International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Operation-A Checks and Balances Model

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 “General Ledger” 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 “General Ledger” 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. Turing’s 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 –A 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 (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 Church–Turing 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 Church–Turing thesis (also known as the Church–Turing 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 Church–Turing thesis states that "everything algorithmically computable is computable by a Turing machine.‖ American mathematician Alonzo Church created a method for defining functions called the λ-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 λ-calculus, and the Turing machine) were shown to be equivalent—all 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 Church–Turing 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 λ-function). The Church–Turing 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 thesis—the www.ijmer.com 2028 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 notion of what it means for a function to be "effectively calculable" (computable)—is "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 ―General Ledger‖ 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 ―General Ledger‖ 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]: qiaj→qi1aj1dk, 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.

www.ijmer.com 2029 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Note that every part of the machine—its state and symbol-collections—and its actions—printing, erasing and tape motion— 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-½ 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. www.ijmer.com 2030 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 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.

www.ijmer.com 2031 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Sublime synthesis not dismal anchorage: Many real-world computational systems compute more than just a single function—the 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 www.ijmer.com 2032 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 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) www.ijmer.com 2033 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 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 1936–1937, 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 Post–Turing 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 performed—i.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 Δ (which we suppose not to appear elsewhere) and then by the note of instructions. This expression is called the 'state formula'. —Undecidable, p.139–140, 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"—the instruction's label—beneath 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 Gödel 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 TABLE—an "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 concepts—e.g. machines with "reset" states and machines with repeating patterns (cf Hill and Peterson p. 244ff)—can 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", Hopcroft– 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. 139–140). , 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 Church–Turing thesis hypothesizes this to be true for any kind of machine: that anything that can be "computed" can be computed by some Turing machine.)

www.ijmer.com 2035 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 A Turing machine is equivalent to a 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"—its "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. —Undecidable, 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 ‡, 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"—an unspecified entity "apart from saying that it cannot be a machine" (Turing (1939), Undecidable p. 166–168). 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"—"U" for short—is 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. —Minsky (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 . 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

www.ijmer.com 2036 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 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"—memory "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.)

―A‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) OR STATE: THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON.

MODULE NUMBERED ONE NOTATION : 퐺13 : CATEGORY ONE OF‖A‖ 퐺14 : CATEGORY TWO OF‖A‖ 퐺15 : CATEGORY THREE OF ‗A‘ 푇13 : CATEGORY ONE OF ‗B‘ 푇14 : CATEGORY TWO OF ‗B‘ 푇15 :CATEGORY THREE OF ‗B‘

―B‖ AND ―A‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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. ======퐺16 : CATEGORY ONE OF ‗B‘ (NOTE THAT THEY REPRESENT CONFIGURATIONS,INSTRUCTIONS OR STATES) 퐺17 : CATEGORY TWO OF ‗B‘ www.ijmer.com 2037 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

퐺18 : CATEGORY THREE OF ‗B‘ 푇16 :CATEGORY ONE OF ‗A‘ 푇17 : CATEGORY TWO OF ‗A‘ 푇18 : CATEGORY THREE OF‘A‘ ======A‖ AND ―C‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) OR STATE: THE CONVENTION SHOWN IS AFTER MCCLUSKY,BOOTH,HILL AND PETERSON. MODULE NUMBERED THREE ======

퐺20 : CATEGORY ONE OF‘A‘ 퐺21 :CATEGORY TWO OF‘A‘ 퐺22 : CATEGORY THREE OF‘A‘ 푇20 : CATEGORY ONE OF ‗C‘ 푇21 :CATEGORY TWO OF ‗C‘ 푇22 : CATEGORY THREE OF ‗C‘

―C‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 : ======

퐺24 : CATEGORY ONE OF ―C‖(EVALUATIVE PARAMETRICIZATION OF SITUATIONAL ORIENTATIONS AND ESSENTIAL COGNITIVE ORIENTATION AND CHOICE VARIABLES OF THE SYSTEM TO WHICH CONFIGURATION IS APPLICABLE) 퐺25 : CATEGORY TWO OF ―C‖ 퐺26 : CATEGORY THREE OF ―C‖ 푇24 :CATEGORY ONE OF ―B‖ 푇25 :CATEGORY TWO OF ―B‖(SYSTEMIC INSTRUMENTAL CHARACTERISATIONS AND ACTION ORIENTATIONS AND FUNCTIONAL IMPERATIVES OF CHANGE MANIFESTED THEREIN ) 푇26 : CATEGORY THREE OF‖B‖ ―C‖ AND ―H‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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: ======퐺28 : CATEGORY ONE OF ―C‖ 퐺29 : CATEGORY TWO OF‖C‖ 퐺30 :CATEGORY THREE OF ―C‖ 푇28 :CATEGORY ONE OF ―H‖ 푇29 :CATEGORY TWO OF ―H‖ 푇30 :CATEGORY THREE OF ―H‖ ======―B‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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

www.ijmer.com 2038 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 EAILY BY REPLACING THE COEFFICIENTS BY EQUALITY SIGN OR GIVING IT THE POSITION OF ZERO:

======퐺32 : CATEGORY ONE OF ―B‖ 퐺33 : CATEGORY TWO OF ―B‖ 퐺34 : CATEGORY THREE OF‖B‖ INTERACTS WITH:ITSELF: 푇32 : CATEGORY ONE OF ―B‖ 푇33 : CATEGORY TWO OF‖B‖ 푇34 : CATEGORY THREE OF ―B‖ ―INPUT‖ AND ―A‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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: ======퐺36 : CATEGORY ONE OF ―INPUT‖ 퐺37 : CATEGORY TWO OF ―INPUT‖ 퐺38 : CATEGORY THREE OF‖INPUT‖ (INPUT FEEDING AND CONCOMITANT GENERATION OF ENERGY DIFFERENTIAL-TIME LAG OR INSTANTANEOUSNESSMIGHT EXISTS WHEREBY ACCENTUATION AND ATTRITIONS MODEL MAY ASSUME ZERO POSITIONS) 푇36 : CATEGORY ONE OF ―A‖ 푇37 : CATEGORY TWO OF "A" 푇38 : CATEGORY THREE OF‖A‖ ======

1 1 1 1 1 1 2 2 2 2 2 2 푎13 , 푎14 , 푎15 , 푏13 , 푏14 , 푏15 푎16 , 푎17 , 푎18 푏16 , 푏17 , 푏18 : 3 3 3 3 3 3 푎20 , 푎21 , 푎22 , 푏20 , 푏21 , 푏22 4 4 4 4 4 4 5 5 5 5 5 5 푎24 , 푎25 , 푎26 , 푏24 , 푏25 , 푏26 , 푏28 , 푏29 , 푏30 , 푎28 , 푎29 , 푎30 , 6 6 6 6 6 6 푎32 , 푎33 , 푎34 , 푏32 , 푏33 , 푏34 are Accentuation coefficients

′ 1 ′ 1 ′ 1 ′ 1 ′ 1 ′ 1 ′ 2 ′ 2 ′ 2 ′ 2 ′ 2 ′ 2 푎13 , 푎14 , 푎15 , 푏13 , 푏14 , 푏15 , 푎16 , 푎17 , 푎18 , 푏16 , 푏17 , 푏18 ′ 3 ′ 3 ′ 3 ′ 3 ′ 3 ′ 3 , 푎20 , 푎21 , 푎22 , 푏20 , 푏21 , 푏22

′ 4 ′ 4 ′ 4 ′ 4 ′ 4 ′ 4 ′ 5 ′ 5 ′ 5 ′ 5 ′ 5 ′ 5 푎24 , 푎25 , 푎26 , 푏24 , 푏25 , 푏26 , 푏28 , 푏29 , 푏30 푎28 , 푎29 , 푎30 , ′ 6 ′ 6 ′ 6 ′ 6 ′ 6 ′ 6 푎32 , 푎33 , 푎34 , 푏32 , 푏33 , 푏34 are Dissipation coefficients-

―A‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 13 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 퐺 -2 푑푡 13 14 13 13 14 13 푑퐺 14 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 퐺 -3 푑푡 14 13 14 14 14 14 푑퐺 1 1 15 = 푎 1 퐺 − 푎′ + 푎′′ 푇 , 푡 퐺 -4 푑푡 15 14 15 15 14 15 푑푇 13 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 푇 -5 푑푡 13 14 13 13 13 푑푇 14 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 푇 -6 푑푡 14 13 14 14 14 푑푇 1 1 15 = 푏 1 푇 − 푏′ − 푏′′ 퐺, 푡 푇 -7 푑푡 15 14 15 15 15 ′′ 1 + 푎13 푇14 , 푡 = First augmentation factor -8 ′′ 1 − 푏13 퐺, 푡 = First detritions factor-

www.ijmer.com 2039 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 ―B‖ AND ―A‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 16 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 -10 푑푡 16 17 16 16 17 16 푑퐺 17 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 -11 푑푡 17 16 17 17 17 17 푑퐺 18 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 -12 푑푡 18 17 18 18 17 18 푑푇 16 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 -13 푑푡 16 17 16 16 19 16 푑푇 17 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 -14 푑푡 17 16 17 17 19 17 푑푇 18 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 -15 푑푡 18 17 18 18 19 18 ′′ 2 + 푎16 푇17 , 푡 = First augmentation factor -16 ′′ 2 − 푏16 퐺19 , 푡 = First detritions factor -17 : ======A‖ AND ―C‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 20 = 푎 3 퐺 − 푎′ 3 + 푎′′ 3 푇 , 푡 퐺 -19 푑푡 20 21 20 20 21 20 푑퐺 21 = 푎 3 퐺 − 푎′ 3 + 푎′′ 3 푇 , 푡 퐺 -20 푑푡 21 20 21 21 21 21 푑퐺 22 = 푎 3 퐺 − 푎′ 3 + 푎′′ 3 푇 , 푡 퐺 -21 푑푡 22 21 22 22 21 22 푑푇 20 = 푏 3 푇 − 푏′ 3 − 푏′′ 3 퐺 , 푡 푇 -22 푑푡 20 21 20 20 23 20 푑푇 21 = 푏 3 푇 − 푏′ 3 − 푏′′ 3 퐺 , 푡 푇 -23 푑푡 21 20 21 21 23 21 푑푇 22 = 푏 3 푇 − 푏′ 3 − 푏′′ 3 퐺 , 푡 푇 -24 푑푡 22 21 22 22 23 22 ′′ 3 + 푎20 푇21 , 푡 = First augmentation factor- ′′ 3 − 푏20 퐺23 , 푡 = First detritions factor -25

―C‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 24 = 푎 4 퐺 − 푎′ 4 + 푎′′ 4 푇 , 푡 퐺 -27 푑푡 24 25 24 24 25 24 푑퐺 4 4 25 = 푎 4 퐺 − 푎′ + 푎′′ 푇 , 푡 퐺 -28 푑푡 25 24 25 25 25 25 푑퐺 26 = 푎 4 퐺 − 푎′ 4 + 푎′′ 4 푇 , 푡 퐺 -29 푑푡 26 25 26 26 25 26 푑푇 24 = 푏 4 푇 − 푏′ 4 − 푏′′ 4 퐺 , 푡 푇 -30 푑푡 24 25 24 24 27 24 푑푇 4 4 25 = 푏 4 푇 − 푏′ − 푏′′ 퐺 , 푡 푇 -31 푑푡 25 24 25 25 27 25 푑푇 26 = 푏 4 푇 − 푏′ 4 − 푏′′ 4 퐺 , 푡 푇 -32 푑푡 26 25 26 26 27 26 ′′ 4 + 푎24 푇25 , 푡 = First augmentation factor-33 ′′ 4 − 푏24 퐺27 , 푡 = First detritions factor -34

www.ijmer.com 2040 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 ―C‖ AND ―H‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 28 = 푎 5 퐺 − 푎′ 5 + 푎′′ 5 푇 , 푡 퐺 -36 푑푡 28 29 28 28 29 28 푑퐺 29 = 푎 5 퐺 − 푎′ 5 + 푎′′ 5 푇 , 푡 퐺 -37 푑푡 29 28 29 29 29 29 푑퐺 30 = 푎 5 퐺 − 푎′ 5 + 푎′′ 5 푇 , 푡 퐺 -38 푑푡 30 29 30 30 29 30 푑푇 28 = 푏 5 푇 − 푏′ 5 − 푏′′ 5 퐺 , 푡 푇 -39 푑푡 28 29 28 28 31 28 푑푇 29 = 푏 5 푇 − 푏′ 5 − 푏′′ 5 퐺 , 푡 푇 -40 푑푡 29 28 29 29 31 29 푑푇 30 = 푏 5 푇 − 푏′ 5 − 푏′′ 5 퐺 , 푡 푇 -41 푑푡 30 29 30 30 31 30 ′′ 5 + 푎28 푇29 , 푡 = First augmentation factor -42 ′′ 5 − 푏28 퐺31 , 푡 = First detritions factor -43

B‖ AND ―B‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 32 = 푎 6 퐺 − 푎′ 6 + 푎′′ 6 푇 , 푡 퐺 -46 푑푡 32 33 32 32 33 32 푑퐺 33 = 푎 6 퐺 − 푎′ 6 + 푎′′ 6 푇 , 푡 퐺 -47 푑푡 33 32 33 33 33 33 푑퐺 34 = 푎 6 퐺 − 푎′ 6 + 푎′′ 6 푇 , 푡 퐺 -48 푑푡 34 33 34 34 33 34 푑푇 32 = 푏 6 푇 − 푏′ 6 − 푏′′ 6 퐺 , 푡 푇 -49 푑푡 32 33 32 32 35 32 푑푇 33 = 푏 6 푇 − 푏′ 6 − 푏′′ 6 퐺 , 푡 푇 -50 푑푡 33 32 33 33 35 33 푑푇 34 = 푏 6 푇 − 푏′ 6 − 푏′′ 6 퐺 , 푡 푇 -51 푑푡 34 33 34 34 35 34 ′′ 6 + 푎32 푇33 , 푡 = First augmentation factor-52

