0 ) 5 :-
(i) If we define (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 by
(π2) 5 = (π0 ) 5 , (π1) 5 = (π1) 5 , ππ (π0) 5 < (π1 ) 5
(π2) 5 = (π1 ) 5 , (π1) 5 = (π 1 ) 5 , ππ (π1 ) 5 < (π0 ) 5 < (π 1 ) 5 ,
and (π0 ) 5 =πΊ28
0
πΊ290
( π2) 5 = (π1) 5 , (π1) 5 = (π0 ) 5 , ππ (π 1 ) 5 < (π0) 5
456
and analogously
(π2) 5 = (π’0) 5 , (π1) 5 = (π’1) 5 , ππ (π’0) 5 < (π’1) 5
(π2) 5 = (π’1) 5 , (π1) 5 = (π’ 1) 5 , ππ (π’1) 5 < (π’0) 5 < (π’ 1) 5 ,
and (π’0) 5 =π28
0
π290
( π2) 5 = (π’1) 5 , (π1) 5 = (π’0) 5 , ππ (π’ 1) 5 < (π’0) 5 where (π’1) 5 , (π’ 1) 5 are defined respectively
457
Then the solution satisfies the inequalities
458
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πΊ280 π (π1) 5 β(π28 ) 5 π‘ β€ πΊ28 (π‘) β€ πΊ28
0 π(π1) 5 π‘
where (ππ) 5 is defined
1
(π5) 5 πΊ280 π (π1) 5 β(π28 ) 5 π‘ β€ πΊ29 (π‘) β€
1
(π2) 5 πΊ280 π(π1) 5 π‘
459 460
(π30 ) 5 πΊ28
0
(π1) 5 (π1) 5 β(π28 ) 5 β(π2) 5 π (π1) 5 β(π28 ) 5 π‘ β πβ(π2) 5 π‘ + πΊ30
0 πβ(π2) 5 π‘ β€ πΊ30 π‘ β€
(π30)5πΊ280(π2)5(π1)5β(π30β²)5π(π1)5π‘βπβ(π30β²)5π‘+ πΊ300πβ(π30β²)5π‘
461
π280 π(π 1) 5 π‘ β€ π28 (π‘) β€ π28
0 π (π 1 ) 5 +(π28 ) 5 π‘
462
1
(π1) 5 π280 π(π 1) 5 π‘ β€ π28 (π‘) β€
1
(π2) 5 π280 π (π 1 ) 5 +(π28 ) 5 π‘
463
(π30 ) 5 π280
(π1) 5 (π 1 ) 5 β(π30β² ) 5
π(π 1) 5 π‘ β πβ(π30β² ) 5 π‘ + π30
0 πβ(π30β² ) 5 π‘ β€ π30 (π‘) β€
(π30 ) 5 π28
0
(π2) 5 (π 1 ) 5 +(π28 ) 5 +(π 2 ) 5 π (π 1 ) 5 +(π28 ) 5 π‘ β πβ(π 2) 5 π‘ + π30
0 πβ(π 2) 5 π‘
464
Definition of (π1) 5 , (π2) 5 , (π 1) 5 , (π 2) 5 :-
Where (π1) 5 = (π28 ) 5 (π2) 5 β (π28β² ) 5
(π2) 5 = (π30 ) 5 β (π30 ) 5
(π 1) 5 = (π28) 5 (π2) 5 β (π28β² ) 5
(π 2) 5 = (π30β² ) 5 β (π30 ) 5
465
Behavior of the solutions If we denote and define
Definition of (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 :
(j) (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 four constants satisfying
β(π2) 6 β€ β(π32β² ) 6 + (π33
β² ) 6 β (π32β²β² ) 6 π33 , π‘ + (π33
β²β² ) 6 π33 , π‘ β€ β(π1) 6
β(π2) 6 β€ β(π32β² ) 6 + (π33
β² ) 6 β (π32β²β² ) 6 πΊ35 , π‘ β (π33
β²β² ) 6 πΊ35 , π‘ β€ β(π1) 6
466
Definition of (π1 ) 6 , (π2 ) 6 , (π’1) 6 , (π’2) 6 ,π 6 ,π’ 6 :
(k) By (π1) 6 > 0 , (π2 ) 6 < 0 and respectively (π’1) 6 > 0 , (π’2) 6 < 0 the roots of the equations
(π33 ) 6 π 6 2
+ (π1) 6 π 6 β (π32 ) 6 = 0
and (π33) 6 π’ 6 2
+ (π1) 6 π’ 6 β (π32 ) 6 = 0 and
467
Definition of (π 1 ) 6 , , (π 2) 6 , (π’ 1) 6 , (π’ 2) 6 :
By (π 1) 6 > 0 , (π 2 ) 6 < 0 and respectively (π’ 1) 6 > 0 , (π’ 2) 6 < 0 the
roots of the equations (π33 ) 6 π 6 2
+ (π2) 6 π 6 β (π32 ) 6 = 0
and (π33 ) 6 π’ 6 2
+ (π2) 6 π’ 6 β (π32 ) 6 = 0
Definition of (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 , (π0 ) 6 :-
(l) If we define (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 by
468
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(π2) 6 = (π0 ) 6 , (π1) 6 = (π1) 6 , ππ (π0) 6 < (π1 ) 6
(π2) 6 = (π1 ) 6 , (π1) 6 = (π 6 ) 6 , ππ (π1 ) 6 < (π0 ) 6 < (π 1 ) 6 ,
and (π0 ) 6 =πΊ32
0
πΊ330
( π2) 6 = (π1) 6 , (π1) 6 = (π0 ) 6 , ππ (π 1 ) 6 < (π0) 6
470
and analogously
(π2) 6 = (π’0) 6 , (π1) 6 = (π’1) 6 , ππ (π’0) 6 < (π’1) 6
(π2) 6 = (π’1) 6 , (π1) 6 = (π’ 1) 6 , ππ (π’1) 6 < (π’0) 6 < (π’ 1) 6 ,
and (π’0) 6 =π32
0
π330
( π2) 6 = (π’1) 6 , (π1) 6 = (π’0) 6 , ππ (π’ 1) 6 < (π’0) 6 where (π’1) 6 , (π’ 1) 6 are defined respectively
471
Then the solution satisfies the inequalities
πΊ320 π (π1) 6 β(π32 ) 6 π‘ β€ πΊ32 (π‘) β€ πΊ32
0 π(π1) 6 π‘
where (ππ) 6 is defined
472
1
(π1) 6 πΊ320 π (π1) 6 β(π32 ) 6 π‘ β€ πΊ33 (π‘) β€
1
(π2) 6 πΊ320 π(π1) 6 π‘
473
(π34 ) 6 πΊ32
0
(π1) 6 (π1) 6 β(π32 ) 6 β(π2) 6 π (π1) 6 β(π32 ) 6 π‘ β πβ(π2) 6 π‘ + πΊ34
0 πβ(π2) 6 π‘ β€ πΊ34 π‘ β€
(π34)6πΊ320(π2)6(π1)6β(π34β²)6π(π1)6π‘βπβ(π34β²)6π‘+ πΊ340πβ(π34β²)6π‘
474
π320 π(π 1) 6 π‘ β€ π32 (π‘) β€ π32
0 π (π 1 ) 6 +(π32 ) 6 π‘
475
1
(π1) 6 π320 π(π 1) 6 π‘ β€ π32 (π‘) β€
1
(π2) 6 π320 π (π 1 ) 6 +(π32 ) 6 π‘
476
(π34 ) 6 π320
(π1) 6 (π 1 ) 6 β(π34β² ) 6
π(π 1) 6 π‘ β πβ(π34β² ) 6 π‘ + π34
0 πβ(π34β² ) 6 π‘ β€ π34 (π‘) β€
(π34 ) 6 π32
0
(π2) 6 (π 1 ) 6 +(π32 ) 6 +(π 2 ) 6 π (π 1 ) 6 +(π32 ) 6 π‘ β πβ(π 2) 6 π‘ + π34
0 πβ(π 2) 6 π‘
477
Definition of (π1) 6 , (π2) 6 , (π 1) 6 , (π 2) 6 :-
Where (π1) 6 = (π32 ) 6 (π2) 6 β (π32β² ) 6
(π2) 6 = (π34 ) 6 β (π34 ) 6
(π 1) 6 = (π32 ) 6 (π2) 6 β (π32β² ) 6
(π 2) 6 = (π34β² ) 6 β (π34 ) 6
478
If we denote and define
Definition of (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 :
(m) (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 four constants satisfying
479
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β(π2) 7 β€ β(π36β² ) 7 + (π37
β² ) 7 β (π36β²β² ) 7 π37 , π‘ + (π37
β²β² ) 7 π37 , π‘ β€ β(π1) 7
β(π2) 7 β€ β(π36β² ) 7 + (π37
β² ) 7 β (π36β²β² ) 7 πΊ39 , π‘ β (π37
β²β² ) 7 πΊ39 , π‘ β€ β(π1) 7
Definition of (π1 ) 7 , (π2 ) 7 , (π’1) 7 , (π’2) 7 ,π 7 ,π’ 7 :
(n) By (π1) 7 > 0 , (π2 ) 7 < 0 and respectively (π’1) 7 > 0 , (π’2) 7 < 0 the roots of the equations
(π37 ) 7 π 7 2
+ (π1) 7 π 7 β (π36 ) 7 = 0
and (π37) 7 π’ 7 2
+ (π1) 7 π’ 7 β (π36 ) 7 = 0 and
480
481
Definition of (π 1 ) 7 , , (π 2) 7 , (π’ 1) 7 , (π’ 2) 7 :
By (π 1) 7 > 0 , (π 2 ) 7 < 0 and respectively (π’ 1) 7 > 0 , (π’ 2) 7 < 0 the
roots of the equations (π37 ) 7 π 7 2
+ (π2) 7 π 7 β (π36 ) 7 = 0
and (π37 ) 7 π’ 7 2
+ (π2) 7 π’ 7 β (π36 ) 7 = 0
Definition of (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 , (π0 ) 7 :-
(o) If we define (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 by
(π2) 7 = (π0 ) 7 , (π1) 7 = (π1) 7 , ππ (π0) 7 < (π1 ) 7
(π2) 7 = (π1 ) 7 , (π1) 7 = (π 1 ) 7 , ππ (π1 ) 7 < (π0 ) 7 < (π 1 ) 7 ,
and (π0 ) 7 =πΊ36
0
πΊ370
( π2) 7 = (π1) 7 , (π1) 7 = (π0 ) 7 , ππ (π 1 ) 7 < (π0) 7
482
and analogously
(π2) 7 = (π’0) 7 , (π1) 7 = (π’1) 7 , ππ (π’0) 7 < (π’1) 7
(π2) 7 = (π’1) 7 , (π1) 7 = (π’ 1) 7 , ππ (π’1) 7 < (π’0) 7 < (π’ 1) 7 ,
and (π’0) 7 =π36
0
π370
( π2) 7 = (π’1) 7 , (π1) 7 = (π’0) 7 , ππ (π’ 1) 7 < (π’0) 7 where (π’1) 7 , (π’ 1) 7
are defined respectively
483
Then the solution satisfies the inequalities
πΊ360 π (π1) 7 β(π36 ) 7 π‘ β€ πΊ36 (π‘) β€ πΊ36
0 π(π1) 7 π‘
where (ππ) 7 is defined
484