―INPUT‖ AND ―A‖(SEE FIGURE REPRESENTS AN ―M CONFIGURATION‖ OR ―INSTRUCTIONS) 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 푑퐺 36 = 푎 7 퐺 − 푎′ 7 + 푎′′ 7 푇 , 푡 퐺 -54 푑푡 36 37 36 36 37 36 푑퐺 37 = 푎 7 퐺 − 푎′ 7 + 푎′′ 7 푇 , 푡 퐺 -55 푑푡 37 36 37 37 37 37

www.ijmer.com 2041 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 푑퐺 38 = 푎 7 퐺 − 푎′ 7 + 푎′′ 7 푇 , 푡 퐺 -56 푑푡 38 37 38 38 37 38 푑푇 36 = 푏 7 푇 − 푏′ 7 − 푏′′ 7 퐺 , 푡 푇 -57 푑푡 36 37 36 36 39 36 푑푇 37 = 푏 7 푇 − 푏′ 7 − 푏′′ 7 퐺 , 푡 푇 -58 푑푡 37 36 37 37 39 37

59 푑푇 38 = 푏 7 푇 − 푏′ 7 − 푏′′ 7 퐺 , 푡 푇 -60 푑푡 38 37 38 38 39 38 ′′ 7 + 푎36 푇37 , 푡 = First augmentation factor -61 ′′ 7 − 푏36 퐺39 , 푡 = First detritions factor

FIRST MODULE CONCATENATION: ′ 1 ′′ 1 ′′ 2,2, ′′ 3,3, 푎13 + 푎13 푇14 , 푡 + 푎16 푇17 , 푡 + 푎20 푇21 , 푡 푑퐺 13 = 푎 1 퐺 − + 푎′′ 4,4,4,4, 푇 , 푡 + 푎′′ 5,5,5,5, 푇 , 푡 + 푎′′ 6,6,6,6, 푇 , 푡 퐺 푑푡 13 14 24 25 28 29 32 33 13 ′′ 7 + 푎36 푇37 , 푡 ′ 1 ′′ 1 ′′ 2,2, ′′ 3,3, 푎14 + 푎14 푇14 , 푡 + 푎17 푇17 , 푡 + 푎21 푇21 , 푡 푑퐺 14 = 푎 1 퐺 − ′′ 4,4,4,4, ′′ 5,5,5,5, ′′ 6,6,6,6, 퐺 푑푡 14 13 + 푎25 푇25 ,푡 + 푎29 푇29 , 푡 + 푎33 푇33 , 푡 14

′′ 7 + 푎37 푇37 , 푡

′ 1 ′′ 1 ′′ 2,2, ′′ 3,3, 푎15 + 푎15 푇14 , 푡 + 푎18 푇17 , 푡 + 푎22 푇21 , 푡

푑퐺15 1 ′′ 4,4,4,4, ′′ 5,5,5,5, ′′ 6,6,6,6, = 푎 퐺 − + 푎26 푇25 , 푡 + 푎30 푇29 , 푡 + 푎34 푇33 , 푡 퐺 푑푡 15 14 15 ′′ 7 + 푎38 푇37 , 푡

′′ 1 ′′ 1 ′′ 1 Where 푎13 푇14 , 푡 , 푎14 푇14 , 푡 , 푎15 푇14 , 푡 are first augmentation coefficients for category 1, 2 and 3 ′′ 2,2, ′′ 2,2, ′′ 2,2, + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are second augmentation coefficient for category 1, 2 and 3 ′′ 3,3, ′′ 3,3, ′′ 3,3, + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 are third augmentation coefficient for category 1, 2 and 3

′′ 4,4,4,4, ′′ 4,4,4,4, ′′ 4,4,4,4, + 푎24 푇25 , 푡 , + 푎25 푇25 , 푡 , + 푎26 푇25 , 푡 are fourth augmentation coefficient for category 1, 2 and 3 ′′ 5,5,5,5, ′′ 5,5,5,5, ′′ 5,5,5,5, + 푎28 푇29 , 푡 , + 푎29 푇29, 푡 , + 푎30 푇29 , 푡 are fifth augmentation coefficient for category 1, 2 and 3 ′′ 6,6,6,6, ′′ 6,6,6,6, ′′ 6,6,6,6, + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 are sixth augmentation coefficient for category 1, 2 and 3 ′′ 7 ′′ 7 ′′ 7 + 푎36 푇37 , 푡 + 푎37 푇37 , 푡 + 푎38 푇37 , 푡 ARESEVENTHAUGMENTATION COEFFICIENTS ′ 1 ′′ 1 ′′ 7, ′′ 3,3, 푏13 − 푏16 퐺, 푡 − 푏36 퐺39, 푡 – 푏20 퐺23 , 푡

푑푇13 1 − 푏′′ 4,4,4,4, 퐺 , 푡 − 푏′′ 5,5,5,5, 퐺 , 푡 – 푏′′ 6,6,6,6, 퐺 ,푡 = 푏13 푇14 − 24 27 28 31 32 35 푇13 푑푡 ′′ 7, − 푏36 퐺39, 푡 ′ 1 ′′ 1 ′′ 2,2, ′′ 3,3, 푏14 − 푏14 퐺, 푡 − 푏17 퐺19, 푡 – 푏21 퐺23 , 푡

푑푇14 1 ′′ 4,4,4,4, ′′ 5,5,5,5, ′′ 6,6,6,6, = 푏14 푇13 − − 푏25 퐺27 , 푡 – 푏29 퐺31 , 푡 – 푏33 퐺35 , 푡 푇14 푑푡 ′′ 7, − 푏37 퐺39 , 푡

′ 1 ′′ 1 ′′ 2,2, ′′ 3,3, 푏 − 푏 퐺, 푡 − 푏18 퐺19, 푡 – 푏22 퐺23 , 푡 15 15 푑푇15 = 푏 1 푇 − ′′ 4,4,4,4, ′′ 5,5,5,5, ′′ 6,6,6,6, 푇 푑푡 15 14 – 푏26 퐺27 , 푡 – 푏30 퐺31 , 푡 – 푏34 퐺35 ,푡 15

′′ 7, − 푏38 퐺39, 푡

′′ 1 ′′ 1 ′′ 1 Where − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are first detritions coefficients for category 1, 2 and 3 ′′ 2,2, ′′ 2,2, ′′ 2,2, − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 are second detritions coefficients for category 1, 2 and 3

www.ijmer.com 2042 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

′′ 3,3, ′′ 3,3, ′′ 3,3, − 푏20 퐺23 , 푡 , − 푏21 퐺23 , 푡 , − 푏22 퐺23 , 푡 are third detritions coefficients for category 1, 2 and 3

′′ 4,4,4,4, ′′ 4,4,4,4, ′′ 4,4,4,4, − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 are fourth detritions coefficients for category 1, 2 and 3 ′′ 5,5,5,5, ′′ 5,5,5,5, ′′ 5,5,5,5, − 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 are fifth detritions coefficients for category 1, 2 and 3 ′′ 6,6,6,6, ′′ 6,6,6,6, ′′ 6,6,6,6, − 푏32 퐺35 , 푡 , − 푏33 퐺35, 푡 , − 푏34 퐺35 ,푡 are sixth detritions coefficients for category 1, 2 and 3 ′′ 7, ′′ 7, ′′ 7, − 푏36 퐺39, 푡 − 푏36 퐺39, 푡 − 푏36 퐺39, 푡 ARE SEVENTH DETRITION COEFFICIENTS -62 1 1 푏′ − 푏′′ 퐺, 푡 − 푏′′ 2,2, 퐺 , 푡 – 푏′′ 3,3, 퐺 , 푡 푑푇15 1 15 15 18 19 22 23 = 푏15 푇14 − 푇15 -63 푑푡 ′′ 4,4,4,4, ′′ 5,5,5,5, ′′ 6,6,6,6, − 푏26 퐺27 , 푡 − 푏30 퐺31 , 푡 − 푏34 퐺35, 푡

′′ 1 ′′ 1 ′′ 1 Where − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are first detrition coefficients for category 1, 2 and 3 ′′ 2,2, ′′ 2,2, ′′ 2,2, − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 are second detritions coefficients for category 1, 2 and 3 ′′ 3,3, ′′ 3,3, ′′ 3,3, − 푏20 퐺23 , 푡 , − 푏21 퐺23 , 푡 , − 푏22 퐺23 , 푡 are third detritions coefficients for category 1, 2 and 3

′′ 4,4,4,4, ′′ 4,4,4,4, ′′ 4,4,4,4, − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 are fourth detritions coefficients for category 1, 2 and 3 ′′ 5,5,5,5, ′′ 5,5,5,5, ′′ 5,5,5,5, − 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 are fifth detritions coefficients for category 1, 2 and 3 ′′ 6,6,6,6, ′′ 6,6,6,6, ′′ 6,6,6,6, − 푏32 퐺35 , 푡 , − 푏33 퐺35, 푡 , − 푏34 퐺35 ,푡 are sixth detritions coefficients for category 1, 2 and 3 -64

SECOND MODULE CONCATENATION:- ′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3 푎16 + 푎16 푇17 , 푡 + 푎13 푇14 , 푡 + 푎20 푇21 , 푡 푑퐺 65 16 = 푎 2 퐺 − + 푎′′ 4,4,4,4,4 푇 , 푡 + 푎′′ 5,5,5,5,5 푇 , 푡 + 푎′′ 6,6,6,6,6 푇 , 푡 퐺 -66 푑푡 16 17 24 25 28 29 32 33 16 ′′ 7,7, + 푎36 푇37 , 푡 ′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3 푎17 + 푎17 푇17 , 푡 + 푎14 푇14 , 푡 + 푎21 푇21 , 푡

푑퐺17 2 ′′ 4,4,4,4,4 ′′ 5,5,5,5,5 ′′ 6,6,6,6,6 = 푎17 퐺16 − + 푎25 푇25 ,푡 + 푎29 푇29 , 푡 + 푎33 푇33 , 푡 퐺17 -67 푑푡

′′ 7,7, + 푎37 푇37 , 푡

′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3 푎18 + 푎18 푇17 , 푡 + 푎15 푇14 , 푡 + 푎22 푇21 , 푡 푑퐺 18 = 푎 2 퐺 − ′′ 4,4,4,4,4 ′′ 5,5,5,5,5 ′′ 6,6,6,6,6 퐺 -68 푑푡 18 17 + 푎26 푇25 , 푡 + 푎30 푇29 , 푡 + 푎34 푇33 , 푡 18

′′ 7,7, + 푎38 푇37 , 푡 ′′ 2 ′′ 2 ′′ 2 Where + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are first augmentation coefficients for category 1, 2 and 3

′′ 1,1, ′′ 1,1, ′′ 1,1, + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14 , 푡 are second augmentation coefficient for category 1, 2 and 3 ′′ 3,3,3 ′′ 3,3,3 ′′ 3,3,3 + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 are third augmentation coefficient for category 1, 2 and 3

′′ 4,4,4,4,4 ′′ 4,4,4,4,4 ′′ 4,4,4,4,4 + 푎24 푇25 , 푡 , + 푎25 푇25 , 푡 , + 푎26 푇25 , 푡 are fourth augmentation coefficient for category 1, 2 and 3 ′′ 5,5,5,5,5 ′′ 5,5,5,5,5 ′′ 5,5,5,5,5 + 푎28 푇29 , 푡 , + 푎29 푇29, 푡 , + 푎30 푇29 , 푡 are fifth augmentation coefficient for category 1, 2 and 3 ′′ 6,6,6,6,6 ′′ 6,6,6,6,6 ′′ 6,6,6,6,6 + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 are sixth augmentation coefficient for category 1, 2 and 3 -69

www.ijmer.com 2043 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 70 ′′ 7,7, ′′ 7,7, ′′ 7,7, + 푎36 푇37 , 푡 + 푎37 푇37 , 푡 + 푎38 푇37 , 푡 ARE SEVENTH DETRITION COEFFICIENTS-71 ′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3, 푏16 − 푏16 퐺19, 푡 − 푏13 퐺, 푡 – 푏20 퐺23 , 푡 푑푇 16 = 푏 2 푇 − ′′ 4,4,4,4,4 ′′ 5,5,5,5,5 ′′ 6,6,6,6,6 푇 -72 푑푡 16 17 − 푏24 퐺27 , 푡 – 푏28 퐺31 , 푡 – 푏32 퐺35 ,푡 16

′′ 7,7 − 푏36 퐺39, 푡 ′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3, 푏17 − 푏17 퐺19, 푡 − 푏14 퐺, 푡 – 푏21 퐺23 , 푡 푑푇 17 = 푏 2 푇 − ′′ 4,4,4,4,4 ′′ 5,5,5,5,5 ′′ 6,6,6,6,6 푇 -73 푑푡 17 16 – 푏25 퐺27 , 푡 – 푏29 퐺31 , 푡 – 푏33 퐺35 ,푡 17

′′ 7,7 − 푏37 퐺39, 푡

′ 2 ′′ 2 ′′ 1,1, ′′ 3,3,3, 푏18 − 푏18 퐺19 , 푡 − 푏 퐺, 푡 – 푏22 퐺23 , 푡 15 푑푇18 = 푏 2 푇 − ′′ 4,4,4,4,4 ′′ 5,5,5,5,5 ′′ 6,6,6,6,6 푇 -74 푑푡 18 17 − 푏26 퐺27 , 푡 – 푏30 퐺31 , 푡 – 푏34 퐺35 ,푡 18

′′ 7,7 − 푏38 퐺39, 푡 ′′ 2 ′′ 2 ′′ 2 where − b16 G19,t , − b17 G19,t , − b18 G19, t are first detrition coefficients for category 1, 2 and 3

′′ 1,1, ′′ 1,1, ′′ 1,1, − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are second detrition coefficients for category 1,2 and 3 ′′ 3,3,3, ′′ 3,3,3, ′′ 3,3,3, − 푏20 퐺23 , 푡 , − 푏21 퐺23 , 푡 , − 푏22 퐺23 , 푡 are third detrition coefficients for category 1,2 and 3

′′ 4,4,4,4,4 ′′ 4,4,4,4,4 ′′ 4,4,4,4,4 − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 are fourth detritions coefficients for category 1,2 and 3 ′′ 5,5,5,5,5 ′′ 5,5,5,5,5 ′′ 5,5,5,5,5 − 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 are fifth detritions coefficients for category 1,2 and 3 ′′ 6,6,6,6,6 ′′ 6,6,6,6,6 ′′ 6,6,6,6,6 − 푏32 퐺35 , 푡 , − 푏33 퐺35 ,푡 , − 푏34 퐺35 ,푡 are sixth detritions coefficients for category 1,2 and 3 ′′ 7,7 ′′ 7,7 ′′ 7,7 − 푏36 퐺39, 푡 − 푏36 퐺39, 푡 − 푏36 퐺39, 푡 푎푟푒 푠푒푣푒푛푡푕 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠

THIRD MODULE CONCATENATION:-75 ′ 3 ′′ 3 ′′ 2,2,2 ′′ 1,1,1, 푎20 + 푎20 푇21 , 푡 + 푎16 푇17 , 푡 + 푎13 푇14 , 푡

푑퐺20 3 + 푎′′ 4,4,4,4,4,4 푇 , 푡 + 푎′′ 5,5,5,5,5,5 푇 , 푡 + 푎′′ 6,6,6,6,6,6 푇 , 푡 = 푎20 퐺21 − 24 25 28 29 32 33 퐺20 -76 푑푡

′′ 7.7.7. + 푎36 푇37 , 푡 ′ 3 ′′ 3 ′′ 2,2,2 ′′ 1,1,1, 푎21 + 푎21 푇21 , 푡 + 푎17 푇17 , 푡 + 푎14 푇14 , 푡 푑퐺 21 = 푎 3 퐺 − ′′ 4,4,4,4,4,4 ′′ 5,5,5,5,5,5 ′′ 6,6,6,6,6,6 퐺 -77 푑푡 21 20 + 푎25 푇25 , 푡 + 푎29 푇29, 푡 + 푎33 푇33 , 푡 21

′′ 7.7.7. + 푎37 푇37 , 푡

′ 3 ′′ 3 ′′ 2,2,2 ′′ 1,1,1, 푎22 + 푎22 푇21 , 푡 + 푎18 푇17 , 푡 + 푎15 푇14 , 푡 푑퐺 22 = 푎 3 퐺 − ′′ 4,4,4,4,4,4 ′′ 5,5,5,5,5,5 ′′ 6,6,6,6,6,6 퐺 -78 푑푡 22 21 + 푎26 푇25 , 푡 + 푎30 푇29, 푡 + 푎34 푇33 , 푡 22

′′ 7.7.7. + 푎38 푇37 , 푡

′′ 3 ′′ 3 ′′ 3 + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 are first augmentation coefficients for category 1, 2 and 3 ′′ 2,2,2 ′′ 2,2,2 ′′ 2,2,2 + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are second augmentation coefficients for category 1, 2 and 3

′′ 1,1,1, ′′ 1,1,1, ′′ 1,1,1, + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14 , 푡 are third augmentation coefficients for category 1, 2 and 3

′′ 4,4,4,4,4,4 ′′ 4,4,4,4,4,4 ′′ 4,4,4,4,4,4 + 푎24 푇25 , 푡 , + 푎25 푇25 , 푡 , + 푎26 푇25 , 푡 are fourth augmentation coefficients for category 1, 2 and 3

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′′ 5,5,5,5,5,5 ′′ 5,5,5,5,5,5 ′′ 5,5,5,5,5,5 + 푎28 푇29 , 푡 , + 푎29 푇29 , 푡 , + 푎30 푇29, 푡 are fifth augmentation coefficients for category 1, 2 and 3 ′′ 6,6,6,6,6,6 ′′ 6,6,6,6,6,6 ′′ 6,6,6,6,6,6 + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 are sixth augmentation coefficients for category 1, 2 and 3 -79

80 ′′ 7.7.7. ′′ 7.7.7. ′′ 7.7.7. + 푎36 푇37 , 푡 + 푎37 푇37 , 푡 + 푎38 푇37 , 푡 are seventh augmentation coefficient-81 ′ 3 ′′ 3 ′′ 7,7,7 ′′ 1,1,1, 푏20 − 푏20 퐺23 , 푡 – 푏36 퐺19, 푡 – 푏13 퐺, 푡 푑푇 20 = 푏 3 푇 − − 푏′′ 4,4,4,4,4,4 퐺 , 푡 – 푏′′ 5,5,5,5,5,5 퐺 , 푡 – 푏′′ 6,6,6,6,6,6 퐺 , 푡 푇 -82 푑푡 20 21 24 27 28 31 32 35 20

′′ 7,7,7 – 푏36 퐺39, 푡 ′ 3 ′′ 3 ′′ 2,2,2 ′′ 1,1,1, 푏21 − 푏21 퐺23 , 푡 – 푏17 퐺19, 푡 – 푏14 퐺, 푡 푑푇 21 = 푏 3 푇 − ′′ 4,4,4,4,4,4 ′′ 5,5,5,5,5,5 ′′ 6,6,6,6,6,6 푇 -83 푑푡 21 20 − 푏25 퐺27 , 푡 – 푏29 퐺31 , 푡 – 푏33 퐺35, 푡 21

′′ 7,7,7 – 푏37 퐺39, 푡

′ 3 ′′ 3 ′′ 2,2,2 ′′ 1,1,1, 푏22 − 푏22 퐺23 , 푡 – 푏18 퐺19, 푡 – 푏15 퐺, 푡 푑푇 22 = 푏 3 푇 − ′′ 4,4,4,4,4,4 ′′ 5,5,5,5,5,5 ′′ 6,6,6,6,6,6 푇 -84 푑푡 22 21 − 푏26 퐺27 , 푡 – 푏30 퐺31 , 푡 – 푏34 퐺35 , 푡 22

′′ 7,7,7 – 푏38 퐺39, 푡 ′′ 3 ′′ 3 ′′ 3 − 푏20 퐺23 , 푡 , − 푏21 퐺23 , 푡 , − 푏22 퐺23 , 푡 are first detritions coefficients for category 1, 2 and 3 ′′ 2,2,2 ′′ 2,2,2 ′′ 2,2,2 − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 are second detritions coefficients for category 1, 2 and 3

′′ 1,1,1, ′′ 1,1,1, ′′ 1,1,1, − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are third detrition coefficients for category 1,2 and 3

′′ 4,4,4,4,4,4 ′′ 4,4,4,4,4,4 ′′ 4,4,4,4,4,4 − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 are fourth detritions coefficients for category 1, 2 and 3 ′′ 5,5,5,5,5,5 ′′ 5,5,5,5,5,5 ′′ 5,5,5,5,5,5 − 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 are fifth detritions coefficients for category 1, 2 and 3 ′′ 6,6,6,6,6,6 ′′ 6,6,6,6,6,6 ′′ 6,6,6,6,6,6 − 푏32 퐺35 ,푡 , − 푏33 퐺35, 푡 , − 푏34 퐺35 ,푡 are sixth detritions coefficients for category 1, 2 and 3 -85 ′′ 7,7,7 ′′ 7,7,7 ′′ 7,7,7 – 푏36 퐺39, 푡 – 푏37 퐺39, 푡 – 푏38 퐺39, 푡 are seventh detritions coefficients ======

FOURTH MODULE CONCATENATION:-86 ′ 4 ′′ 4 ′′ 5,5, ′′ 6,6, 푎24 + 푎24 푇25 , 푡 + 푎28 푇29 , 푡 + 푎32 푇33 , 푡 푑퐺 24 = 푎 4 퐺 − + 푎′′ 1,1,1,1 푇 , 푡 + 푎′′ 2,2,2,2 푇 , 푡 + 푎′′ 3,3,3,3 푇 , 푡 퐺 -87 푑푡 24 25 13 14 16 17 20 21 24 ′′ 7,7,7,7, + 푎36 푇37 , 푡

′ 4 ′′ 4 ′′ 5,5, ′′ 6,6 푎25 + 푎25 푇25 , 푡 + 푎29 푇29 , 푡 + 푎33 푇33 , 푡

푑퐺25 4 ′′ 1,1,1,1 ′′ 2,2,2,2 ′′ 3,3,3,3 = 푎25 퐺24 − + 푎14 푇14 , 푡 + 푎17 푇17 , 푡 + 푎21 푇21 , 푡 퐺25 -88 푑푡

′′ 7,7,7,7, + 푎37 푇37 , 푡 ′ 4 ′′ 4 ′′ 5,5, ′′ 6,6, 푎26 + 푎26 푇25 , 푡 + 푎30 푇29 , 푡 + 푎34 푇33 , 푡 1,1,1,1 푑퐺26 4 ′′ ′′ 2,2,2,2 ′′ 3,3,3,3 = 푎 퐺 − + 푎15 푇14 , 푡 + 푎18 푇17 , 푡 + 푎22 푇21 , 푡 퐺 -89 푑푡 26 25 26 ′′ 7,7,7,7, + 푎38 푇37 , 푡

′′ 4 ′′ 4 ′′ 4 푊푕푒푟푒 푎24 푇25 , 푡 , 푎25 푇25 , 푡 , 푎26 푇25 , 푡 푎푟푒 푓푖푟푠푡 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 5,5, ′′ 5,5, ′′ 5,5, + 푎28 푇29, 푡 , + 푎29 푇29, 푡 , + 푎30 푇29 , 푡 푎푟푒 푠푒푐표푛푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

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′′ 6,6, ′′ 6,6, ′′ 6,6, + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 푎푟푒 푡푕푖푟푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 1,1,1,1 ′′ 1,1,1,1 ′′ 1,1,1,1 + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14 , 푡 are fourth augmentation coefficients for category 1, 2,and 3 ′′ 2,2,2,2 ′′ 2,2,2,2 ′′ 2,2,2,2 + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are fifth augmentation coefficients for category 1, 2,and 3 ′′ 3,3,3,3 ′′ 3,3,3,3 ′′ 3,3,3,3 + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 are sixth augmentation coefficients for category 1, 2,and 3 ′′ 7,7,7,7, ′′ 7,7,7,7, ′′ 7,7,7,7, + 푎36 푇37 , 푡 + 푎36 푇37 , 푡 + 푎36 푇37 , 푡 ARE SEVENTH augmentation coefficients-90

91 -92 ′ 4 ′′ 4 ′′ 5,5, ′′ 6,6, 푏24 − 푏24 퐺27 , 푡 − 푏28 퐺31 , 푡 – 푏32 퐺35, 푡 푑푇 24 = 푏 4 푇 − ′′ 1,1,1,1 ′′ 2,2,2,2 ′′ 3,3,3,3 푇 -93 푑푡 24 25 − 푏13 퐺, 푡 − 푏16 퐺19, 푡 – 푏20 퐺23 , 푡 24

′′ 7,7,7,7,,, − 푏36 퐺39, 푡

′ 4 ′′ 4 ′′ 5,5, ′′ 6,6, 푏25 − 푏25 퐺27 , 푡 − 푏29 퐺31 , 푡 – 푏33 퐺35, 푡 푑푇25 = 푏 4 푇 − ′′ 1,1,1,1 ′′ 2,2,2,2 ′′ 3,3,3,3 푇 -94 푑푡 25 24 − 푏14 퐺, 푡 − 푏17 퐺19, 푡 – 푏21 퐺23 , 푡 25

′′ 7,7,7,77,, − 푏37 퐺39, 푡 ′ 4 ′′ 4 ′′ 5,5, ′′ 6,6, 푏26 − 푏26 퐺27 , 푡 − 푏30 퐺31 , 푡 – 푏34 퐺35 ,푡 푑푇 26 = 푏 4 푇 − ′′ 1,1,1,1 ′′ 2,2,2,2 ′′ 3,3,3,3 푇 -95 푑푡 26 25 − 푏15 퐺, 푡 − 푏18 퐺19, 푡 – 푏22 퐺23 , 푡 26

′′ 7,7,7,,7,, − 푏38 퐺39, 푡

′′ 4 ′′ 4 ′′ 4 푊푕푒푟푒 − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 푎푟푒 푓푖푟푠푡 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 5,5, ′′ 5,5, ′′ 5,5, − 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 푎푟푒 푠푒푐표푛푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 6,6, ′′ 6,6, ′′ 6,6, − 푏32 퐺35 ,푡 , − 푏33 퐺35 ,푡 , − 푏34 퐺35 ,푡 푎푟푒 푡푕푖푟푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 1,1,1,1 ′′ 1,1,1,1 ′′ 1,1,1,1 − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 푎푟푒 푓표푢푟푡푕 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 2,2,2,2 ′′ 2,2,2,2 ′′ 2,2,2,2 − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 푎푟푒 푓푖푓푡푕 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 3,3,3,3 ′′ 3,3,3,3 ′′ 3,3,3,3 – 푏20 퐺23 , 푡 , – 푏21 퐺23 , 푡 , – 푏22 퐺23 , 푡 푎푟푒 푠푖푥푡푕 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 7,7,7,7,7,, ′′ 7,7,7,7,7,, ′′ 7,7,7,7,7,, − 푏36 퐺39, 푡 − 푏37 퐺39, 푡 − 푏38 퐺39, 푡 퐴푅퐸 SEVENTH DETRITION COEFFICIENTS-96 -97 FIFTH MODULE CONCATENATION:- ′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푎28 + 푎28 푇29 , 푡 + 푎24 푇25 , 푡 + 푎32 푇33 , 푡 푑퐺 98 28 = 푎 5 퐺 − + 푎′′ 1,1,1,1,1 푇 , 푡 + 푎′′ 2,2,2,2,2 푇 , 푡 + 푎′′ 3,3,3,3,3 푇 , 푡 퐺 -99 푑푡 28 29 13 14 16 17 20 21 28 ′′ 7,7,,7,,7,7 + 푎36 푇37 , 푡