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485 1
(π7) 7 πΊ360 π (π1) 7 β(π36 ) 7 π‘ β€ πΊ37 (π‘) β€
1
(π2) 7 πΊ360 π(π1) 7 π‘
486
(
(π38 ) 7 πΊ360
(π1) 7 (π1) 7 β(π36 ) 7 β(π2) 7 π (π1) 7 β(π36 ) 7 π‘ β πβ(π2) 7 π‘ + πΊ38
0 πβ(π2) 7 π‘ β€ πΊ38 (π‘) β€
(π38 ) 7 πΊ360
(π2) 7 (π1) 7 β(π38β² ) 7
[π(π1) 7 π‘ β πβ(π38β² ) 7 π‘ ] + πΊ38
0 πβ(π38β² ) 7 π‘)
487
π360 π(π 1) 7 π‘ β€ π36 (π‘) β€ π36
0 π (π 1 ) 7 +(π36 ) 7 π‘ 488
1
(π1) 7 π360 π(π 1) 7 π‘ β€ π36 (π‘) β€
1
(π2) 7 π360 π (π 1 ) 7 +(π36 ) 7 π‘ 489
(π38 ) 7 π360
(π1) 7 (π 1 ) 7 β(π38β² ) 7
π(π 1) 7 π‘ β πβ(π38β² ) 7 π‘ + π38
0 πβ(π38β² ) 7 π‘ β€ π38 (π‘) β€
(π38 ) 7 π360
(π2) 7 (π 1 ) 7 +(π36 ) 7 +(π 2 ) 7 π (π 1 ) 7 +(π36 ) 7 π‘ β πβ(π 2) 7 π‘ + π38
0 πβ(π 2) 7 π‘
490
Definition of (π1) 7 , (π2) 7 , (π 1) 7 , (π 2) 7 :-
Where (π1) 7 = (π36 ) 7 (π2) 7 β (π36β² ) 7
(π2) 7 = (π38 ) 7 β (π38 ) 7 (π 1) 7 = (π36) 7 (π2) 7 β (π36
β² ) 7
(π 2) 7 = (π38β² ) 7 β (π38 ) 7
491
From GLOBAL EQUATIONS we obtain ππ 7
ππ‘= (π36 ) 7 β (π36
β² ) 7 β (π37β² ) 7 + (π36
β²β² ) 7 π37 , π‘ β
(π37β²β² ) 7 π37 , π‘ π 7 β (π37 ) 7 π 7
Definition of π 7 :- π 7 =πΊ36
πΊ37
It follows
β (π37 ) 7 π 7 2
+ (π2) 7 π 7 β (π36 ) 7 β€ππ 7
ππ‘β€
β (π37 ) 7 π 7 2
+ (π1) 7 π 7 β (π36 ) 7
From which one obtains
Definition of (π 1) 7 , (π0) 7 :-
(a) For 0 < (π0 ) 7 =πΊ36
0
πΊ370 < (π1 ) 7 < (π 1 ) 7
π 7 (π‘) β₯(π1) 7 +(πΆ) 7 (π2) 7 π
β π37 7 (π1) 7 β(π0) 7 π‘
1+(πΆ) 7 π β π37 7 (π1) 7 β(π0) 7 π‘
, (πΆ) 7 =(π1) 7 β(π0) 7
(π0) 7 β(π2) 7
it follows (π0 ) 7 β€ π 7 (π‘) β€ (π1) 7
492
In the same manner , we get
π 7 (π‘) β€(π 1) 7 +(πΆ ) 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+(πΆ ) 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
, (πΆ ) 7 =(π 1) 7 β(π0) 7
(π0) 7 β(π 2) 7
From which we deduce (π0 ) 7 β€ π 7 (π‘) β€ (π 1) 7
493
(b) If 0 < (π1 ) 7 < (π0 ) 7 =πΊ36
0
πΊ370 < (π 1) 7 we find like in the previous case, 494
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(π1 ) 7 β€(π1) 7 + πΆ 7 (π2) 7 π
β π37 7 (π1) 7 β(π2) 7 π‘
1+ πΆ 7 π β π37 7 (π1) 7 β(π2) 7 π‘
β€ π 7 π‘ β€
(π 1) 7 + πΆ 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+ πΆ 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
β€ (π 1) 7
(c) If 0 < (π1 ) 7 β€ (π 1 ) 7 β€ (π0) 7 =πΊ36
0
πΊ370 , we obtain
(π1) 7 β€ π 7 π‘ β€(π 1) 7 + πΆ 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+ πΆ 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
β€ (π0) 7
And so with the notation of the first part of condition (c) , we have
Definition of π 7 π‘ :-
(π2) 7 β€ π 7 π‘ β€ (π1) 7 , π 7 π‘ =πΊ36 π‘
πΊ37 π‘
In a completely analogous way, we obtain
Definition of π’ 7 π‘ :-
(π2) 7 β€ π’ 7 π‘ β€ (π1) 7 , π’ 7 π‘ =π36 π‘
π37 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π36β²β² ) 7 = (π37
β²β² ) 7 , π‘πππ (π1) 7 = (π2) 7 and in this case (π1) 7 = (π 1 ) 7 if in addition (π0) 7 = (π1) 7
then π 7 π‘ = (π0 ) 7 and as a consequence πΊ36 (π‘) = (π0) 7 πΊ37 (π‘) this also defines (π0 ) 7 for the special case .
Analogously if (π36β²β² ) 7 = (π37
β²β² ) 7 , π‘πππ (π1) 7 = (π2) 7 and then
(π’1) 7 = (π’ 1) 7 if in addition (π’0) 7 = (π’1) 7 then π36 (π‘) = (π’0) 7 π37 (π‘) This is an important consequence
of the relation between (π1) 7 and (π 1) 7 , and definition of (π’0) 7 .
495
We can prove the following
If (ππβ²β² ) 7 πππ (ππ
β²β² ) 7 are independent on π‘ , and the conditions
(π36β² ) 7 (π37
β² ) 7 β π36 7 π37
7 < 0 (π36
β² ) 7 (π37β² ) 7 β π36
7 π37 7 + π36
7 π36 7 + (π37
β² ) 7 π37 7 + π36
7 π37 7 > 0
(π36β² ) 7 (π37
β² ) 7 β π36 7 π37
7 > 0 ,
(π36β² ) 7 (π37
β² ) 7 β π36 7 π37
7 β (π36β² ) 7 π37
7 β (π37β² ) 7 π37
7 + π36 7 π37
7 < 0
π€ππ‘π π36 7 , π37
7 as defined are satisfied , then the system WITH THE SATISFACTION OF THE FOLLOWING PROPERTIES HAS A SOLUTION AS DERIVED BELOW.
496
496A
496B
496C
497C
497D
497E
497F
497G
Particular case :
If (π16β²β² ) 2 = (π17
β²β² ) 2 , π‘πππ (Ο1) 2 = (Ο2) 2 and in this case (π1) 2 = (π 1) 2 if in addition (π0 ) 2 = (π1) 2
then π 2 π‘ = (π0 ) 2 and as a consequence πΊ16 (π‘) = (π0 ) 2 πΊ17 (π‘)
Analogously if (π16β²β² ) 2 = (π17
β²β² ) 2 , π‘πππ (Ο1) 2 = (Ο2) 2 and then
(π’1) 2 = (π’ 1) 2 if in addition (π’0) 2 = (π’1) 2 then π16 (π‘) = (π’0) 2 π17 (π‘) This is an important consequence
of the relation between (π1 ) 2 and (π 1) 2
498
499
From GLOBAL EQUATIONS we obtain 500
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ππ 3
ππ‘= (π20 ) 3 β (π20
β² ) 3 β (π21β² ) 3 + (π20
β²β² ) 3 π21 , π‘ β (π21β²β² ) 3 π21 , π‘ π 3 β (π21 ) 3 π 3
Definition of π 3 :- π 3 =πΊ20
πΊ21
It follows
β (π21 ) 3 π 3 2
+ (π2) 3 π 3 β (π20 ) 3 β€ππ 3
ππ‘β€ β (π21 ) 3 π 3
2+ (π1) 3 π 3 β (π20 ) 3
501
From which one obtains
(a) For 0 < (π0) 3 =πΊ20
0
πΊ210 < (π1 ) 3 < (π 1 ) 3
π 3 (π‘) β₯(π1) 3 +(πΆ) 3 (π2) 3 π
β π21 3 (π1) 3 β(π0) 3 π‘
1+(πΆ) 3 π β π21 3 (π1) 3 β(π0) 3 π‘
, (πΆ) 3 =(π1) 3 β(π0) 3
(π0) 3 β(π2) 3
it follows (π0 ) 3 β€ π 3 (π‘) β€ (π1) 3
502
In the same manner , we get
π 3 (π‘) β€(π 1) 3 +(πΆ ) 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+(πΆ ) 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
, (πΆ ) 3 =(π 1) 3 β(π0) 3
(π0) 3 β(π 2) 3
Definition of (π 1) 3 :-
From which we deduce (π0 ) 3 β€ π 3 (π‘) β€ (π 1 ) 3
503
(b) If 0 < (π1 ) 3 < (π0) 3 =πΊ20
0
πΊ210 < (π 1 ) 3 we find like in the previous case,
(π1 ) 3 β€(π1) 3 + πΆ 3 (π2) 3 π
β π21 3 (π1) 3 β(π2) 3 π‘
1+ πΆ 3 π β π21 3 (π1) 3 β(π2) 3 π‘
β€ π 3 π‘ β€
(π 1) 3 + πΆ 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+ πΆ 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
β€ (π 1) 3
504
(c) If 0 < (π1 ) 3 β€ (π 1) 3 β€ (π0 ) 3 =πΊ20
0
πΊ210 , we obtain
(π1) 3 β€ π 3 π‘ β€(π 1) 3 + πΆ 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+ πΆ 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
β€ (π0) 3
And so with the notation of the first part of condition (c) , we have
Definition of π 3 π‘ :-
(π2) 3 β€ π 3 π‘ β€ (π1) 3 , π 3 π‘ =πΊ20 π‘
πΊ21 π‘
In a completely analogous way, we obtain
Definition of π’ 3 π‘ :-
(π2) 3 β€ π’ 3 π‘ β€ (π1) 3 , π’ 3 π‘ =π20 π‘
π21 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem.
Particular case :
If (π20β²β² ) 3 = (π21
β²β² ) 3 , π‘πππ (π1) 3 = (π2) 3 and in this case (π1 ) 3 = (π 1) 3 if in addition (π0 ) 3 = (π1 ) 3
then π 3 π‘ = (π0 ) 3 and as a consequence πΊ20 (π‘) = (π0) 3 πΊ21 (π‘)
Analogously if (π20β²β² ) 3 = (π21
β²β² ) 3 , π‘πππ (π1) 3 = (π2) 3 and then
(π’1) 3 = (π’ 1) 3 if in addition (π’0) 3 = (π’1) 3 then π20 (π‘) = (π’0) 3 π21 (π‘) This is an important consequence
of the relation between (π1 ) 3 and (π 1) 3
505
506
: From GLOBAL EQUATIONS we obtain
ππ 4
ππ‘= (π24 ) 4 β (π24
β² ) 4 β (π25β² ) 4 + (π24
β²β² ) 4 π25 , π‘ β (π25β²β² ) 4 π25 , π‘ π 4 β (π25 ) 4 π 4
Definition of π 4 :- π 4 =πΊ24
πΊ25
It follows
β (π25 ) 4 π 4 2
+ (π2) 4 π 4 β (π24 ) 4 β€ππ 4
ππ‘β€ β (π25 ) 4 π 4
2+ (π4) 4 π 4 β (π24 ) 4
507 508
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From which one obtains
Definition of (π 1) 4 , (π0) 4 :-
(d) For 0 < (π0 ) 4 =πΊ24
0
πΊ250 < (π1 ) 4 < (π 1 ) 4
π 4 π‘ β₯(π1) 4 + πΆ 4 (π2) 4 π
β π25 4 (π1) 4 β(π0) 4 π‘
4+ πΆ 4 π β π25 4 (π1) 4 β(π0) 4 π‘
, πΆ 4 =(π1) 4 β(π0) 4
(π0) 4 β(π2) 4
it follows (π0 ) 4 β€ π 4 (π‘) β€ (π1) 4
In the same manner , we get
π 4 π‘ β€(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
4+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
, (πΆ ) 4 =(π 1) 4 β(π0) 4
(π0) 4 β(π 2) 4
From which we deduce (π0 ) 4 β€ π 4 (π‘) β€ (π 1) 4
509
(e) If 0 < (π1 ) 4 < (π0 ) 4 =πΊ24
0
πΊ250 < (π 1) 4 we find like in the previous case,
(π1 ) 4 β€(π1) 4 + πΆ 4 (π2) 4 π
β π25 4 (π1) 4 β(π2) 4 π‘
1+ πΆ 4 π β π25 4 (π1) 4 β(π2) 4 π‘
β€ π 4 π‘ β€
(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
1+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
β€ (π 1) 4
510
511
(f) If 0 < (π1 ) 4 β€ (π 1 ) 4 β€ (π0) 4 =πΊ24
0
πΊ250 , we obtain
(π1) 4 β€ π 4 π‘ β€(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
1+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
β€ (π0) 4
And so with the notation of the first part of condition (c) , we have
Definition of π 4 π‘ :-
(π2) 4 β€ π 4 π‘ β€ (π1) 4 , π 4 π‘ =πΊ24 π‘
πΊ25 π‘
In a completely analogous way, we obtain
Definition of π’ 4 π‘ :-
(π2) 4 β€ π’ 4 π‘ β€ (π1) 4 , π’ 4 π‘ =π24 π‘
π25 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π24β²β² ) 4 = (π25
β²β² ) 4 , π‘πππ (π1) 4 = (π2) 4 and in this case (π1) 4 = (π 1 ) 4 if in addition (π0) 4 = (π1) 4
then π 4 π‘ = (π0 ) 4 and as a consequence πΊ24 (π‘) = (π0) 4 πΊ25 (π‘) this also defines (π0 ) 4 for the special case .
Analogously if (π24β²β² ) 4 = (π25
β²β² ) 4 , π‘πππ (π1) 4 = (π2) 4 and then
(π’1) 4 = (π’ 4) 4 if in addition (π’0) 4 = (π’1) 4 then π24 (π‘) = (π’0) 4 π25 (π‘) This is an important consequence
of the relation between (π1) 4 and (π 1) 4 , and definition of (π’0) 4 .
512 513
514
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From GLOBAL EQUATIONS we obtain
ππ 5
ππ‘= (π28 ) 5 β (π28
β² ) 5 β (π29β² ) 5 + (π28
β²β² ) 5 π29 , π‘ β (π29β²β² ) 5 π29 , π‘ π 5 β (π29) 5 π 5
Definition of π 5 :- π 5 =πΊ28
πΊ29
It follows
β (π29) 5 π 5 2
+ (π2) 5 π 5 β (π28 ) 5 β€ππ 5
ππ‘β€ β (π29) 5 π 5
2+ (π1) 5 π 5 β (π28 ) 5
From which one obtains
Definition of (π 1) 5 , (π0) 5 :-
(g) For 0 < (π0 ) 5 =πΊ28
0
πΊ290 < (π1 ) 5 < (π 1 ) 5
π 5 (π‘) β₯(π1) 5 +(πΆ) 5 (π2) 5 π
β π29 5 (π1) 5 β(π0) 5 π‘
5+(πΆ) 5 π β π29 5 (π1) 5 β(π0) 5 π‘
, (πΆ) 5 =(π1) 5 β(π0) 5
(π0) 5 β(π2) 5
it follows (π0 ) 5 β€ π 5 (π‘) β€ (π1) 5
515
In the same manner , we get
π 5 (π‘) β€(π 1) 5 +(πΆ ) 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
5+(πΆ ) 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
, (πΆ ) 5 =(π 1) 5 β(π0) 5
(π0) 5 β(π 2) 5
From which we deduce (π0 ) 5 β€ π 5 (π‘) β€ (π 5) 5
516
(h) If 0 < (π1 ) 5 < (π0 ) 5 =πΊ28
0
πΊ290 < (π 1) 5 we find like in the previous case,
(π1 ) 5 β€(π1) 5 + πΆ 5 (π2) 5 π
β π29 5 (π1) 5 β(π2) 5 π‘
1+ πΆ 5 π β π29 5 (π1) 5 β(π2) 5 π‘
β€ π 5 π‘ β€
(π 1) 5 + πΆ 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
1+ πΆ 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
β€ (π 1) 5
517
(i) If 0 < (π1 ) 5 β€ (π 1 ) 5 β€ (π0) 5 =πΊ28
0
πΊ290 , we obtain
(π1) 5 β€ π 5 π‘ β€(π 1) 5 + πΆ 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
1+ πΆ 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
β€ (π0) 5
And so with the notation of the first part of condition (c) , we have
Definition of π 5 π‘ :-
(π2) 5 β€ π 5 π‘ β€ (π1) 5 , π 5 π‘ =πΊ28 π‘
πΊ29 π‘
In a completely analogous way, we obtain
Definition of π’ 5 π‘ :-
(π2) 5 β€ π’ 5 π‘ β€ (π1) 5 , π’ 5 π‘ =π28 π‘
π29 π‘
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Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π28β²β² ) 5 = (π29
β²β² ) 5 , π‘πππ (π1) 5 = (π2) 5 and in this case (π1) 5 = (π 1) 5 if in addition (π0 ) 5 = (π5) 5
then π 5 π‘ = (π0 ) 5 and as a consequence πΊ28 (π‘) = (π0) 5 πΊ29(π‘) this also defines (π0) 5 for the special case .
Analogously if (π28β²β² ) 5 = (π29
β²β² ) 5 , π‘πππ (π1) 5 = (π2) 5 and then
(π’1) 5 = (π’ 1) 5 if in addition (π’0) 5 = (π’1) 5 then π28 (π‘) = (π’0) 5 π29 (π‘) This is an important consequence
of the relation between (π1) 5 and (π 1) 5 , and definition of (π’0) 5 .