′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푎29 + 푎29 푇29 , 푡 + 푎25 푇25 , 푡 + 푎33 푇33 , 푡 푑퐺 29 = 푎 5 퐺 − ′′ 1,1,1,1,1 ′′ 2,2,2,2,2 ′′ 3,3,3,3,3 퐺 -100 푑푡 29 28 + 푎14 푇14 , 푡 + 푎17 푇17 , 푡 + 푎21 푇21 , 푡 29

′′ 7,7,,,7,,7,7 + 푎37 푇37 , 푡 ′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푎30 + 푎30 푇29, 푡 + 푎26 푇25 , 푡 + 푎34 푇33 , 푡 1,1,1,1,1 푑퐺30 5 ′′ ′′ 2,2,2,2,2 ′′ 3,3,3,3,3 = 푎30 퐺29 − + 푎15 푇14 , 푡 + 푎18 푇17 , 푡 + 푎22 푇21 , 푡 퐺30 -101 푑푡 ′′ 7,7,,7,,7,7 + 푎38 푇37 , 푡

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′′ 5 ′′ 5 ′′ 5 푊푕푒푟푒 + 푎28 푇29, 푡 , + 푎29 푇29, 푡 , + 푎30 푇29 , 푡 푎푟푒 푓푖푟푠푡 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 4,4, ′′ 4,4, ′′ 4,4, 퐴푛푑 + 푎24 푇25 , 푡 , + 푎25 푇25 , 푡 , + 푎26 푇25 , 푡 푎푟푒 푠푒푐표푛푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 6,6,6 ′′ 6,6,6 ′′ 6,6,6 + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 푎푟푒 푡푕푖푟푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 1,1,1,1,1 ′′ 1,1,1,1,1 ′′ 1,1,1,1,1 + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14 , 푡 are fourth augmentation coefficients for category 1,2, and 3 ′′ 2,2,2,2,2 ′′ 2,2,2,2,2 ′′ 2,2,2,2,2 + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are fifth augmentation coefficients for category 1,2,and 3 ′′ 3,3,3,3,3 ′′ 3,3,3,3,3 ′′ 3,3,3,3,3 + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 are sixth augmentation coefficients for category 1,2, 3 -102 -103 ′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푏28 − 푏28 퐺31 , 푡 − 푏24 퐺23 , 푡 – 푏32 퐺35, 푡 푑푇 28 = 푏 5 푇 − ′′ 1,1,1,1,1 ′′ 2,2,2,2,2 ′′ 3,3,3,3,3 푇 -104 푑푡 28 29 − 푏13 퐺, 푡 − 푏16 퐺19 , 푡 – 푏20 퐺23 , 푡 28

′′ 7,7,,7,7,7, − 푏36 퐺38 , 푡

′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푏29 − 푏29 퐺31 , 푡 − 푏25 퐺27 , 푡 – 푏33 퐺35 , 푡

푑푇29 = 푏 5 푇 − ′′ 1,1,1,1,1 ′′ 2,2,2,2,2 ′′ 3,3,3,3,3 푇 -105 푑푡 29 28 − 푏14 퐺, 푡 − 푏17 퐺19 , 푡 – 푏21 퐺23 , 푡 29

′′ 7,7,7,7,7, − 푏37 퐺38 , 푡 ′ 5 ′′ 5 ′′ 4,4, ′′ 6,6,6 푏30 − 푏30 퐺31 , 푡 − 푏26 퐺27 , 푡 – 푏34 퐺35, 푡 푑푇 30 = 푏 5 푇 − ′′ 1,1,1,1,1, ′′ 2,2,2,2,2 ′′ 3,3,3,3,3 푇 -106 푑푡 30 29 − 푏15 퐺, 푡 − 푏18 퐺19, 푡 – 푏22 퐺23 , 푡 30

′′ 7,7,7,7,7, − 푏38 퐺38 , 푡 ′′ 5 ′′ 5 ′′ 5 푤푕푒푟푒 – 푏28 퐺31 , 푡 , − 푏29 퐺31 , 푡 , − 푏30 퐺31 , 푡 푎푟푒 푓푖푟푠푡 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 4,4, ′′ 4,4, ′′ 4,4, − 푏24 퐺27 , 푡 , − 푏25 퐺27 , 푡 , − 푏26 퐺27 , 푡 푎푟푒 푠푒푐표푛푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1,2 푎푛푑 3 ′′ 6,6,6 ′′ 6,6,6 ′′ 6,6,6 − 푏32 퐺35, 푡 , − 푏33 퐺35, 푡 , − 푏34 퐺35, 푡 푎푟푒 푡푕푖푟푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1,2 푎푛푑 3

′′ 1,1,1,1,1 ′′ 1,1,1,1,1 ′′ 1,1,1,1,1, − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are fourth detrition coefficients for category 1,2, and 3 ′′ 2,2,2,2,2 ′′ 2,2,2,2,2 ′′ 2,2,2,2,2 − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 are fifth detrition coefficients for category 1,2, and 3 ′′ 3,3,3,3,3 ′′ 3,3,3,3,3 ′′ 3,3,3,3,3 – 푏20 퐺23 , 푡 , – 푏21 퐺23 , 푡 , – 푏22 퐺23 , 푡 are sixth detrition coefficients for category 1,2, and 3-107

SIXTH MODULE CONCATENATION-108 ′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푎32 + 푎32 푇33 , 푡 + 푎28 푇29, 푡 + 푎24 푇25 , 푡 푑퐺 32 = 푎 6 퐺 − + 푎′′ 1,1,1,1,1,1 푇 , 푡 + 푎′′ 2,2,2,2,2,2 푇 , 푡 + 푎′′ 3,3,3,3,3,3 푇 , 푡 퐺 -109 푑푡 32 33 13 14 16 17 20 21 32 ′′ 7,7,7,7,7,7, + 푎36 푇37 , 푡

′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푎33 + 푎33 푇33 , 푡 + 푎29 푇29 , 푡 + 푎25 푇25 , 푡 푑퐺 33 = 푎 6 퐺 − ′′ 1,1,1,1,1,1 ′′ 2,2,2,2,2,2 ′′ 3,3,3,3,3,3 퐺 -110 푑푡 33 32 + 푎14 푇14 , 푡 + 푎17 푇17 , 푡 + 푎21 푇21 , 푡 33

′′ 7,7,7,,7,7,7, + 푎37 푇37 , 푡 ′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푎34 + 푎34 푇33 , 푡 + 푎30 푇29 , 푡 + 푎26 푇25 , 푡 푑퐺 34 = 푎 6 퐺 − ′′ 1,1,1,1,1,1 ′′ 2,2,2,2,2,2 ′′ 3,3,3,3,3,3 퐺 -111 푑푡 34 33 + 푎15 푇14 , 푡 + 푎18 푇17 , 푡 + 푎22 푇21 , 푡 34

′′ 7,7,7,7,7,7, + 푎38 푇37 , 푡 ′′ 6 ′′ 6 ′′ 6 + 푎32 푇33 , 푡 , + 푎33 푇33 , 푡 , + 푎34 푇33 , 푡 푎푟푒 푓푖푟푠푡 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 ′′ 5,5,5 ′′ 5,5,5 ′′ 5,5,5 + 푎28 푇29 , 푡 , + 푎29 푇29 , 푡 , + 푎30 푇29, 푡 푎푟푒 푠푒푐표푛푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3 www.ijmer.com 2047 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

′′ 4,4,4, ′′ 4,4,4, ′′ 4,4,4, + 푎24 푇25 , 푡 , + 푎25 푇25 ,푡 , + 푎26 푇25 , 푡 푎푟푒 푡푕푖푟푑 푎푢푔푚푒푛푡푎푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 1,1,1,1,1,1 ′′ 1,1,1,1,1,1 ′′ 1,1,1,1,1,1 + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14 , 푡 - are fourth augmentation coefficients ′′ 2,2,2,2,2,2 ′′ 2,2,2,2,2,2 ′′ 2,2,2,2,2,2 + 푎16 푇17 , 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 - fifth augmentation coefficients ′′ 3,3,3,3,3,3 ′′ 3,3,3,3,3,3 ′′ 3,3,3,3,3,3 + 푎20 푇21 , 푡 , + 푎21 푇21 , 푡 , + 푎22 푇21 , 푡 sixth augmentation coefficients ′′ 7,7,7,7,7,7, ′′ 7,7,,77,7,,7, ′′ 7,7,7,,7,7,7, + 푎36 푇37 , 푡 + 푎36 푇37 , 푡 + 푎36 푇37 , 푡 ARE SVENTH AUGMENTATION COEFFICIENTS-112 -113 ′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푏32 − 푏32 퐺35 , 푡 – 푏28 퐺31 , 푡 – 푏24 퐺27 , 푡 푑푇 32 = 푏 6 푇 − − 푏′′ 1,1,1,1,1,1 퐺, 푡 − 푏′′ 2,2,2,2,2,2 퐺 , 푡 – 푏′′ 3,3,3,3,3,3 퐺 , 푡 푇 -114 푑푡 32 33 13 16 19 20 23 32

′′ 7,7,7,,7,7,7 – 푏36 퐺39, 푡 ′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푏 33 − 푏 33 퐺 35,푡 – 푏 29 퐺 31,푡 – 푏 25 퐺 27, 푡

푑 푇 33 6 1,1,1,1,1,1 2,2,2,2,2,2 3,3,3,3,3,3 = 푏 33 푇 32 − ′′ ′′ ′′ 푇 33 -115 푑푡 − 푏 14 퐺 , 푡 − 푏 17 퐺 19, 푡 – 푏 21 퐺 23,푡

′′ 7,7,7,,7,7,7 – 푏 37 퐺 39,푡 ′ 6 ′′ 6 ′′ 5,5,5 ′′ 4,4,4, 푏 34 − 푏 34 퐺 35,푡 – 푏 30 퐺 31,푡 – 푏 26 퐺 27, 푡

푑 푇 34 6 ′′ 1,1,1,1,1,1 ′′ 2,2,2,2,2,2 ′′ 3,3,3,3,3,3 = 푏 34 푇 33 − − 푏 퐺 , 푡 − 푏 퐺 , 푡 – 푏 퐺 ,푡 푇 34 -116 푑푡 15 18 19 22 23 ′′ 7,7,7,,7,7,7 – 푏 38 퐺 39,푡 ′′ 6 ′′ 6 ′′ 6 − 푏 32 퐺 35,푡 , − 푏 33 퐺 35, 푡 , − 푏 34 퐺 35, 푡 푎푟푒 푓푖푟푠푡 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡 푠 푓표푟 푐푎푡푒푔표푟푦 1, 2 푎푛푑 3

′′ 5,5,5 ′′ 5,5,5 ′′ 5,5,5 − 푏 28 퐺 31,푡 , − 푏 29 퐺 31, 푡 , − 푏 30 퐺 31,푡 푎푟푒 푠푒푐표푛푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1,2 푎푛푑 3

′′ 4,4,4, ′′ 4,4,4, ′′ 4,4,4, − 푏 24 퐺 27,푡 , − 푏 25 퐺 27, 푡 , − 푏 26 퐺 27,푡 푎푟푒 푡 푕 푖푟푑 푑푒푡푟푖푡푖표푛 푐표푒푓푓푖푐푖푒푛푡푠 푓표푟 푐푎푡푒푔표푟푦 1,2 푎푛푑 3

′′ 1,1,1,1,1,1 ′′ 1,1,1,1,1,1 ′′ 1,1,1,1,1,1 − 푏 13 퐺 ,푡 , − 푏 14 퐺 ,푡 , − 푏 15 퐺 ,푡 are fourth detrition coefficients for category 1, 2, and 3 ′′ 2,2,2,2,2,2 ′′ 2,2,2,2,2,2 ′′ 2,2,2,2,2,2 − 푏 16 퐺 19,푡 , − 푏 17 퐺 19,푡 , − 푏 18 퐺 19,푡 are fifth detrition coefficients for category 1, 2, and 3 ′′ 3,3,3,3,3,3 ′′ 3,3,3,3,3,3 ′′ 3,3,3,3,3,3 – 푏 20 퐺 23, 푡 , – 푏 21 퐺 23, 푡 , – 푏 22 퐺 23,푡 are sixth detrition coefficients for category 1, 2, and 3 ′′ 7,7,7,7,7,7 ′′ 7,7,7,7,7,7 ′′ 7,7,7,7,7,7 – 푏 36 퐺 39, 푡 – 푏 36 퐺 39, 푡 – 푏 36 퐺 39, 푡 ARE SEVENTH DETRITION COEFFICIENTS-117 -118

SEVENTH MODULE CONCATENATION:-119 푑 퐺 36 = 푑푡

7 ′ 7 ′′ 7 ′′ 7 ′′ 7 ′′ 7 푎 36 퐺 37 − 푎 36 + 푎 36 푇 37,푡 + 푎 16 푇 17, 푡 + 푎 20 푇 21,푡 + 푎 24 푇 23, 푡 퐺 36 + �28′′7�29,� + �32′′7�33,� +�13′′7�14,� �36-120

121 푑 퐺 37 = 푑푡

7 ′ 7 ′′ 7 ′′ 7 ′′ 7 ′′ 7 푎 37 퐺 36 − 푎 37 + 푎 37 푇 37,푡 + 푎 14 푇 14, 푡 + 푎 21 푇 21, 푡 + 푎 17 푇 17, 푡 + �25′′7�25,� +�33′′7�33,� + �29′′7�29,� �37

-122

www.ijmer.com 2048 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 푑 퐺 38 = 푑푡

7 7 7 7 7 7 ′ ′′ ′′ ′′ ′′ 푎 38 퐺 37 − 푎 38 + 푎 38 푇 37, 푡 + 푎 15 푇 14 , 푡 + 푎 22 푇 21, 푡 + + 푎 18 푇 17,푡 + �26′′7�25,� + �34′′7�33,� + �30′′7�29,� �38