520 we obtain
ππ 6
ππ‘= (π32 ) 6 β (π32
β² ) 6 β (π33β² ) 6 + (π32
β²β² ) 6 π33 , π‘ β (π33β²β² ) 6 π33 , π‘ π 6 β (π33 ) 6 π 6
Definition of π 6 :- π 6 =πΊ32
πΊ33
It follows
β (π33 ) 6 π 6 2
+ (π2) 6 π 6 β (π32 ) 6 β€ππ 6
ππ‘β€ β (π33 ) 6 π 6
2+ (π1) 6 π 6 β (π32 ) 6
From which one obtains
Definition of (π 1) 6 , (π0) 6 :-
(j) For 0 < (π0 ) 6 =πΊ32
0
πΊ330 < (π1 ) 6 < (π 1 ) 6
π 6 (π‘) β₯(π1) 6 +(πΆ) 6 (π2) 6 π
β π33 6 (π1) 6 β(π0) 6 π‘
1+(πΆ) 6 π β π33 6 (π1) 6 β(π0) 6 π‘
, (πΆ) 6 =(π1) 6 β(π0) 6
(π0) 6 β(π2) 6
it follows (π0 ) 6 β€ π 6 (π‘) β€ (π1) 6
521
In the same manner , we get
π 6 (π‘) β€(π 1) 6 +(πΆ ) 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+(πΆ ) 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
, (πΆ ) 6 =(π 1) 6 β(π0) 6
(π0) 6 β(π 2) 6
From which we deduce (π0 ) 6 β€ π 6 (π‘) β€ (π 1) 6
522 523
(k) If 0 < (π1 ) 6 < (π0 ) 6 =πΊ32
0
πΊ330 < (π 1) 6 we find like in the previous case,
(π1 ) 6 β€(π1) 6 + πΆ 6 (π2) 6 π
β π33 6 (π1) 6 β(π2) 6 π‘
1+ πΆ 6 π β π33 6 (π1) 6 β(π2) 6 π‘
β€ π 6 π‘ β€
(π 1) 6 + πΆ 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+ πΆ 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
β€ (π 1) 6
524
(l) If 0 < (π1 ) 6 β€ (π 1 ) 6 β€ (π0) 6 =πΊ32
0
πΊ330 , we obtain
525
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(π1) 6 β€ π 6 π‘ β€(π 1) 6 + πΆ 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+ πΆ 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
β€ (π0) 6
And so with the notation of the first part of condition (c) , we have
Definition of π 6 π‘ :-
(π2) 6 β€ π 6 π‘ β€ (π1) 6 , π 6 π‘ =πΊ32 π‘
πΊ33 π‘
In a completely analogous way, we obtain
Definition of π’ 6 π‘ :-
(π2) 6 β€ π’ 6 π‘ β€ (π1) 6 , π’ 6 π‘ =π32 π‘
π33 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π32β²β² ) 6 = (π33
β²β² ) 6 , π‘πππ (π1) 6 = (π2) 6 and in this case (π1) 6 = (π 1 ) 6 if in addition (π0) 6 = (π1) 6
then π 6 π‘ = (π0 ) 6 and as a consequence πΊ32 (π‘) = (π0) 6 πΊ33 (π‘) this also defines (π0 ) 6 for the special case .
Analogously if (π32β²β² ) 6 = (π33
β²β² ) 6 , π‘πππ (π1) 6 = (π2) 6 and then
(π’1) 6 = (π’ 1) 6 if in addition (π’0) 6 = (π’1) 6 then π32 (π‘) = (π’0) 6 π33 (π‘) This is an important consequence
of the relation between (π1) 6 and (π 1) 6 , and definition of (π’0) 6 . 526
Behavior of the solutions
If we denote and define
Definition of (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 :
(p) (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 four constants satisfying
β(π2) 7 β€ β(π36β² ) 7 + (π37
β² ) 7 β (π36β²β² ) 7 π37 , π‘ + (π37
β²β² ) 7 π37 , π‘ β€ β(π1) 7
β(π2) 7 β€ β(π36β² ) 7 + (π37
β² ) 7 β (π36β²β² ) 7 πΊ39 , π‘ β (π37
β²β² ) 7 πΊ39 , π‘ β€ β(π1) 7
527
Definition of (π1 ) 7 , (π2) 7 , (π’1) 7 , (π’2) 7 ,π 7 ,π’ 7 :
(q) By (π1) 7 > 0 , (π2 ) 7 < 0 and respectively (π’1) 7 > 0 , (π’2) 7 < 0 the roots of the equations
(π37 ) 7 π 7 2
+ (π1) 7 π 7 β (π36 ) 7 = 0
and (π37 ) 7 π’ 7 2
+ (π1) 7 π’ 7 β (π36 ) 7 = 0 and
528
529
Definition of (π 1 ) 7 , , (π 2 ) 7 , (π’ 1) 7 , (π’ 2) 7 :
By (π 1) 7 > 0 , (π 2 ) 7 < 0 and respectively (π’ 1) 7 > 0 , (π’ 2) 7 < 0 the
roots of the equations (π37 ) 7 π 7 2
+ (π2) 7 π 7 β (π36 ) 7 = 0
and (π37 ) 7 π’ 7 2
+ (π2) 7 π’ 7 β (π36 ) 7 = 0
Definition of (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 , (π0 ) 7 :-
(r) If we define (π1) 7 , (π2) 7 , (π1) 7 , (π2) 7 by
530.
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(π2) 7 = (π0 ) 7 , (π1) 7 = (π1) 7 , ππ (π0) 7 < (π1 ) 7
(π2) 7 = (π1) 7 , (π1) 7 = (π 1 ) 7 , ππ (π1 ) 7 < (π0 ) 7 < (π 1 ) 7 ,
and (π0 ) 7 =πΊ36
0
πΊ370
( π2) 7 = (π1) 7 , (π1) 7 = (π0 ) 7 , ππ (π 1 ) 7 < (π0) 7
and analogously
(π2) 7 = (π’0) 7 , (π1) 7 = (π’1) 7 , ππ (π’0) 7 < (π’1) 7
(π2) 7 = (π’1) 7 , (π1) 7 = (π’ 1) 7 , ππ (π’1) 7 < (π’0) 7 < (π’ 1) 7 ,
and (π’0) 7 =π36
0
π370
( π2) 7 = (π’1) 7 , (π1) 7 = (π’0) 7 , ππ (π’ 1) 7 < (π’0) 7 where (π’1) 7 , (π’ 1) 7
are defined by 59 and 67 respectively
531
Then the solution of GLOBAL EQUATIONS satisfies the inequalities
πΊ360 π (π1) 7 β(π36 ) 7 π‘ β€ πΊ36 (π‘) β€ πΊ36
0 π(π1) 7 π‘
where (ππ) 7 is defined
532
1
(π7) 7 πΊ360 π (π1) 7 β(π36 ) 7 π‘ β€ πΊ37 (π‘) β€
1
(π2) 7 πΊ360 π(π1) 7 π‘
533
(
(π38 ) 7 πΊ360
(π1) 7 (π1) 7 β(π36 ) 7 β(π2) 7 π (π1) 7 β(π36 ) 7 π‘ β πβ(π2) 7 π‘ + πΊ38
0 πβ(π2) 7 π‘ β€ πΊ38 (π‘) β€
(π38 ) 7 πΊ360
(π2) 7 (π1) 7 β(π38β² ) 7
[π(π1) 7 π‘ β πβ(π38β² ) 7 π‘ ] + πΊ38
0 πβ(π38β² ) 7 π‘)
534
π360 π(π 1) 7 π‘ β€ π36 (π‘) β€ π36
0 π (π 1 ) 7 +(π36 ) 7 π‘ 535
1
(π1) 7 π360 π(π 1) 7 π‘ β€ π36 (π‘) β€
1
(π2) 7 π360 π (π 1 ) 7 +(π36 ) 7 π‘ 536
(π38 ) 7 π360
(π1) 7 (π 1 ) 7 β(π38β² ) 7
π(π 1) 7 π‘ β πβ(π38β² ) 7 π‘ + π38
0 πβ(π38β² ) 7 π‘ β€ π38 (π‘) β€
(π38 ) 7 π360
(π2) 7 (π 1 ) 7 +(π36 ) 7 +(π 2 ) 7 π (π 1 ) 7 +(π36 ) 7 π‘ β πβ(π 2) 7 π‘ + π38
0 πβ(π 2) 7 π‘
537
Definition of (π1) 7 , (π2) 7 , (π 1) 7 , (π 2) 7 :-
Where (π1) 7 = (π36 ) 7 (π2) 7 β (π36β² ) 7
(π2) 7 = (π38 ) 7 β (π38 ) 7
(π 1) 7 = (π36) 7 (π2) 7 β (π36β² ) 7
(π 2) 7 = (π38β² ) 7 β (π38 ) 7
538
539
From CONCATENATED GLOBAL EQUATIONS we obtain
ππ 7
ππ‘= (π36 ) 7 β (π36
β² ) 7 β (π37β² ) 7 + (π36
β²β² ) 7 π37 , π‘ β
(π37β²β² ) 7 π37 , π‘ π 7 β (π37 ) 7 π 7
540
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Definition of π 7 :- π 7 =πΊ36
πΊ37
It follows
β (π37 ) 7 π 7 2
+ (π2) 7 π 7 β (π36 ) 7 β€ππ 7
ππ‘β€
β (π37 ) 7 π 7 2
+ (π1) 7 π 7 β (π36 ) 7
From which one obtains
Definition of (π 1) 7 , (π0 ) 7 :-
(m) For 0 < (π0 ) 7 =πΊ36
0
πΊ370 < (π1) 7 < (π 1 ) 7
π 7 (π‘) β₯(π1) 7 +(πΆ) 7 (π2) 7 π
β π37 7 (π1) 7 β(π0) 7 π‘
1+(πΆ) 7 π β π37 7 (π1) 7 β(π0) 7 π‘
, (πΆ) 7 =(π1) 7 β(π0) 7
(π0) 7 β(π2) 7
it follows (π0 ) 7 β€ π 7 (π‘) β€ (π1) 7
In the same manner , we get
π 7 (π‘) β€(π 1) 7 +(πΆ ) 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+(πΆ ) 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
, (πΆ ) 7 =(π 1) 7 β(π0) 7
(π0) 7 β(π 2) 7
From which we deduce (π0 ) 7 β€ π 7 (π‘) β€ (π 1 ) 7
541
(n) If 0 < (π1 ) 7 < (π0) 7 =πΊ36
0
πΊ370 < (π 1 ) 7 we find like in the previous case,
(π1 ) 7 β€(π1) 7 + πΆ 7 (π2) 7 π
β π37 7 (π1) 7 β(π2) 7 π‘
1+ πΆ 7 π β π37 7 (π1) 7 β(π2) 7 π‘
β€ π 7 π‘ β€
(π 1) 7 + πΆ 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+ πΆ 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
β€ (π 1) 7
542
(o) If 0 < (π1) 7 β€ (π 1) 7 β€ (π0 ) 7 =πΊ36
0
πΊ370 , we obtain
(π1) 7 β€ π 7 π‘ β€(π 1) 7 + πΆ 7 (π 2) 7 π
β π37 7 (π 1) 7 β(π 2) 7 π‘
1+ πΆ 7 π β π37 7 (π 1) 7 β(π 2) 7 π‘
β€ (π0) 7
And so with the notation of the first part of condition (c) , we have
Definition of π 7 π‘ :-
(π2) 7 β€ π 7 π‘ β€ (π1) 7 , π 7 π‘ =πΊ36 π‘
πΊ37 π‘
In a completely analogous way, we obtain
Definition of π’ 7 π‘ :-
(π2) 7 β€ π’ 7 π‘ β€ (π1) 7 , π’ 7 π‘ =π36 π‘
π37 π‘
Now, using this result and replacing it in CONCATENATED GLOBAL EQUATIONS we get easily the result
543
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stated in the theorem.
Particular case :
If (π36β²β² ) 7 = (π37
β²β² ) 7 , π‘πππ (π1) 7 = (π2) 7 and in this case (π1 ) 7 = (π 1) 7 if in addition (π0 ) 7 = (π1 ) 7
then π 7 π‘ = (π0 ) 7 and as a consequence πΊ36 (π‘) = (π0) 7 πΊ37 (π‘) this also defines (π0) 7 for the special
case .
Analogously if (π36β²β² ) 7 = (π37
β²β² ) 7 , π‘πππ (π1) 7 = (π2) 7 and then
(π’1) 7 = (π’ 1) 7 if in addition (π’0) 7 = (π’1) 7 then π36 (π‘) = (π’0) 7 π37 (π‘) This is an important consequence
of the relation between (π1 ) 7 and (π 1) 7 , and definition of (π’0) 7 .
π14 1 π13 β [(π14
β² ) 1 β (π14β²β² ) 1 πΊ ]π14 = 0 544
π15 1 π14 β [(π15
β² ) 1 β (π15β²β² ) 1 πΊ ]π15 = 0 545
has a unique positive solution , which is an equilibrium solution for the system 546
π16 2 πΊ17 β (π16
β² ) 2 + (π16β²β² ) 2 π17 πΊ16 = 0 547
π17 2 πΊ16 β (π17
β² ) 2 + (π17β²β² ) 2 π17 πΊ17 = 0 548
π18 2 πΊ17 β (π18
β² ) 2 + (π18β²β² ) 2 π17 πΊ18 = 0 549
π16 2 π17 β [(π16
β² ) 2 β (π16β²β² ) 2 πΊ19 ]π16 = 0 550
π17 2 π16 β [(π17
β² ) 2 β (π17β²β² ) 2 πΊ19 ]π17 = 0 551
π18 2 π17 β [(π18
β² ) 2 β (π18β²β² ) 2 πΊ19 ]π18 = 0 552
has a unique positive solution , which is an equilibrium solution for 553
π20 3 πΊ21 β (π20
β² ) 3 + (π20β²β² ) 3 π21 πΊ20 = 0 554
π21 3 πΊ20 β (π21
β² ) 3 + (π21β²β² ) 3 π21 πΊ21 = 0 555
π22 3 πΊ21 β (π22
β² ) 3 + (π22β²β² ) 3 π21 πΊ22 = 0 556
π20 3 π21 β [(π20
β² ) 3 β (π20β²β² ) 3 πΊ23 ]π20 = 0 557
π21 3 π20 β [(π21
β² ) 3 β (π21β²β² ) 3 πΊ23 ]π21 = 0 558
π22 3 π21 β [(π22
β² ) 3 β (π22β²β² ) 3 πΊ23 ]π22 = 0 559
has a unique positive solution , which is an equilibrium solution 560
π24 4 πΊ25 β (π24
β² ) 4 + (π24β²β² ) 4 π25 πΊ24 = 0
561
π25 4 πΊ24 β (π25
β² ) 4 + (π25β²β² ) 4 π25 πΊ25 = 0 563
π26 4 πΊ25 β (π26
β² ) 4 + (π26β²β² ) 4 π25 πΊ26 = 0
564
π24 4 π25 β [(π24
β² ) 4 β (π24β²β² ) 4 πΊ27 ]π24 = 0
565
π25 4 π24 β [(π25
β² ) 4 β (π25β²β² ) 4 πΊ27 ]π25 = 0
566
π26 4 π25 β [(π26
β² ) 4 β (π26β²β² ) 4 πΊ27 ]π26 = 0
567
has a unique positive solution , which is an equilibrium solution for the system 568
π28 5 πΊ29 β (π28
β² ) 5 + (π28β²β² ) 5 π29 πΊ28 = 0
569
π29 5 πΊ28 β (π29
β² ) 5 + (π29β²β² ) 5 π29 πΊ29 = 0
570
π30 5 πΊ29 β (π30
β² ) 5 + (π30β²β² ) 5 π29 πΊ30 = 0
571
π28 5 π29 β [(π28
β² ) 5 β (π28β²β² ) 5 πΊ31 ]π28 = 0
572
π29 5 π28 β [(π29
β² ) 5 β (π29β²β² ) 5 πΊ31 ]π29 = 0
573
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π30 5 π29 β [(π30
β² ) 5 β (π30β²β² ) 5 πΊ31 ]π30 = 0
574
has a unique positive solution , which is an equilibrium solution for the system 575
π32 6 πΊ33 β (π32
β² ) 6 + (π32β²β² ) 6 π33 πΊ32 = 0
576
π33 6 πΊ32 β (π33
β² ) 6 + (π33β²β² ) 6 π33 πΊ33 = 0
577
π34 6 πΊ33 β (π34
β² ) 6 + (π34β²β² ) 6 π33 πΊ34 = 0
578
π32 6 π33 β [(π32
β² ) 6 β (π32β²β² ) 6 πΊ35 ]π32 = 0
579
π33 6 π32 β [(π33
β² ) 6 β (π33β²β² ) 6 πΊ35 ]π33 = 0
580
π34 6 π33 β [(π34
β² ) 6 β (π34β²β² ) 6 πΊ35 ]π34 = 0
584
has a unique positive solution , which is an equilibrium solution for the system 582
π36 7 πΊ37 β (π36
β² ) 7 + (π36β²β² ) 7 π37 πΊ36 = 0 583
π37 7 πΊ36 β (π37
β² ) 7 + (π37β²β² ) 7 π37 πΊ37 = 0 584
π38 7 πΊ37 β (π38
β² ) 7 + (π38β²β² ) 7 π37 πΊ38 = 0 585
586
π36 7 π37 β [(π36
β² ) 7 β (π36β²β² ) 7 πΊ39 ]π36 = 0 587
π37 7 π36 β [(π37
β² ) 7 β (π37β²β² ) 7 πΊ39 ]π37 = 0 588
π38 7 π37 β [(π38
β² ) 7 β (π38β²β² ) 7 πΊ39 ]π38 = 0
589
has a unique positive solution , which is an equilibrium solution for the system
(a) Indeed the first two equations have a nontrivial solution πΊ36 , πΊ37 if
πΉ π39 =
(π36β² ) 7 (π37
β² ) 7 β π36 7 π37
7 + (π36β² ) 7 (π37
β²β² ) 7 π37 + (π37β² ) 7 (π36
β²β² ) 7 π37 +
(π36β²β² ) 7 π37 (π37
β²β² ) 7 π37 = 0
560
Definition and uniqueness of T37β :-
After hypothesis π 0 < 0,π β > 0 and the functions (ππβ²β² ) 7 π37 being increasing, it follows that there
exists a unique π37β for which π π37
β = 0. With this value , we obtain from the three first equations
πΊ36 = π36 7 πΊ37
(π36β² ) 7 +(π36
β²β² ) 7 π37β
, πΊ38 = π38 7 πΊ37
(π38β² ) 7 +(π38
β²β² ) 7 π37β
(e) By the same argument, the equations( SOLUTIONAL) admit solutions πΊ36 , πΊ37 if
π πΊ39 = (π36β² ) 7 (π37
β² ) 7 β π36 7 π37
7 β
(π36β² ) 7 (π37
β²β² ) 7 πΊ39 + (π37β² ) 7 (π36
β²β² ) 7 πΊ39 +(π36β²β² ) 7 πΊ39 (π37
β²β² ) 7 πΊ39 = 0
561
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Where in πΊ39 πΊ36 , πΊ37 , πΊ38 ,πΊ36 ,πΊ38 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ37 taking into account the hypothesis π 0 > 0 ,π β < 0 it follows that there exists
a unique πΊ37β such that π πΊβ = 0
Finally we obtain the unique solution OF THE SYSTEM
πΊ37β given by π πΊ39
β = 0 , π37β given by π π37
β = 0 and
πΊ36β =
π36 7 πΊ37β
(π36β² ) 7 +(π36
β²β² ) 7 π37β
, πΊ38β =
π38 7 πΊ37β
(π38β² ) 7 +(π38
β²β² ) 7 π37β
562
π36β =
π36 7 π37β
(π36β² ) 7 β(π36
β²β² ) 7 πΊ39 β , π38
β = π38 7 π37
β
(π38β² ) 7 β(π38
β²β² ) 7 πΊ39 β
563
Definition and uniqueness of T21β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β²) 1 π21 being increasing, it follows that there
exists a unique π21β for which π π21
β = 0. With this value , we obtain from the three first equations
πΊ20 = π20 3 πΊ21
(π20β² ) 3 +(π20
β²β² ) 3 π21β
, πΊ22 = π22 3 πΊ21
(π22β² ) 3 +(π22
β²β² ) 3 π21β
564
565
Definition and uniqueness of T25β :-
After hypothesis π 0 < 0,π β > 0 and the functions (ππβ²β² ) 4 π25 being increasing, it follows that there
exists a unique π25β for which π π25
β = 0. With this value , we obtain from the three first equations
πΊ24 = π24 4 πΊ25
(π24β² ) 4 +(π24
β²β² ) 4 π25β
, πΊ26 = π26 4 πΊ25
(π26β² ) 4 +(π26
β²β² ) 4 π25β
566
Definition and uniqueness of T29β :-
After hypothesis π 0 < 0,π β > 0 and the functions (ππβ²β² ) 5 π29 being increasing, it follows that there
exists a unique π29β for which π π29
β = 0. With this value , we obtain from the three first equations
πΊ28 = π28 5 πΊ29
(π28β² ) 5 +(π28
β²β² ) 5 π29β
, πΊ30 = π30 5 πΊ29
(π30β² ) 5 +(π30