-123 124 125 푑 푇 7 7 7 7 36 = 푏 7 푇 − 푏 ′ − 푏 ′′ 퐺 , 푡 − 푏 ′′ 퐺 , 푡 − 푏 ′′ 퐺 , 푡 − 푑푡 36 37 36 36 39 16 19 13 14 �20′′7�231,� − �24′′7�27,� −�28′′7�31,� −�32′′7�35,� �36

-126 푑 푇 7 7 7 7 37 = 푏 7 푇 − 푏 ′ − 푏 ′′ 퐺 , 푡 − 푏 ′′ 퐺 , 푡 − 푏 ′′ 퐺 , 푡 − 푑푡 37 36 36 37 39 17 19 19 14 �21′′7�231,� − �25′′7�27,� −�29′′7�31,� −�33′′7�35,� �37

-127 Where we suppose

1 ′ 1 ′′ 1 1 ′ 1 ′′ 1 (A) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 13,14,15

′′ 1 ′′ 1 (B) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 1 1 Definition of (푝푖 ) , (푟푖 ) :

′′ 1 1 (1) 푎푖 (푇14 , 푡) ≤ (푝푖 ) ≤ ( 퐴13 )

′′ 1 1 ′ 1 (1) 푏푖 (퐺, 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵13 )

′′ 1 1 (C) 푙푖푚푇2→∞ 푎푖 푇14 , 푡 = (푝푖 ) ′′ 1 1 limG→∞ 푏푖 퐺, 푡 = (푟푖 )

(1) (1) Definition of ( 퐴 13 ) ,( 퐵 13 ) :

(1) (1) 1 1 Where ( 퐴 13 ) ,( 퐵 13 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 13,14,15

They satisfy Lipschitz condition: (1) ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 |(푎푖 ) 푇14 , 푡 − (푎푖 ) 푇14 , 푡 | ≤ ( 푘13 ) |푇14 − 푇14 |푒

(1) ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 |(푏푖 ) 퐺 , 푡 − (푏푖 ) 퐺, 푇 | < ( 푘13 ) ||퐺 − 퐺 ||푒

′′ 1 ′ ′′ 1 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇14 , 푡 and(푎푖 ) 푇14 , 푡 . 푇14 , 푡 and (1) (1) ′′ 1 푇14 , 푡 are points belonging to the interval ( 푘13 ) , ( 푀13 ) . It is to be noted that (푎푖 ) 푇14 , 푡 is uniformly continuous. (1) ′′ 1 In the eventuality of the fact, that if ( 푀13 ) = 1 then the function (푎푖 ) 푇14 , 푡 , the first augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous.

(1) (1) Definition of ( 푀 13 ) ,( 푘13 ) :

(1) (1) (D) ( 푀 13 ) , ( 푘13 ) , are positive constants

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1 1 (푎푖) (푏푖) (1) , (1) < 1 ( 푀 13 ) ( 푀 13 )

(1) (1) Definition of ( 푃 13 ) , ( 푄 13 ) :

(1) (1) (1) (1) (1) (1) (E) There exists two constants ( 푃 13 ) and ( 푄 13 ) which together with ( 푀 13 ) , ( 푘13 ) , (퐴 13 ) 푎푛푑 ( 퐵 13 ) and 1 ′ 1 1 ′ 1 1 1 the constants (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 13,14,15, satisfy the inequalities

1 1 ′ 1 (1) (1) (1) (1) [ (푎푖 ) + (푎푖 ) + ( 퐴13 ) + ( 푃13 ) ( 푘13 ) ] < 1 ( 푀 13 )

1 1 ′ 1 (1) (1) (1) (1) [ (푏푖 ) + (푏푖 ) + ( 퐵13 ) + ( 푄13 ) ( 푘13 ) ] < 1 ( 푀 13 )

푑푇 38 = 푏 7 푇 − 푏′ 7 − 푏′′ 7 퐺 , 푡 − 푏′′ 7 퐺 , 푡 − 푏′′ 7 퐺 , 푡 − 128 푑푡 38 37 38 38 39 18 19 20 14 푏22′′7퐺23,푡 − 푏26′′7퐺27,푡 −푏30′′7퐺31,푡 −푏34′′7퐺35,푡 129 푇38 130

131

132

′′ 7 + 푎36 푇37 , 푡 = First augmentation factor 134

2 ′ 2 ′′ 2 2 ′ 2 ′′ 2 (1) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 16,17,18 135

′′ 2 ′′ 2 (F) (2) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 136

2 2 Definition of (pi) , (ri) : 137

′′ 2 2 2 138 푎푖 푇17 , 푡 ≤ (푝푖 ) ≤ 퐴16

′′ 2 2 ′ 2 (2) 푏푖 (퐺19 , 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵16 ) 139

′′ 2 2 140 (G) (3) lim푇2→∞ 푎푖 푇17 , 푡 = (푝푖 )

′′ 2 2 lim퐺→∞ 푏푖 퐺19 , 푡 = (푟푖 ) 141

(2) (2) Definition of ( 퐴 16 ) , ( 퐵 16 ) : 142

(2) (2) 2 2 Where ( 퐴 16 ) , ( 퐵 16 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 16,17,18

They satisfy Lipschitz condition: 143

(2) ′′ 2 ′ ′′ 2 (2) ′ −( 푀 16 ) 푡 144 |(푎푖 ) 푇17 , 푡 − (푎푖 ) 푇17 , 푡 | ≤ ( 푘16 ) |푇17 − 푇17 |푒

(2) ′′ 2 ′ ′′ 2 (2) ′ −( 푀 16 ) 푡 145 |(푏푖 ) 퐺19 , 푡 − (푏푖 ) 퐺19 , 푡 | < ( 푘16 ) || 퐺19 − 퐺19 ||푒

′′ 2 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇17 , 푡 146 ′′ 2 ′ (2) (2) and(푎푖 ) 푇17 , 푡 . 푇17 , 푡 And 푇17 , 푡 are points belonging to the interval ( 푘16 ) , ( 푀16 ) . It is to be ′′ 2 (2) noted that (푎푖 ) 푇17 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀16 ) = 1 then the ′′ 2 function (푎푖 ) 푇17 , 푡 , the SECOND augmentation coefficient would be absolutely continuous.

(2) (2) Definition of ( 푀 16 ) , ( 푘16 ) : 147

(2) (2) (H) (4) ( 푀 16 ) , ( 푘16 ) , are positive constants 148

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2 2 (푎푖) (푏푖) (2) , (2) < 1 ( 푀 16 ) ( 푀 16 )

(2) (2) Definition of ( 푃 13 ) , ( 푄 13 ) : 149

(2) (2) There exists two constants ( 푃 16 ) and ( 푄 16 ) which together (2) (2) (2) (2) with ( 푀 16 ) , ( 푘16 ) , (퐴 16 ) 푎푛푑 ( 퐵 16 ) and the constants 2 ′ 2 2 ′ 2 2 2 (푎푖 ) , (푎푖 ) , (푏푖 ) ,(푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 16,17,18, satisfy the inequalities

1 2 ′ 2 (2) (2) (2) 150 (2) [ (ai) + (ai) + ( A16 ) + ( P16 ) ( k16 ) ] < 1 ( M 16 )

1 2 ′ 2 (2) (2) (2) 151 (2) [ (푏푖 ) + (푏푖 ) + ( 퐵16 ) + ( 푄16 ) ( 푘16 ) ] < 1 ( 푀 16 )

Where we suppose 152

3 ′ 3 ′′ 3 3 ′ 3 ′′ 3 (I) (5) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 20,21,22 153

′′ 3 ′′ 3 The functions 푎푖 , 푏푖 are positive continuous increasing and bounded.

3 3 Definition of (푝푖 ) , (ri ) :

′′ 3 3 (3) 푎푖 (푇21 , 푡) ≤ (푝푖 ) ≤ ( 퐴20 )

′′ 3 3 ′ 3 (3) 푏푖 (퐺23 , 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵20 )

′′ 3 3 154 푙푖푚푇2→∞ 푎푖 푇21 , 푡 = (푝푖 )

′′ 3 3 155 limG→∞ 푏푖 퐺23 , 푡 = (푟푖 )

(3) (3) 156 Definition of ( 퐴 20 ) ,( 퐵 20 ) :

(3) (3) 3 3 Where ( 퐴 20 ) ,( 퐵 20 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 20,21,22

They satisfy Lipschitz condition: 157

(3) ′′ 3 ′ ′′ 3 (3) ′ −( 푀 20 ) 푡 158 |(푎푖 ) 푇21 , 푡 − (푎푖 ) 푇21 , 푡 | ≤ ( 푘20 ) |푇21 − 푇21 |푒

(3) 159 ′′ 3 ′ ′′ 3 (3) ′ −( 푀 20 ) 푡 |(푏푖 ) 퐺23 , 푡 − (푏푖 ) 퐺23 , 푡 | < ( 푘20 ) ||퐺23 − 퐺23 ||푒

′′ 3 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇21 , 푡 160 ′′ 3 ′ (3) (3) and(푎푖 ) 푇21 , 푡 . 푇21 , 푡 And 푇21 , 푡 are points belonging to the interval ( 푘20 ) , ( 푀20 ) . It is to be ′′ 3 (3) noted that (푎푖 ) 푇21 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀20 ) = 1 then the ′′ 3 function (푎푖 ) 푇21 , 푡 , the THIRD augmentation coefficient, would be absolutely continuous.

(3) (3) Definition of ( 푀 20 ) , ( 푘20 ) : 161

(3) (3) (J) (6) ( 푀 20 ) , ( 푘20 ) , are positive constants

3 3 (푎푖) (푏푖) (3) , (3) < 1 ( 푀 20 ) ( 푀 20 )

(3) (3) There exists two constants There exists two constants ( 푃 20 ) and ( 푄 20 ) which together with 162 (3) (3) (3) (3) 3 ′ 3 3 ′ 3 3 3 ( 푀20 ) , ( 푘20 ) , (퐴20 ) 푎푛푑 ( 퐵20 ) and the constants (푎푖 ) , (푎푖 ) , (푏푖 ) ,(푏푖 ) ,(푝푖 ) , (푟푖 ) , 푖 = 20,21,22, 163 satisfy the inequalities 164 1 3 ′ 3 (3) (3) (3) (3) [ (푎푖 ) + (푎푖 ) + ( 퐴20 ) + ( 푃20 ) ( 푘20 ) ] < 1 ( 푀 20 ) 165

www.ijmer.com 2051 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 1 3 ′ 3 (3) (3) (3) 166 (3) [ (푏푖 ) + (푏푖 ) + ( 퐵20 ) + ( 푄20 ) ( 푘20 ) ] < 1 ( 푀 20 ) 167

Where we suppose 168

4 ′ 4 ′′ 4 4 ′ 4 ′′ 4 (K) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 24,25,26 169

′′ 4 ′′ 4 (L) (7) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded.

4 4 Definition of (푝푖 ) , (푟푖 ) :

′′ 4 4 (4) 푎푖 (푇25 , 푡) ≤ (푝푖 ) ≤ ( 퐴24 )

′′ 4 4 ′ 4 (4) 푏푖 퐺27 , 푡 ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵24 )

170

′′ 4 4 (M) (8) 푙푖푚푇2→∞ 푎푖 푇25 , 푡 = (푝푖 ) ′′ 4 4 limG→∞ 푏푖 퐺27 , 푡 = (푟푖 )

(4) (4) Definition of ( 퐴 24 ) , ( 퐵 24 ) :

(4) (4) 4 4 Where ( 퐴 24 ) , ( 퐵 24 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 24,25,26

They satisfy Lipschitz condition: 171

(4) ′′ 4 ′ ′′ 4 (4) ′ −( 푀 24 ) 푡 |(푎푖 ) 푇25 , 푡 − (푎푖 ) 푇25 , 푡 | ≤ ( 푘24 ) |푇25 − 푇25 |푒

(4) ′′ 4 ′ ′′ 4 (4) ′ −( 푀 24 ) 푡 |(푏푖 ) 퐺27 , 푡 − (푏푖 ) 퐺27 , 푡 | < ( 푘24 ) || 퐺27 − 퐺27 ||푒

′′ 4 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇25 , 푡 172 ′′ 4 ′ (4) (4) and(푎푖 ) 푇25 , 푡 . 푇25 , 푡 And 푇25 , 푡 are points belonging to the interval ( 푘24 ) , ( 푀24 ) . It is to be

′′ 4 (4) noted that (푎푖 ) 푇25 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀24 ) = 4 then the ′′ 4 function (푎푖 ) 푇25 , 푡 , the FOURTH augmentation coefficient WOULD be absolutely continuous.