β²β² ) 5 π29β
567
Definition and uniqueness of T33β :-
After hypothesis π 0 < 0,π β > 0 and the functions (ππβ²β² ) 6 π33 being increasing, it follows that there
exists a unique π33β for which π π33
β = 0. With this value , we obtain from the three first equations
πΊ32 = π32 6 πΊ33
(π32β² ) 6 +(π32
β²β² ) 6 π33β
, πΊ34 = π34 6 πΊ33
(π34β² ) 6 +(π34
β²β² ) 6 π33β
568
(f) By the same argument, the equations 92,93 admit solutions πΊ13 , πΊ14 if
π πΊ = (π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 β
(π13β² ) 1 (π14
β²β² ) 1 πΊ + (π14β² ) 1 (π13
β²β² ) 1 πΊ +(π13β²β² ) 1 πΊ (π14
β²β² ) 1 πΊ = 0
Where in πΊ πΊ13 ,πΊ14 ,πΊ15 , πΊ13 , πΊ15 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ14 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there exists a
unique πΊ14β such that π πΊβ = 0
569
(g) By the same argument, the equations 92,93 admit solutions πΊ16 , πΊ17 if
Ο πΊ19 = (π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 β
570
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(π16β² ) 2 (π17
β²β² ) 2 πΊ19 + (π17β² ) 2 (π16
β²β² ) 2 πΊ19 +(π16β²β² ) 2 πΊ19 (π17
β²β² ) 2 πΊ19 = 0
Where in πΊ19 πΊ16 ,πΊ17 ,πΊ18 ,πΊ16 , πΊ18 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ17 taking into account the hypothesis Ο 0 > 0 ,π β < 0 it follows that there exists a
unique G14β such that Ο πΊ19
β = 0
571
(a) By the same argument, the concatenated equations admit solutions πΊ20 ,πΊ21 if
π πΊ23 = (π20
β² ) 3 (π21β² ) 3 β π20
3 π21 3 β
(π20β² ) 3 (π21
β²β² ) 3 πΊ23 + (π21β² ) 3 (π20
β²β² ) 3 πΊ23 +(π20β²β² ) 3 πΊ23 (π21
β²β² ) 3 πΊ23 = 0
Where in πΊ23 πΊ20 ,πΊ21 ,πΊ22 ,πΊ20 ,πΊ22 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ21 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there exists a
unique πΊ21β such that π πΊ23
β = 0
572
573
(b) By the same argument, the equations of modules admit solutions πΊ24 ,πΊ25 if
π πΊ27 = (π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 β
(π24β² ) 4 (π25
β²β² ) 4 πΊ27 + (π25β² ) 4 (π24
β²β² ) 4 πΊ27 +(π24β²β² ) 4 πΊ27 (π25
β²β² ) 4 πΊ27 = 0
Where in πΊ27 πΊ24 , πΊ25 ,πΊ26 , πΊ24 , πΊ26 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ25 taking into account the hypothesis π 0 > 0 ,π β < 0 it follows that there exists
a unique πΊ25β such that π πΊ27
β = 0
574
(c) By the same argument, the equations (modules) admit solutions πΊ28 , πΊ29 if
π πΊ31 = (π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 β
(π28β² ) 5 (π29
β²β² ) 5 πΊ31 + (π29β² ) 5 (π28
β²β² ) 5 πΊ31 +(π28β²β² ) 5 πΊ31 (π29
β²β² ) 5 πΊ31 = 0
Where in πΊ31 πΊ28 , πΊ29 , πΊ30 ,πΊ28 ,πΊ30 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ29 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there exists
a unique πΊ29β such that π πΊ31
β = 0
575
(d) By the same argument, the equations (modules) admit solutions πΊ32 ,πΊ33 if
π πΊ35 = (π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 β
(π32β² ) 6 (π33
β²β² ) 6 πΊ35 + (π33β² ) 6 (π32
β²β² ) 6 πΊ35 +(π32β²β² ) 6 πΊ35 (π33
β²β² ) 6 πΊ35 = 0
Where in πΊ35 πΊ32 , πΊ33 , πΊ34 , πΊ32 , πΊ34 must be replaced by their values It is easy to see that Ο is a decreasing
function in πΊ33 taking into account the hypothesis π 0 > 0 ,π β < 0 it follows that there exists a unique
πΊ33β such that π πΊβ = 0
578
579
580
581
Finally we obtain the unique solution of 89 to 94
πΊ14β given by π πΊβ = 0 , π14
β given by π π14β = 0 and
πΊ13β =
π13 1 πΊ14β
(π13β² ) 1 +(π13
β²β² ) 1 π14β
, πΊ15β =
π15 1 πΊ14β
(π15β² ) 1 +(π15
β²β² ) 1 π14β
π13β =
π13 1 π14β
(π13β² ) 1 β(π13
β²β² ) 1 πΊβ , π15
β = π15 1 π14
β
(π15β² ) 1 β(π15
β²β² ) 1 πΊβ
582
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Obviously, these values represent an equilibrium solution
Finally we obtain the unique solution 583
G17β given by Ο πΊ19
β = 0 , T17β given by π T17
β = 0 and 584
G16β =
a16 2 G17β
(a16β² ) 2 +(a16
β²β² ) 2 T17β
, G18β =
a18 2 G17β
(a18β² ) 2 +(a18
β²β² ) 2 T17β
585
T16β =
b16 2 T17β
(b16β² ) 2 β(b16
β²β² ) 2 πΊ19 β , T18
β = b18 2 T17
β
(b18β² ) 2 β(b18
β²β² ) 2 πΊ19 β
586
Obviously, these values represent an equilibrium solution 587
Finally we obtain the unique solution
πΊ21β given by π πΊ23
β = 0 , π21β given by π π21
β = 0 and
πΊ20β =
π20 3 πΊ21β
(π20β² ) 3 +(π20
β²β² ) 3 π21β
, πΊ22β =
π22 3 πΊ21β
(π22β² ) 3 +(π22
β²β² ) 3 π21β
π20β =
π20 3 π21β
(π20β² ) 3 β(π20
β²β² ) 3 πΊ23β
, π22β =
π22 3 π21β
(π22β² ) 3 β(π22
β²β² ) 3 πΊ23β
Obviously, these values represent an equilibrium solution
588
Finally we obtain the unique solution
πΊ25β given by π πΊ27 = 0 , π25
β given by π π25β = 0 and
πΊ24β =
π24 4 πΊ25β
(π24β² ) 4 +(π24
β²β² ) 4 π25β
, πΊ26β =
π26 4 πΊ25β
(π26β² ) 4 +(π26
β²β² ) 4 π25β
589
π24β =
π24 4 π25β
(π24β² ) 4 β(π24
β²β² ) 4 πΊ27 β , π26
β = π26 4 π25
β
(π26β² ) 4 β(π26
β²β² ) 4 πΊ27 β
Obviously, these values represent an equilibrium solution
590
Finally we obtain the unique solution
πΊ29β given by π πΊ31
β = 0 , π29β given by π π29
β = 0 and
πΊ28β =
π28 5 πΊ29β
(π28β² ) 5 +(π28
β²β² ) 5 π29β
, πΊ30β =
π30 5 πΊ29β
(π30β² ) 5 +(π30
β²β² ) 5 π29β
591
π28β =
π28 5 π29β
(π28β² ) 5 β(π28
β²β² ) 5 πΊ31 β , π30
β = π30 5 π29
β
(π30β² ) 5 β(π30
β²β² ) 5 πΊ31 β
Obviously, these values represent an equilibrium solution
592
Finally we obtain the unique solution
πΊ33β given by π πΊ35
β = 0 , π33β given by π π33
β = 0 and
πΊ32β =
π32 6 πΊ33β
(π32β² ) 6 +(π32
β²β² ) 6 π33β
, πΊ34β =
π34 6 πΊ33β
(π34β² ) 6 +(π34
β²β² ) 6 π33β
593
π32β =
π32 6 π33β
(π32β² ) 6 β(π32
β²β² ) 6 πΊ35 β , π34
β = π34 6 π33
β
(π34β² ) 6 β(π34
β²β² ) 6 πΊ35 β
Obviously, these values represent an equilibrium solution
594
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ASYMPTOTIC STABILITY ANALYSIS
Theorem 4: If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 1 πππ (ππ
β²β² ) 1
Belong to πΆ 1 ( β+) then the above equilibrium point is asymptotically stable.
Proof: Denote
Definition of πΎπ ,ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π14
β²β² ) 1
ππ14 π14
β = π14 1 ,
π(ππβ²β² ) 1
ππΊπ πΊβ = π ππ
595
596
Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597
ππΎ13
ππ‘= β (π13
β² ) 1 + π13 1 πΎ13 + π13
1 πΎ14 β π13 1 πΊ13
β π14 598
ππΎ14
ππ‘= β (π14
β² ) 1 + π14 1 πΎ14 + π14
1 πΎ13 β π14 1 πΊ14
β π14 599
ππΎ15
ππ‘= β (π15
β² ) 1 + π15 1 πΎ15 + π15
1 πΎ14 β π15 1 πΊ15
β π14 600
ππ13
ππ‘= β (π13
β² ) 1 β π13 1 π13 + π13
1 π14 + π 13 π π13β πΎπ
15π=13 601
ππ14
ππ‘= β (π14
β² ) 1 β π14 1 π14 + π14
1 π13 + π 14 (π )π14β πΎπ
15π=13 602
ππ15
ππ‘= β (π15
β² ) 1 β π15 1 π15 + π15
1 π14 + π 15 (π )π15β πΎπ
15π=13 603
If the conditions of the previous theorem are satisfied and if the functions (aπβ²β² ) 2 and (bπ
β²β² ) 2 Belong to
C 2 ( β+) then the above equilibrium point is asymptotically stable
604
Denote
Definition of πΎπ ,ππ :-
605
Gπ = Gπβ + πΎπ , Tπ = Tπ
β + ππ 606
β(π17β²β² ) 2
βT17 T17
β = π17 2 ,
β(ππβ²β² ) 2
βGπ πΊ19
β = π ππ 607
taking into account equations (global)and neglecting the terms of power 2, we obtain 608
dπΎ16
dt= β (π16
β² ) 2 + π16 2 πΎ16 + π16
2 πΎ17 β π16 2 G16
β π17 609
dπΎ17
dt= β (π17
β² ) 2 + π17 2 πΎ17 + π17
2 πΎ16 β π17 2 G17
β π17 610
dπΎ18
dt= β (π18
β² ) 2 + π18 2 πΎ18 + π18
2 πΎ17 β π18 2 G18
β π17 611
dπ16
dt= β (π16
β² ) 2 β π16 2 π16 + π16
2 π17 + π 16 π T16β πΎπ
18π=16 612
dπ17
dt= β (π17
β² ) 2 β π17 2 π17 + π17
2 π16 + π 17 (π )T17β πΎπ
18π=16 613
dπ18
dt= β (π18
β² ) 2 β π18 2 π18 + π18
2 π17 + π 18 (π )T18β πΎπ
18π=16 614
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 3 πππ (ππ
β²β² ) 3 Belong to
πΆ 3 ( β+) then the above equilibrium point is asymptotically stabl
615
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Denote
Definition of πΎπ ,ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π21
β²β² ) 3
ππ21 π21
β = π21 3 ,
π(ππβ²β² ) 3
ππΊπ πΊ23
β = π ππ
616
Then taking into account equations (global) and neglecting the terms of power 2, we obtain 617
ππΎ20
ππ‘= β (π20
β² ) 3 + π20 3 πΎ20 + π20
3 πΎ21 β π20 3 πΊ20
β π21 618
ππΎ21
ππ‘= β (π21
β² ) 3 + π21 3 πΎ21 + π21
3 πΎ20 β π21 3 πΊ21
β π21 619
ππΎ22
ππ‘= β (π22
β² ) 3 + π22 3 πΎ22 + π22
3 πΎ21 β π22 3 πΊ22
β π21 6120
ππ20
ππ‘= β (π20
β² ) 3 β π20 3 π20 + π20
3 π21 + π 20 π π20β πΎπ
22π=20 621
ππ21
ππ‘= β (π21
β² ) 3 β π21 3 π21 + π21
3 π20 + π 21 (π )π21β πΎπ
22π=20 622
ππ22
ππ‘= β (π22
β² ) 3 β π22 3 π22 + π22
3 π21 + π 22 (π )π22β πΎπ
22π=20 623
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 4 πππ (ππ
β²β² ) 4 Belong to
πΆ 4 ( β+) then the above equilibrium point is asymptotically stabl
Denote
624
Definition of πΎπ ,ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π25
β²β² ) 4
ππ25 π25
β = π25 4 ,
π(ππβ²β² ) 4
ππΊπ πΊ27
β = π ππ
625
Then taking into account equations (global) and neglecting the terms of power 2, we obtain 626
ππΎ24
ππ‘= β (π24
β² ) 4 + π24 4 πΎ24 + π24
4 πΎ25 β π24 4 πΊ24
β π25 627
ππΎ25
ππ‘= β (π25
β² ) 4 + π25 4 πΎ25 + π25
4 πΎ24 β π25 4 πΊ25
β π25 628
ππΎ26
ππ‘= β (π26
β² ) 4 + π26 4 πΎ26 + π26
4 πΎ25 β π26 4 πΊ26
β π25 629
ππ24
ππ‘= β (π24
β² ) 4 β π24 4 π24 + π24
4 π25 + π 24 π π24β πΎπ
26π=24 630
ππ25
ππ‘= β (π25
β² ) 4 β π25 4 π25 + π25
4 π24 + π 25 π π25β πΎπ
26π=24 631
ππ26
ππ‘= β (π26
β² ) 4 β π26 4 π26 + π26
4 π25 + π 26 (π )π26β πΎπ
26π=24 632
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 5 πππ (ππ
β²β² ) 5 Belong to
πΆ 5 ( β+) then the above equilibrium point is asymptotically stable
633
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Denote
Definition of πΎπ ,ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π29
β²β² ) 5
ππ29 π29
β = π29 5 ,
π(ππβ²β² ) 5
ππΊπ πΊ31
β = π ππ
634
Then taking into account equations (global) and neglecting the terms of power 2, we obtain 635
ππΎ28
ππ‘= β (π28
β² ) 5 + π28 5 πΎ28 + π28
5 πΎ29 β π28 5 πΊ28
β π29 636
ππΎ29
ππ‘= β (π29
β² ) 5 + π29 5 πΎ29 + π29
5 πΎ28 β π29 5 πΊ29
β π29 637
ππΎ30
ππ‘= β (π30
β² ) 5 + π30 5 πΎ30 + π30
5 πΎ29 β π30 5 πΊ30
β π29 638
ππ28
ππ‘= β (π28
β² ) 5 β π28 5 π28 + π28
5 π29 + π 28 π π28β πΎπ
30π=28 639
ππ29
ππ‘= β (π29
β² ) 5 β π29 5 π29 + π29
5 π28 + π 29 π π29β πΎπ
30π =28 640
ππ30
ππ‘= β (π30
β² ) 5 β π30 5 π30 + π30
5 π29 + π 30 (π )π30β πΎπ
30π=28 641
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 6 πππ (ππ
β²β² ) 6 Belong to
πΆ 6 ( β+) then the above equilibrium point is asymptotically stable
Denote
642
Definition of πΎπ ,ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π33
β²β² ) 6
ππ33 π33
β = π33 6 ,
π(ππβ²β² ) 6
ππΊπ πΊ35
β = π ππ
643
Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644
ππΎ32
ππ‘= β (π32
β² ) 6 + π32 6 πΎ32 + π32
6 πΎ33 β π32 6 πΊ32
β π33 645
ππΎ33
ππ‘= β (π33
β² ) 6 + π33 6 πΎ33 + π33
6 πΎ32 β π33 6 πΊ33
β π33 646
ππΎ34
ππ‘= β (π34
β² ) 6 + π34 6 πΎ34 + π34
6 πΎ33 β π34 6 πΊ34
β π33 647
ππ32
ππ‘= β (π32
β² ) 6 β π32 6 π32 + π32
6 π33 + π 32 π π32β πΎπ
34π=32 648
ππ33
ππ‘= β (π33
β² ) 6 β π33 6 π33 + π33
6 π32 + π 33 π π33β πΎπ
34π=32 649
ππ34
ππ‘= β (π34
β² ) 6 β π34 6 π34 + π34
6 π33 + π 34 (π )π34β πΎπ
34π=32 650
Obviously, these values represent an equilibrium solution of 79,20,36,22,23,
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 7 πππ (ππ
β²β² ) 7 Belong to
πΆ 7 ( β+) then the above equilibrium point is asymptotically stable.
Proof: Denote
651
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Definition of πΎπ ,ππ :- πΊπ = πΊπ
β + πΎπ , ππ = ππβ + ππ
π(π37
β²β² ) 7
ππ37 π37
β = π37 7 ,
π(ππβ²β² ) 7
ππΊπ πΊ39
ββ = π ππ
652
653
Then taking into account equations(SOLUTIONAL) and neglecting the terms of power 2, we obtain
654
655 ππΎ36
ππ‘= β (π36
β² ) 7 + π36 7 πΎ36 + π36
7 πΎ37 β π36 7 πΊ36
β π37 656
ππΎ37
ππ‘= β (π37
β² ) 7 + π37 7 πΎ37 + π37
7 πΎ36 β π37 7 πΊ37
β π37 657
ππΎ38
ππ‘= β (π38
β² ) 7 + π38 7 πΎ38 + π38
7 πΎ37 β π38 7 πΊ38
β π37 658
ππ36
ππ‘= β (π36
β² ) 7 β π36 7 π36 + π36
7 π37 + π 36 π π36β πΎπ
38π=36 659
ππ37
ππ‘= β (π37
β² ) 7 β π37 7 π37 + π37
7 π36 + π 37 π π37β πΎπ
38π=36 660
ππ38
ππ‘= β (π38
β² ) 7 β π38 7 π38 + π38
7 π37 + π 38 (π )π38β πΎπ
38π=36 661
2.