173

(4) (4) Definition of ( 푀 24 ) , ( 푘24 ) : 174

175(4) (4) (N) ( 푀 24 )176 , ( 푘24 ) , are positive constants (O) 4 4 (푎푖) (푏푖) (4) , (4) < 1 ( 푀 24 ) ( 푀 24 )

(4) (4) Definition of ( 푃 24 ) , ( 푄 24 ) : 175

(4) (4) (P) (9) There exists two constants ( 푃 24 ) and ( 푄 24 ) which together with (4) (4) (4) (4) ( 푀 24 ) , ( 푘24 ) , (퐴 24 ) 푎푛푑 ( 퐵 24 ) and the constants 4 ′ 4 4 ′ 4 4 4 (푎푖 ) , (푎푖 ) , (푏푖 ) ,(푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 24,25,26, satisfy the inequalities

1 4 ′ 4 (4) (4) (4) (4) [ (푎푖 ) + (푎푖 ) + ( 퐴24 ) + ( 푃24 ) ( 푘24 ) ] < 1 ( 푀 24 )

1 4 ′ 4 (4) (4) (4) (4) [ (푏푖 ) + (푏푖 ) + ( 퐵24 ) + ( 푄24 ) ( 푘24 ) ] < 1 ( 푀 24 )

www.ijmer.com 2052 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Where we suppose 176

5 ′ 5 ′′ 5 5 ′ 5 ′′ 5 (Q) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 28,29,30 177 ′′ 5 ′′ 5 (R) (10) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 5 5 Definition of (푝푖 ) , (푟푖 ) :

′′ 5 5 (5) 푎푖 (푇29 , 푡) ≤ (푝푖 ) ≤ ( 퐴28 )

′′ 5 5 ′ 5 (5) 푏푖 퐺31 , 푡 ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵28 )

178

′′ 5 5 (S) (11) 푙푖푚푇2→∞ 푎푖 푇29, 푡 = (푝푖 ) ′′ 5 5 limG→∞ 푏푖 퐺31 , 푡 = (푟푖 )

(5) (5) Definition of ( 퐴 28 ) , ( 퐵 28 ) :

(5) (5) 5 5 Where ( 퐴 28 ) , ( 퐵 28 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 28,29,30

They satisfy Lipschitz condition: 179

(5) ′′ 5 ′ ′′ 5 (5) ′ −( 푀 28 ) 푡 |(푎푖 ) 푇29, 푡 − (푎푖 ) 푇29 , 푡 | ≤ ( 푘28 ) |푇29 − 푇29 |푒

(5) ′′ 5 ′ ′′ 5 (5) ′ −( 푀 28 ) 푡 |(푏푖 ) 퐺31 , 푡 − (푏푖 ) 퐺31 , 푡 | < ( 푘28 ) || 퐺31 − 퐺31 ||푒

′′ 5 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇29 , 푡 180 ′′ 5 ′ (5) (5) and(푎푖 ) 푇29, 푡 . 푇29 , 푡 and 푇29 , 푡 are points belonging to the interval ( 푘28 ) , ( 푀28 ) . It is to be ′′ 5 (5) noted that (푎푖 ) 푇29 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀28 ) = 5 then the ′′ 5 function (푎푖 ) 푇29 , 푡 , theFIFTH augmentation coefficient attributable would be absolutely continuous.

(5) (5) Definition of ( 푀 28 ) , ( 푘28 ) : 181

(5) (5) (T) ( 푀 28 ) , ( 푘28 ) , are positive constants 5 5 (푎푖) (푏푖) (5) , (5) < 1 ( 푀 28 ) ( 푀 28 )

(5) (5) Definition of ( 푃 28 ) , ( 푄 28 ) : 182

(5) (5) (U) There exists two constants ( 푃 28 ) and ( 푄 28 ) which together with (5) (5) (5) (5) ( 푀 28 ) , ( 푘28 ) , (퐴 28 ) 푎푛푑 ( 퐵 28 ) and the constants 5 ′ 5 5 ′ 5 5 5 (푎푖 ) , (푎푖 ) , (푏푖 ) ,(푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 28,29,30, satisfy the inequalities

1 5 ′ 5 (5) (5) (5) (5) [ (푎푖 ) + (푎푖 ) + ( 퐴28 ) + ( 푃28 ) ( 푘28 ) ] < 1 ( 푀 28 )

1 5 ′ 5 (5) (5) (5) (5) [ (푏푖 ) + (푏푖 ) + ( 퐵28 ) + ( 푄28 ) ( 푘28 ) ] < 1 ( 푀 28 )

Where we suppose 183

6 ′ 6 ′′ 6 6 ′ 6 ′′ 6 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 32,33,34 184 ′′ 6 ′′ 6 (12) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 6 6 Definition of (푝푖 ) , (푟푖 ) :

′′ 6 6 (6) 푎푖 (푇33 , 푡) ≤ (푝푖 ) ≤ ( 퐴32 )

′′ 6 6 ′ 6 (6) 푏푖 ( 퐺35 , 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵32 )

www.ijmer.com 2053 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 185

′′ 6 6 (13) 푙푖푚푇2→∞ 푎푖 푇33 , 푡 = (푝푖 ) ′′ 6 6 limG→∞ 푏푖 퐺35 , 푡 = (푟푖 )

(6) (6) Definition of ( 퐴 32 ) , ( 퐵 32 ) :

(6) (6) 6 6 Where ( 퐴 32 ) , ( 퐵 32 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 32,33,34

They satisfy Lipschitz condition: 186

(6) ′′ 6 ′ ′′ 6 (6) ′ −( 푀 32 ) 푡 |(푎푖 ) 푇33 , 푡 − (푎푖 ) 푇33 , 푡 | ≤ ( 푘32 ) |푇33 − 푇33 |푒

(6) ′′ 6 ′ ′′ 6 (6) ′ −( 푀 32 ) 푡 |(푏푖 ) 퐺35 , 푡 − (푏푖 ) 퐺35 , 푡 | < ( 푘32 ) || 퐺35 − 퐺35 ||푒

′′ 6 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇33 , 푡 187 ′′ 6 ′ (6) (6) and(푎푖 ) 푇33 , 푡 . 푇33 , 푡 and 푇33 , 푡 are points belonging to the interval ( 푘32 ) , ( 푀32 ) . It is to be ′′ 6 (6) noted that (푎푖 ) 푇33 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀32 ) = 6 then the ′′ 6 function (푎푖 ) 푇33 , 푡 , the SIXTH augmentation coefficient would be absolutely continuous.

(6) (6) Definition of ( 푀 32 ) , ( 푘32 ) : 188

(6) (6) ( 푀 32 ) , ( 푘32 ) , are positive constants 6 6 (푎푖) (푏푖) (6) , (6) < 1 ( 푀 32 ) ( 푀 32 )

(6) (6) Definition of ( 푃 32 ) , ( 푄 32 ) : 189

(6) (6) There exists two constants ( 푃 32 ) and ( 푄 32 ) which together with (6) (6) (6) (6) 6 ′ 6 6 ′ 6 6 6 ( 푀32 ) , ( 푘32 ) , (퐴32 ) 푎푛푑 ( 퐵32 ) and the constants (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 32,33,34, satisfy the inequalities

1 6 ′ 6 (6) (6) (6) (6) [ (푎푖 ) + (푎푖 ) + ( 퐴32 ) + ( 푃32 ) ( 푘32 ) ] < 1 ( 푀 32 )

1 6 ′ 6 (6) (6) (6) (6) [ (푏푖 ) + (푏푖 ) + ( 퐵32 ) + ( 푄32 ) ( 푘32 ) ] < 1 ( 푀 32 )

Where we suppose 190

7 ′ 7 ′′ 7 7 ′ 7 ′′ 7 191 (V) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 36,37,38

′′ 7 ′′ 7 (W) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 7 7 Definition of (푝푖 ) , (푟푖 ) :

′′ 7 7 (7) 푎푖 (푇37 , 푡) ≤ (푝푖 ) ≤ ( 퐴36 )

′′ 7 7 ′ 7 (7) 푏푖 (퐺, 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵36 )

7 ′′ 7 192 lim푇2→∞ 푎푖 푇37 , 푡 = (푝푖 )

′′ 7 7 limG→∞ 푏푖 퐺39 , 푡 = (푟푖 )

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(7) (7) Definition of ( 퐴 36 ) , ( 퐵 36 ) :

(7) (7) 7 7 Where ( 퐴 36 ) ,( 퐵 36 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 36,37,38

They satisfy Lipschitz condition: 193

(7) ′′ 7 ′ ′′ 7 (7) ′ −( 푀 36 ) 푡 |(푎푖 ) 푇37 , 푡 − (푎푖 ) 푇37 , 푡 | ≤ ( 푘36 ) |푇37 − 푇37 |푒

(7) ′′ 7 ′ ′′ 7 (7) ′ −( 푀 36 ) 푡 |(푏푖 ) 퐺39 , 푡 − (푏푖 ) 퐺39 , 푇39 | < ( 푘36 ) || 퐺39 − 퐺39 ||푒

′′ 7 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇37 , 푡 194 ′′ 7 ′ (7) (7) and(푎푖 ) 푇37 , 푡 . 푇37 , 푡 and 푇37 , 푡 are points belonging to the interval ( 푘36 ) , ( 푀36 ) . It is to be ′′ 7 (7) noted that (푎푖 ) 푇37 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀36 ) = 7 then the ′′ 7 function (푎푖 ) 푇37 , 푡 , the first augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous.

(7) (7) Definition of ( 푀 36 ) , ( 푘36 ) : 195

(7) (7) (X) ( 푀 36 ) , ( 푘36 ) , are positive constants 7 7 (푎푖) (푏푖) (7) , (7) < 1 ( 푀 36 ) ( 푀 36 )

(7) (7) Definition of ( 푃 36 ) , ( 푄 36 ) : 196

(7) (7) (Y) There exists two constants ( 푃 36 ) and ( 푄 36 ) which together with (7) (7) (7) (7) ( 푀 36 ) , ( 푘36 ) , (퐴 36 ) 푎푛푑 ( 퐵 36 ) and the constants 7 ′ 7 7 ′ 7 7 7 (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 36,37,38, satisfy the inequalities

1 7 ′ 7 (7) (7) (7) (7) [ (푎푖 ) + (푎푖 ) + ( 퐴36 ) + ( 푃36 ) ( 푘36 ) ] < 1 ( 푀 36 )

1 7 ′ 7 (7) (7) (7) (7) [ (푏푖 ) + (푏푖 ) + ( 퐵36 ) + ( 푄36 ) ( 푘36 ) ] < 1 ( 푀 36 )

Definition of 퐺푖 0 , 푇푖 0 : 197

5 5 푀 28 푡 0 퐺푖 푡 ≤ 푃28 푒 , 퐺푖 0 = 퐺푖 > 0

(5) 198 (5) ( 푀 28 ) 푡 0 푇푖 (푡) ≤ ( 푄28 ) 푒 , 푇푖 0 = 푇푖 > 0

Definition of 퐺푖 0 , 푇푖 0 : 199

6 6 푀 32 푡 0 퐺푖 푡 ≤ 푃32 푒 , 퐺푖 0 = 퐺푖 > 0

(6) (6) ( 푀 32 ) 푡 0 푇푖 (푡) ≤ ( 푄32 ) 푒 , 푇푖 0 = 푇푖 > 0

======

Definition of 퐺푖 0 , 푇푖 0 :

7 7 푀 36 푡 0 퐺푖 푡 ≤ 푃36 푒 , 퐺푖 0 = 퐺푖 > 0

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(7) (7) ( 푀 36 ) 푡 0 푇푖 (푡) ≤ ( 푄36 ) 푒 , 푇푖 0 = 푇푖 > 0

(1) Proof: Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ 200 which satisfy

0 0 0 (1) 0 (1) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃13 ) , 푇푖 ≤ ( 푄13 ) , 201

(1) 0 (1) ( 푀 13 ) 푡 202 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃13 ) 푒

(1) 0 (1) ( 푀 13 ) 푡 203 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄13 ) 푒 By 204

푡 0 1 ′ 1 ′′ 1 퐺13 푡 = 퐺13 + 0 (푎13 ) 퐺14 푠 13 − (푎13 ) + 푎13 ) 푇14 푠 13 , 푠 13 퐺13 푠 13 푑푠 13

푡 0 1 ′ 1 ′′ 1 205 퐺14 푡 = 퐺14 + 0 (푎14 ) 퐺13 푠 13 − (푎14 ) + (푎14 ) 푇14 푠 13 , 푠 13 퐺14 푠 13 푑푠 13

푡 0 1 ′ 1 ′′ 1 206 퐺15 푡 = 퐺15 + 0 (푎15 ) 퐺14 푠 13 − (푎15 ) + (푎15 ) 푇14 푠 13 , 푠 13 퐺15 푠 13 푑푠 13

푡 0 1 ′ 1 ′′ 1 207 푇13 푡 = 푇13 + 0 (푏13 ) 푇14 푠 13 − (푏13 ) − (푏13) 퐺 푠 13 , 푠 13 푇13 푠 13 푑푠 13

푡 0 1 ′ 1 ′′ 1 208 푇14 푡 = 푇14 + 0 (푏14 ) 푇13 푠 13 − (푏14 ) − (푏14) 퐺 푠 13 , 푠 13 푇14 푠 13 푑푠 13

푡 0 1 ′ 1 ′′ 1 209 T15 t = T15 + 0 (푏15 ) 푇14 푠 13 − (푏15) − (푏15 ) 퐺 푠 13 , 푠 13 푇15 푠 13 푑푠 13

Where 푠 13 is the integrand that is integrated over an interval 0, 푡

210

if the conditions IN THE FOREGOING above are fulfilled, there exists a solution satisfying the conditions

Definition of 퐺푖 0 , 푇푖 0 :

7 7 푀 36 푡 0 퐺푖 푡 ≤ 푃36 푒 , 퐺푖 0 = 퐺푖 > 0 (7) (7) ( 푀 36 ) 푡 0 푇푖 (푡) ≤ ( 푄36 ) 푒 , 푇푖 0 = 푇푖 > 0 (7) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which satisfy

0 0 0 (7) 0 (7) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃36 ) , 푇푖 ≤ ( 푄36 ) ,

(7) 0 (7) ( 푀 36 ) 푡 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃36 ) 푒

(7) 0 (7) ( 푀 36 ) 푡 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄36 ) 푒

By

푡 0 7 ′ 7 ′′ 7 퐺36 푡 = 퐺36 + 0 (푎36 ) 퐺37 푠 36 − (푎36 ) + 푎36 ) 푇37 푠 36 , 푠 36 퐺36 푠 36 푑푠 36

0 퐺37 푡 = 퐺37 + 푡 7 ′ 7 ′′ 7 0 (푎37 ) 퐺36 푠 36 − (푎37 ) + (푎37 ) 푇37 푠 36 , 푠 36 퐺37 푠 36 푑푠 36

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0 퐺38 푡 = 퐺38 + 푡 7 ′ 7 ′′ 7 0 (푎38 ) 퐺37 푠 36 − (푎38) + (푎38 ) 푇37 푠 36 , 푠 36 퐺38 푠 36 푑푠 36

푡 0 7 ′ 7 ′′ 7 푇36 푡 = 푇36 + 0 (푏36 ) 푇37 푠 36 − (푏36) − (푏36 ) 퐺 푠 36 , 푠 36 푇36 푠 36 푑푠 36