The characteristic equation of this system is
π 1 + (π15β² )
1 β π15
1 { π 1 + (π15β² )
1 + π
15 1
π 1 + (π13β² )
1 + π
13 1
π14 1
πΊ14β + π14
1 π13 1
πΊ13β
π 1 + (π13β² )
1 β π13
1 π 14 , 14 π14β + π14
1 π 13 , 14 π14β
+ π 1 + (π14β² ) 1 + π14
1 π13 1 πΊ13
β + π13 1 π14
1 πΊ14β
π 1 + (π13β² )
1 β π13
1 π 14 , 13 π14β + π14
1 π 13 , 13 π13β
π 1 2
+ (π13β² )
1 + (π
14β² )
1 + π
13 1
+ π14 1
π 1
π 1 2
+ (π13β² )
1 + (π
14β² )
1 β π13
1 + π14 1 π 1
+ π 1 2
+ (π13β² ) 1 + (π14
β² ) 1 + π13 1 + π14
1 π 1 π15 1 πΊ15
+ π 1 + (π13β² ) 1 + π13
1 π15 1 π14
1 πΊ14β + π14
1 π15 1 π13
1 πΊ13β
π 1 + (π13β² )
1 β π13
1 π 14 , 15 π14β + π14
1 π 13 , 15 π13β } = 0
+
π 2 + (π18β² )
2 β π18
2 { π 2 + (π18β² )
2 + π
18 2
π 2 + (π16β² )
2 + π
16 2
π17 2
G17β + π17
2 π16 2
G16β
π 2 + (π16β² )
2 β π16
2 π 17 , 17 T17β + π17
2 π 16 , 17 T17β
+ π 2 + (π17β² ) 2 + π17
2 π16 2 G16
β + π16 2 π17
2 G17β
π 2 + (π16β² )
2 β π16
2 π 17 , 16 T17β + π17
2 π 16 , 16 T16β
π 2 2
+ (π16β² )
2 + (π
17β² )
2 + π
16 2
+ π17 2
π 2
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π 2 2
+ (π16β² ) 2 + (π17
β² ) 2 β π16 2 + π17
2 π 2
+ π 2 2
+ (π16β² ) 2 + (π17
β² ) 2 + π16 2 + π17
2 π 2 π18 2 G18
+ π 2 + (π16β² ) 2 + π16
2 π18 2 π17
2 G17β + π17
2 π18 2 π16
2 G16β
π 2 + (π16β² )
2 β π16
2 π 17 , 18 T17β + π17
2 π 16 , 18 T16β } = 0
+
π 3 + (π22β² )
3 β π22
3 { π 3 + (π22β² )
3 + π
22 3
π 3 + (π20β² )
3 + π
20 3
π21 3
πΊ21β + π21
3 π20 3
πΊ20β
π 3 + (π20β² )
3 β π20
3 π 21 , 21 π21β + π21
3 π 20 , 21 π21β
+ π 3 + (π21β² ) 3 + π21
3 π20 3 πΊ20
β + π20 3 π21
1 πΊ21β
π 3 + (π20β² )
3 β π20
3 π 21 , 20 π21β + π21
3 π 20 , 20 π20β
π 3 2
+ (π20β² )
3 + (π
21β² )
3 + π
20 3
+ π21 3
π 3
π 3 2
+ (π20β² )
3 + (π
21β² )
3 β π20
3 + π21 3 π 3
+ π 3 2
+ (π20β² ) 3 + (π21
β² ) 3 + π20 3 + π21
3 π 3 π22 3 πΊ22
+ π 3 + (π20β² ) 3 + π20
3 π22 3 π21
3 πΊ21β + π21
3 π22 3 π20
3 πΊ20β
π 3 + (π20β² )
3 β π20
3 π 21 , 22 π21β + π21
3 π 20 , 22 π20β } = 0
+
π 4 + (π26β² )
4 β π26
4 { π 4 + (π26β² )
4 + π
26 4
π 4 + (π24β² )
4 + π
24 4
π25 4
πΊ25β + π25
4 π24 4
πΊ24β
π 4 + (π24β² )
4 β π24
4 π 25 , 25 π25β + π25
4 π 24 , 25 π25β
+ π 4 + (π25β² ) 4 + π25
4 π24 4 πΊ24
β + π24 4 π25
4 πΊ25β
π 4 + (π24β² )
4 β π24
4 π 25 , 24 π25β + π25
4 π 24 , 24 π24β
π 4 2
+ (π24β² )
4 + (π
25β² )
4 + π
24 4
+ π25 4
π 4
π 4 2
+ (π24β² ) 4 + (π25
β² ) 4 β π24 4 + π25
4 π 4
+ π 4 2
+ (π24β² ) 4 + (π25
β² ) 4 + π24 4 + π25
4 π 4 π26 4 πΊ26
+ π 4 + (π24β² ) 4 + π24
4 π26 4 π25
4 πΊ25β + π25
4 π26 4 π24
4 πΊ24β
π 4 + (π24β² )
4 β π24
4 π 25 , 26 π25β + π25
4 π 24 , 26 π24β } = 0
+
π 5 + (π30β² )
5 β π30
5 { π 5 + (π30β² )
5 + π
30 5
π 5 + (π28β² )
5 + π
28 5
π29 5
πΊ29β + π29
5 π28 5
πΊ28β
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π 5 + (π28β² )
5 β π28
5 π 29 , 29 π29β + π29
5 π 28 , 29 π29β
+ π 5 + (π29β² ) 5 + π29
5 π28 5 πΊ28
β + π28 5 π29
5 πΊ29β
π 5 + (π28β² )
5 β π28
5 π 29 , 28 π29β + π29
5 π 28 , 28 π28β
π 5 2
+ (π28β² )
5 + (π
29β² )
5 + π
28 5
+ π29 5
π 5
π 5 2
+ (π28β² ) 5 + (π29
β² ) 5 β π28 5 + π29
5 π 5
+ π 5 2
+ (π28β² ) 5 + (π29
β² ) 5 + π28 5 + π29
5 π 5 π30 5 πΊ30
+ π 5 + (π28β² ) 5 + π28
5 π30 5 π29
5 πΊ29β + π29
5 π30 5 π28
5 πΊ28β
π 5 + (π28β² )
5 β π28
5 π 29 , 30 π29β + π29
5 π 28 , 30 π28β } = 0
+
π 6 + (π34β² )
6 β π34
6 { π 6 + (π34β² )
6 + π
34 6
π 6 + (π32β² )
6 + π
32 6
π33 6
πΊ33β + π33
6 π32 6
πΊ32β
π 6 + (π32β² )
6 β π32
6 π 33 , 33 π33β + π33
6 π 32 , 33 π33β
+ π 6 + (π33β² ) 6 + π33
6 π32 6 πΊ32
β + π32 6 π33
6 πΊ33β
π 6 + (π32β² )
6 β π32
6 π 33 , 32 π33β + π33
6 π 32 , 32 π32β
π 6 2
+ (π32β² )
6 + (π
33β² )
6 + π
32 6
+ π33 6
π 6
π 6 2
+ (π32β² ) 6 + (π33
β² ) 6 β π32 6 + π33
6 π 6
+ π 6 2
+ (π32β² ) 6 + (π33
β² ) 6 + π32 6 + π33
6 π 6 π34 6 πΊ34
+ π 6 + (π32β² ) 6 + π32
6 π34 6 π33
6 πΊ33β + π33
6 π34 6 π32
6 πΊ32β
π 6 + (π32β² )
6 β π32
6 π 33 , 34 π33β + π33
6 π 32 , 34 π32β } = 0
+
π 7 + (π38β² ) 7 β π38
7 { π 7 + (π38β² ) 7 + π38
7
π 7 + (π36β² ) 7 + π36
7 π37 7 πΊ37
β + π37 7 π36
7 πΊ36β
π 7 + (π36β² ) 7 β π36
7 π 37 , 37 π37β + π37
7 π 36 , 37 π37β
+ π 7 + (π37β² ) 7 + π37
7 π36 7 πΊ36
β + π36 7 π37
7 πΊ37β
π 7 + (π36β² ) 7 β π36
7 π 37 , 36 π37β + π37
7 π 36 , 36 π36β
π 7 2
+ (π36β² ) 7 + (π37
β² ) 7 + π36 7 + π37
7 π 7
π 7 2
+ (π36β² ) 7 + (π37
β² ) 7 β π36 7 + π37
7 π 7
+ π 7 2
+ (π36β² ) 7 + (π37
β² ) 7 + π36 7 + π37
7 π 7 π38 7 πΊ38
+ π 7 + (π36β² ) 7 + π36
7 π38 7 π37
7 πΊ37β + π37
7 π38 7 π36
7 πΊ36β
π 7 + (π36β² ) 7 β π36
7 π 37 , 38 π37β + π37
7 π 36 , 38 π36β } = 0
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(20)^ [2] Cockcroft-Walton experiment
(21)^ a b c Conversions used: 1956 International (Steam) Table (IT) values where one calorie β‘ 4.1868 J and one BTU β‘ 1055.05585262 J. Weapons designers' conversion value of one gram TNT β‘
1000 calories used.
(22)^ Assuming the dam is generating at its peak capacity of 6,809 MW.
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=======================================================================AAcknowled
gments:
The introduction is a collection of information from various articles, Books, News Paper
reports, Home Pages Of authors, Journal Reviews, Nature βs L:etters,Article
Abstracts, Research papers, Abstracts Of Research Papers, Stanford Encyclopedia,
Web Pages, Ask a Physicist Column, Deliberations with Professors, the internet
including Wikipedia. We acknowledge all authors who have contributed to the same. In
the eventuality of the fact that there has been any act of omission on the part of the
authors, we regret with great deal of compunction, contrition, regret, trepidation and
remorse. As Newton said, it is only because erudite and eminent people allowed one
to piggy ride on their backs; probably an attempt has been made to look slightly
further. Once again, it is stated that the references are only illustrative and not
comprehensive
First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics, Political Science.
Thesis was based on Mathematical Modeling. He was recently awarded D.litt. for his work on βMathematical Models in
Political Scienceβ--- Department of studies in Mathematics, Kuvempu University, Shimoga, Karnataka, India Corresponding
Author:[email protected]
Second Author: 2Prof. B.S Kiranagi is the Former Chairman of the Department of Studies in Mathematics, Manasa
Gangotri and present Professor Emeritus of UGC in the Department. Professor Kiranagi has guided over 25 students and he
has received many encomiums and laurels for his contribution to Co homology Groups and Mathematical Sciences. Known
for his prolific writing, and one of the senior most Professors of the country, he has over 150 publications to his credit. A
prolific writer and a prodigious thinker, he has to his credit several books on Lie Groups, Co Homology Groups, and other
mathematical application topics, and excellent publication history.-- UGC Emeritus Professor (Department of studies in
Mathematics), Manasagangotri, University of Mysore, Karnataka, India
Third Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Department of Studies in
Computer Science and has guided over 25 students. He has published articles in both national and international journals.
Professor Bagewadi specializes in Differential Geometry and its wide-ranging ramifications. He has to his credit more than
159 research papers. Several Books on Differential Geometry, Differential Equations are coauthored by him--- Chairman,
Department of studies in Mathematics and Computer science, Jnanasahyadri Kuvempu University, Shankarghatta, Shimoga
district, Karnataka, India
==============================================================================
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