푡 0 7 ′ 7 ′′ 7 푇37 푡 = 푇37 + 0 (푏37 ) 푇36 푠 36 − (푏37) − (푏37 ) 퐺 푠 36 , 푠 36 푇37 푠 36 푑푠 36

0 T 38 t = T38 + 푡 7 ′ 7 ′′ 7 0 (푏38) 푇37 푠 36 − (푏38 ) − (푏38) 퐺 푠 36 , 푠 36 푇38 푠 36 푑푠 36

Where 푠 36 is the integrand that is integrated over an interval 0, 푡

(2) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which 211 satisfy

0 0 0 (2) 0 (2) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃16 ) , 푇푖 ≤ ( 푄16 ) , 212

(2) 0 (2) ( 푀 16 ) 푡 213 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃16 ) 푒

(2) 0 (2) ( 푀 16 ) 푡 214 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄16 ) 푒 By 215

푡 0 2 ′ 2 ′′ 2 퐺16 푡 = 퐺16 + 0 (푎16 ) 퐺17 푠 16 − (푎16 ) + 푎16 ) 푇17 푠 16 , 푠 16 퐺16 푠 16 푑푠 16

푡 0 2 ′ 2 ′′ 2 216 퐺17 푡 = 퐺17 + 0 (푎17 ) 퐺16 푠 16 − (푎17 ) + (푎17 ) 푇17 푠 16 , 푠 17 퐺17 푠 16 푑푠 16

푡 0 2 ′ 2 ′′ 2 217 퐺18 푡 = 퐺18 + 0 (푎18 ) 퐺17 푠 16 − (푎18 ) + (푎18 ) 푇17 푠 16 , 푠 16 퐺18 푠 16 푑푠 16

푡 0 2 ′ 2 ′′ 2 218 푇16 푡 = 푇16 + 0 (푏16 ) 푇17 푠 16 − (푏16 ) − (푏16) 퐺 푠 16 , 푠 16 푇16 푠 16 푑푠 16

푡 0 2 ′ 2 ′′ 2 219 푇17 푡 = 푇17 + 0 (푏17 ) 푇16 푠 16 − (푏17 ) − (푏17) 퐺 푠 16 , 푠 16 푇17 푠 16 푑푠 16

푡 0 2 ′ 2 ′′ 2 220 푇18 푡 = 푇18 + 0 (푏18 ) 푇17 푠 16 − (푏18 ) − (푏18) 퐺 푠 16 , 푠 16 푇18 푠 16 푑푠 16

Where 푠 16 is the integrand that is integrated over an interval 0, 푡

(3) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which 221 satisfy

0 0 0 (3) 0 (3) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃20 ) , 푇푖 ≤ ( 푄20 ) , 222

(3) 0 (3) ( 푀 20 ) 푡 223 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃20 ) 푒

(3) 0 (3) ( 푀 20 ) 푡 224 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄20 ) 푒 By 225

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푡 0 3 ′ 3 ′′ 3 퐺20 푡 = 퐺20 + 0 (푎20 ) 퐺21 푠 20 − (푎20 ) + 푎20 ) 푇21 푠 20 , 푠 20 퐺20 푠 20 푑푠 20

푡 0 3 ′ 3 ′′ 3 226 퐺21 푡 = 퐺21 + 0 (푎21 ) 퐺20 푠 20 − (푎21 ) + (푎21 ) 푇21 푠 20 , 푠 20 퐺21 푠 20 푑푠 20

푡 0 3 ′ 3 ′′ 3 227 퐺22 푡 = 퐺22 + 0 (푎22 ) 퐺21 푠 20 − (푎22 ) + (푎22 ) 푇21 푠 20 , 푠 20 퐺22 푠 20 푑푠 20

푡 0 3 ′ 3 ′′ 3 228 푇20 푡 = 푇20 + 0 (푏20 ) 푇21 푠 20 − (푏20) − (푏20 ) 퐺 푠 20 , 푠 20 푇20 푠 20 푑푠 20

푡 0 3 ′ 3 ′′ 3 229 푇21 푡 = 푇21 + 0 (푏21 ) 푇20 푠 20 − (푏21) − (푏21 ) 퐺 푠 20 , 푠 20 푇21 푠 20 푑푠 20

푡 0 3 ′ 3 ′′ 3 230 T22 t = T22 + 0 (푏22) 푇21 푠 20 − (푏22 ) − (푏22 ) 퐺 푠 20 , 푠 20 푇22 푠 20 푑푠 20

Where 푠 20 is the integrand that is integrated over an interval 0, 푡

(4) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which 231 satisfy

0 0 0 (4) 0 (4) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃24 ) , 푇푖 ≤ ( 푄24 ) , 232

(4) 0 (4) ( 푀 24 ) 푡 233 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃24 ) 푒

(4) 0 (4) ( 푀 24 ) 푡 234 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄24 ) 푒

By 235

푡 0 4 ′ 4 ′′ 4 퐺24 푡 = 퐺24 + 0 (푎24 ) 퐺25 푠 24 − (푎24 ) + 푎24 ) 푇25 푠 24 , 푠 24 퐺24 푠 24 푑푠 24

푡 0 4 ′ 4 ′′ 4 236 퐺25 푡 = 퐺25 + 0 (푎25 ) 퐺24 푠 24 − (푎25 ) + (푎25 ) 푇25 푠 24 , 푠 24 퐺25 푠 24 푑푠 24

푡 0 4 ′ 4 ′′ 4 237 퐺26 푡 = 퐺26 + 0 (푎26 ) 퐺25 푠 24 − (푎26 ) + (푎26 ) 푇25 푠 24 ,푠 24 퐺26 푠 24 푑푠 24

푡 0 4 ′ 4 ′′ 4 238 푇24 푡 = 푇24 + 0 (푏24 ) 푇25 푠 24 − (푏24) − (푏24 ) 퐺 푠 24 , 푠 24 푇24 푠 24 푑푠 24

푡 0 4 ′ 4 ′′ 4 239 푇25 푡 = 푇25 + 0 (푏25 ) 푇24 푠 24 − (푏25) − (푏25 ) 퐺 푠 24 , 푠 24 푇25 푠 24 푑푠 24

푡 0 4 ′ 4 ′′ 4 240 T26 t = T26 + 0 (푏26) 푇25 푠 24 − (푏26 ) − (푏26 ) 퐺 푠 24 , 푠 24 푇26 푠 24 푑푠 24

Where 푠 24 is the integrand that is integrated over an interval 0, 푡

(5) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which 241 satisfy 242

0 0 0 (5) 0 (5) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃28 ) , 푇푖 ≤ ( 푄28 ) , 243

(5) 0 (5) ( 푀 28 ) 푡 244 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃28 ) 푒

(5) 0 (5) ( 푀 28 ) 푡 245 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄28 ) 푒

By 246

푡 0 5 ′ 5 ′′ 5 퐺28 푡 = 퐺28 + 0 (푎28 ) 퐺29 푠 28 − (푎28 ) + 푎28 ) 푇29 푠 28 , 푠 28 퐺28 푠 28 푑푠 28

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푡 0 5 ′ 5 ′′ 5 247 퐺29 푡 = 퐺29 + 0 (푎29) 퐺28 푠 28 − (푎29) + (푎29) 푇29 푠 28 , 푠 28 퐺29 푠 28 푑푠 28

푡 0 5 ′ 5 ′′ 5 248 퐺30 푡 = 퐺30 + 0 (푎30 ) 퐺29 푠 28 − (푎30 ) + (푎30 ) 푇29 푠 28 , 푠 28 퐺30 푠 28 푑푠 28

푡 0 5 ′ 5 ′′ 5 249 푇28 푡 = 푇28 + 0 (푏28 ) 푇29 푠 28 − (푏28) − (푏28) 퐺 푠 28 , 푠 28 푇28 푠 28 푑푠 28

푡 0 5 ′ 5 ′′ 5 250 푇29 푡 = 푇29 + 0 (푏29) 푇28 푠 28 − (푏29) − (푏29) 퐺 푠 28 , 푠 28 푇29 푠 28 푑푠 28

푡 0 5 ′ 5 ′′ 5 251 T30 t = T30 + 0 (푏30) 푇29 푠 28 − (푏30) − (푏30 ) 퐺 푠 28 , 푠 28 푇30 푠 28 푑푠 28

Where 푠 28 is the integrand that is integrated over an interval 0, 푡

(6) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which 252 satisfy

0 0 0 (6) 0 (6) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃32 ) , 푇푖 ≤ ( 푄32 ) , 253

(6) 0 (6) ( 푀 32 ) 푡 254 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃32 ) 푒

(6) 0 (6) ( 푀 32 ) 푡 255 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄32 ) 푒

By 256

푡 0 6 ′ 6 ′′ 6 퐺32 푡 = 퐺32 + 0 (푎32 ) 퐺33 푠 32 − (푎32 ) + 푎32 ) 푇33 푠 32 , 푠 32 퐺32 푠 32 푑푠 32

푡 0 6 ′ 6 ′′ 6 257 퐺33 푡 = 퐺33 + 0 (푎33 ) 퐺32 푠 32 − (푎33 ) + (푎33 ) 푇33 푠 32 , 푠 32 퐺33 푠 32 푑푠 32

푡 0 6 ′ 6 ′′ 6 258 퐺34 푡 = 퐺34 + 0 (푎34 ) 퐺33 푠 32 − (푎34 ) + (푎34 ) 푇33 푠 32 ,푠 32 퐺34 푠 32 푑푠 32

푡 0 6 ′ 6 ′′ 6 259 푇32 푡 = 푇32 + 0 (푏32 ) 푇33 푠 32 − (푏32) − (푏32 ) 퐺 푠 32 , 푠 32 푇32 푠 32 푑푠 32

푡 0 6 ′ 6 ′′ 6 260 푇33 푡 = 푇33 + 0 (푏33 ) 푇32 푠 32 − (푏33) − (푏33 ) 퐺 푠 32 , 푠 32 푇33 푠 32 푑푠 32

푡 0 6 ′ 6 ′′ 6 261 T34 t = T34 + 0 (푏34) 푇33 푠 32 − (푏34 ) − (푏34 ) 퐺 푠 32 , 푠 32 푇34 푠 32 푑푠 32

Where 푠 32 is the integrand that is integrated over an interval 0, 푡

: if the conditions IN THE FOREGOING are fulfilled, there exists a solution satisfying the conditions 262

Definition of 퐺푖 0 , 푇푖 0 :

7 7 푀 36 푡 0 퐺푖 푡 ≤ 푃36 푒 , 퐺푖 0 = 퐺푖 > 0

(7) (7) ( 푀 36 ) 푡 0 푇푖 (푡) ≤ ( 푄36 ) 푒 , 푇푖 0 = 푇푖 > 0

Proof:

(7) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which satisfy

www.ijmer.com 2059 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 0 0 0 (7) 0 (7) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃36 ) , 푇푖 ≤ ( 푄36 ) , 263

(7) 0 (7) ( 푀 36 ) 푡 264 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃36 ) 푒

(7) 0 (7) ( 푀 36 ) 푡 265 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄36 ) 푒

By 266

푡 0 7 ′ 7 ′′ 7 퐺36 푡 = 퐺36 + 0 (푎36 ) 퐺37 푠 36 − (푎36 ) + 푎36 ) 푇37 푠 36 , 푠 36 퐺36 푠 36 푑푠 36

267

0 퐺37 푡 = 퐺37 +

푡 7 ′ 7 ′′ 7 0 (푎37 ) 퐺36 푠 36 − (푎37 ) + (푎37 ) 푇37 푠 36 , 푠 36 퐺37 푠 36 푑푠 36

0 퐺38 푡 = 퐺38 + 268

푡 7 ′ 7 ′′ 7 0 (푎38 ) 퐺37 푠 36 − (푎38 ) + (푎38 ) 푇37 푠 36 , 푠 36 퐺38 푠 36 푑푠 36

269

푡 0 7 ′ 7 ′′ 7 푇36 푡 = 푇36 + 0 (푏36 ) 푇37 푠 36 − (푏36) − (푏36 ) 퐺 푠 36 , 푠 36 푇36 푠 36 푑푠 36

푡 0 7 ′ 7 ′′ 7 270 푇37 푡 = 푇37 + 0 (푏37 ) 푇36 푠 36 − (푏37) − (푏37 ) 퐺 푠 36 , 푠 36 푇37 푠 36 푑푠 36

0 T 38 t = T38 + 271

푡 7 ′ 7 ′′ 7 0 (푏38) 푇37 푠 36 − (푏38 ) − (푏38) 퐺 푠 36 , 푠 36 푇38 푠 36 푑푠 36

Where 푠 36 is the integrand that is integrated over an interval 0, 푡

Analogous inequalities hold also for 퐺21 , 퐺22 , 푇20 , 푇21 , 푇22 272

(a) The operator 풜(4) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is 273 obvious that

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푡 (4) 0 4 0 (4) ( 푀24 ) 푠 24 퐺24 푡 ≤ 퐺24 + 0 (푎24 ) 퐺25 +( 푃24 ) 푒 푑푠 24 =

(푎 ) 4 ( 푃 )(4) (4) 4 0 24 24 ( 푀 24 ) 푡 1 + (푎24 ) 푡 퐺25 + (4) 푒 − 1 ( 푀 24 )

From which it follows that 274

(4) 0 ( 푃 24 ) +퐺 4 − 25 (4) (푎24) 0 0 −( 푀24 ) 푡 (4) 0 퐺25 (4) 퐺24 푡 − 퐺24 푒 ≤ (4) ( 푃24 ) + 퐺25 푒 + ( 푃24 ) ( 푀 24 )

0 퐺푖 is as defined in the statement of theorem 1

(b) The operator 풜(5) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is 275 obvious that 푡 (5) 0 5 0 (5) ( 푀28 ) 푠 28 퐺28 푡 ≤ 퐺28 + 0 (푎28 ) 퐺29+( 푃28 ) 푒 푑푠 28 =

(푎 ) 5 ( 푃 )(5) (5) 5 0 28 28 ( 푀 28 ) 푡 1 + (푎28 ) 푡 퐺29 + (5) 푒 − 1 ( 푀 28 )

From which it follows that 276

( 푃 )(5)+퐺0 5 − 28 29 (5) (푎28) 0 0 −( 푀28 ) 푡 (5) 0 퐺29 (5) 퐺28 푡 − 퐺28 푒 ≤ (5) ( 푃28 ) + 퐺29 푒 + ( 푃28 ) ( 푀 28 )

0 퐺푖 is as defined in the statement of theorem 1

(c) The operator 풜(6) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is 277 obvious that

푡 (6) 0 6 0 (6) ( 푀32 ) 푠 32 퐺32 푡 ≤ 퐺32 + 0 (푎32 ) 퐺33 +( 푃32 ) 푒 푑푠 32 =

(푎 ) 6 ( 푃 )(6) (6) 6 0 32 32 ( 푀 32 ) 푡 1 + (푎32 ) 푡 퐺33 + (6) 푒 − 1 ( 푀 32 )

From which it follows that 278

( 푃 )(6)+퐺0 6 − 32 33 (6) (푎32) 0 0 −( 푀32 ) 푡 (6) 0 퐺33 (6) 퐺32 푡 − 퐺32 푒 ≤ (6) ( 푃32 ) + 퐺33 푒 + ( 푃32 ) ( 푀 32 )

0 퐺푖 is as defined in the statement of theorem1

Analogous inequalities hold also for 퐺25 , 퐺26 , 푇24 , 푇25 , 푇26

(d) The operator 풜(7) maps the space of functions satisfying 37,35,36 into itself .Indeed it is obvious that 279

푡 (7) 0 7 0 (7) ( 푀36 ) 푠 36 퐺36 푡 ≤ 퐺36 + 0 (푎36 ) 퐺37 +( 푃36 ) 푒 푑푠 36 =

(푎 ) 7 ( 푃 )(7) (7) 7 0 36 36 ( 푀 36 ) 푡 1 + (푎36 ) 푡 퐺37 + (7) 푒 − 1 ( 푀 36 )

280

www.ijmer.com 2061 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 From which it follows that

( 푃 )(7)+퐺0 7 − 36 37 (7) (푎36) 0 0 −( 푀36 ) 푡 (7) 0 퐺37 (7) 퐺36 푡 − 퐺36 푒 ≤ (7) ( 푃36 ) + 퐺37 푒 + ( 푃36 ) ( 푀 36 )

0 퐺푖 is as defined in the statement of theorem 7

1 1 (푎푖) (푏푖) 281 It is now sufficient to take (1) , (1) < 1 and to choose ( 푀 13 ) ( 푀 13 )

(1) (1) ( P 13 ) and ( Q 13 ) large to have 282

(1) 0 ( 푃13 ) +퐺푗 283 − (푎 ) 1 0 푖 1 (1) 0 퐺푗 (1) 1 ( 푃13 ) + ( 푃13 ) + 퐺푗 푒 ≤ ( 푃13 ) (푀 13)

(1) 0 ( 푄13 ) +푇푗 284 − (푏 ) 1 0 푖 (1) 0 푇푗 (1) (1) 1 ( 푄13 ) + 푇푗 푒 + ( 푄13 ) ≤ ( 푄13 ) (푀 13)

(1) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying GLOBAL 285 EQUATIONS into itself

The operator 풜(1) is a contraction with respect to the metric 286

푑 퐺 1 , 푇 1 , 퐺 2 , 푇 2 =

1 1 1 2 −(푀 13) 푡 1 2 −(푀 13) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote 287

Definition of 퐺 , 푇 :

퐺 , 푇 = 풜(1)(퐺, 푇)

It results

1 2 푡 1 2 1 1 1 −( 푀13) 푠 13 ( 푀13) 푠 13 퐺13 − 퐺푖 ≤ 0 (푎13 ) 퐺14 − 퐺14 푒 푒 푑푠 13 +

푡 1 2 1 1 ′ 1 −( 푀13) 푠 13 −( 푀13) 푠 13 0 {(푎13 ) 퐺13 − 퐺13 푒 푒 +

1 1 ′′ 1 1 1 2 −( 푀13) 푠 13 ( 푀13) 푠 13 (푎13 ) 푇14 , 푠 13 퐺13 − 퐺13 푒 푒 +

1 1 2 ′′ 1 1 ′′ 1 2 −( 푀13) 푠 13 ( 푀13) 푠 13 퐺13 |(푎13 ) 푇14 , 푠 13 − (푎13 ) 푇14 , 푠 13 | 푒 푒 }푑푠 13

Where 푠 13 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows

1 퐺 1 − 퐺 2 푒−( 푀13) 푡 ≤ 288 1 1 ′ 1 1 1 1 1 1 2 2 1 (푎13 ) + (푎13 ) + ( 퐴13 ) + ( 푃13 ) ( 푘13 ) 푑 퐺 ,푇 ; 퐺 , 푇 ( 푀13)

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

′′ 1 ′′ 1 Remark 1: The fact that we supposed (푎13 ) and (푏13 ) depending also on t can be considered as not 289

www.ijmer.com 2062 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 1 1 1 ( 푀13) 푡 1 ( 푀13) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃13 ) 푒 푎푛푑 ( 푄13 ) 푒 respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices to ′′ 1 ′′ 1 consider that (푎푖 ) and (푏푖 ) , 푖 = 13,14,15 depend only on T14 and respectively on 퐺(푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 290 From 19 to 24 it results

푡 ′ 1 ′′ 1 0 − 0 (푎푖 ) −(푎푖 ) 푇14 푠 13 ,푠 13 푑푠 13 291 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0

′ 1 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 1 , 푎푛푑 ( 푀 ) 1 : 292 13 1 13 3

Remark 3: if 퐺13 is bounded, the same property have also 퐺14 푎푛푑 퐺15 . indeed if

푑퐺 퐺 < ( 푀 ) 1 it follows 14 ≤ ( 푀 ) 1 − (푎′ ) 1 퐺 and by integrating 13 13 푑푡 13 1 14 14

퐺 ≤ ( 푀 ) 1 = 퐺0 + 2(푎 ) 1 ( 푀 ) 1 /(푎′ ) 1 14 13 2 14 14 13 1 14 In the same way , one can obtain

퐺 ≤ ( 푀 ) 1 = 퐺0 + 2(푎 ) 1 ( 푀 ) 1 /(푎′ ) 1 15 13 3 15 15 13 2 15

If 퐺14 표푟 퐺15 is bounded, the same property follows for 퐺13 , 퐺15 and 퐺13 , 퐺14 respectively.

Remark 4: If 퐺13 푖푠 bounded, from below, the same property holds for 퐺14 푎푛푑 퐺15 . The proof is analogous 293 with the preceding one. An analogous property is true if 퐺14 is bounded from below.

′′ 1 ′ 1 Remark 5: If T13 is bounded from below and lim푡→∞ ((푏푖 ) (퐺 푡 , 푡)) = (푏14 ) then 푇14 → ∞. 294

1 Definition of 푚 and 휀1 :

Indeed let 푡1 be so that for 푡 > 푡1

1 ′′ 1 1 (푏14 ) − (푏푖 ) (퐺 푡 , 푡) < 휀1, 푇13 (푡) > 푚

푑푇 Then 14 ≥ (푎 ) 1 푚 1 − 휀 푇 which leads to 295 푑푡 14 1 14

(푎 ) 1 푚 1 1 14 −휀1푡 0 −휀1푡 −휀1푡 푇14 ≥ 1 − 푒 + 푇14 푒 If we take t such that 푒 = it results 휀1 2

1 1 (푎14) 푚 2 푇14 ≥ , 푡 = 푙표푔 By taking now 휀1 sufficiently small one sees that T14 is unbounded. The 2 휀1 ′′ 1 ′ 1 same property holds for 푇15 if lim푡→∞(푏15 ) 퐺 푡 , 푡 = (푏15 ) We now state a more precise theorem about the behaviors at infinity of the solutions

296

2 2 (푎푖) (푏푖) 297 It is now sufficient to take (2) , (2) < 1 and to choose ( 푀 16 ) ( 푀 16 )

(2) (2) ( 푃 16 ) 푎푛푑 ( 푄 16 ) large to have

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(2) 0 ( 푃16 ) +퐺푗 298 − (푎 ) 2 0 푖 2 (2) 0 퐺푗 (2) 2 ( 푃16 ) + ( 푃16 ) + 퐺푗 푒 ≤ ( 푃16 ) (푀 16)

299

(2) 0 ( 푄16 ) +푇푗 − (푏 ) 2 0 푖 (2) 0 푇푗 (2) (2) 2 ( 푄16 ) + 푇푗 푒 + ( 푄16 ) ≤ ( 푄16 ) (푀 16)

(2) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying 300

The operator 풜(2) is a contraction with respect to the metric 301

1 1 2 2 푑 퐺19 , 푇19 , 퐺19 , 푇19 =

2 2 1 2 −(푀 16) 푡 1 2 −(푀 16) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote 302

(2) Definition of 퐺 19 , 푇 19 : 퐺 19 , 푇 19 = 풜 (퐺19 , 푇19 )

It results 303

1 2 푡 1 2 2 2 2 −( 푀16) 푠 16 ( 푀16) 푠 16 퐺16 − 퐺푖 ≤ 0 (푎16 ) 퐺17 − 퐺17 푒 푒 푑푠 16 +

푡 1 2 2 2 ′ 2 −( 푀16) 푠 16 −( 푀16) 푠 16 0 {(푎16 ) 퐺16 − 퐺16 푒 푒 +

2 2 ′′ 2 1 1 2 −( 푀16) 푠 16 ( 푀16) 푠 16 (푎16 ) 푇17 , 푠 16 퐺16 − 퐺16 푒 푒 +

2 2 2 ′′ 2 1 ′′ 2 2 −( 푀16) 푠 16 ( 푀16) 푠 16 퐺16 |(푎16 ) 푇17 , 푠 16 − (푎16 ) 푇17 , 푠 16 | 푒 푒 }푑푠 16

Where 푠 16 represents integrand that is integrated over the interval 0, 푡 304

From the hypotheses it follows

2 1 2 −( M 16) t 305 퐺19 − 퐺19 e ≤ 1 2 ′ 2 2 2 2 1 1 2 2 2 (푎16 ) + (푎16 ) + ( A16 ) + ( P16 ) ( 푘16 ) d 퐺19 , 푇19 ; 퐺19 , 푇19 ( M16)

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

′′ 2 ′′ 2 Remark 1: The fact that we supposed (푎16 ) and (푏16 ) depending also on t can be considered as not 307 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 2 2 2 ( M16) t 2 ( M 16) t necessary to prove the uniqueness of the solution bounded by ( P16 ) e and ( Q16 ) e respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices to ′′ 2 ′′ 2 consider that (푎푖 ) and (푏푖 ) , 푖 = 16,17,18 depend only on T17 and respectively on 퐺19 (and not on t) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any t where G푖 t = 0 and T푖 t = 0 308 From 19 to 24 it results

t ′ 2 ′′ 2 0 − 0 (푎푖 ) −(푎푖 ) T17 푠 16 ,푠 16 d푠 16 G푖 t ≥ G푖 e ≥ 0

′ 2 0 −(푏푖 ) t T푖 t ≥ T푖 e > 0 for t > 0

www.ijmer.com 2064 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Definition of ( M ) 2 , ( M ) 2 and ( M ) 2 : 309 16 1 16 2 16 3

Remark 3: if G16 is bounded, the same property have also G17 and G18 . indeed if

dG G < ( M ) 2 it follows 17 ≤ ( M ) 2 − (푎′ ) 2 G and by integrating 16 16 dt 16 1 17 17

G ≤ ( M ) 2 = G0 + 2(푎 ) 2 ( M ) 2 /(푎′ ) 2 17 16 2 17 17 16 1 17

In the same way , one can obtain

G ≤ ( M ) 2 = G0 + 2(푎 ) 2 ( M ) 2 /(푎′ ) 2 18 16 3 18 18 16 2 18 310

If G17 or G18 is bounded, the same property follows for G16 , G18 and G16 , G17 respectively.

Remark 4: If G16 is bounded, from below, the same property holds for G17 and G18 . The proof is analogous 311 with the preceding one. An analogous property is true if G17 is bounded from below.

′′ 2 ′ 2 Remark 5: If T16 is bounded from below and limt→∞ ((푏푖 ) ( 퐺19 t , t)) = (푏17) then T17 → ∞. 312

2 Definition of 푚 and ε2 :

Indeed let t2 be so that for t > t2

2 ′′ 2 2 (푏17 ) − (푏푖 ) ( 퐺19 t , t) < ε2, T16 (t) > 푚

dT Then 17 ≥ (푎 ) 2 푚 2 − ε T which leads to 313 dt 17 2 17

(푎 ) 2 푚 2 1 17 −ε2t 0 −ε2t −ε2t T17 ≥ 1 − e + T17 e If we take t such that e = it results ε2 2

2 2 (푎17) 푚 2 314 T17 ≥ , 푡 = log By taking now ε2 sufficiently small one sees that T17 is unbounded. The 2 ε2 ′′ 2 ′ 2 same property holds for T18 if lim푡→∞ (푏18 ) 퐺19 t , t = (푏18 )

We now state a more precise theorem about the behaviors at infinity of the solutions

315

3 3 (푎푖) (푏푖) 316 It is now sufficient to take (3) , (3) < 1 and to choose ( 푀 20 ) ( 푀 20 )

(3) (3) ( P 20 ) and ( Q 20 ) large to have

(3) 0 ( 푃20 ) +퐺푗 317 − (푎 ) 3 0 푖 3 (3) 0 퐺푗 (3) 3 ( 푃20 ) + ( 푃20 ) + 퐺푗 푒 ≤ ( 푃20 ) (푀 20)

(3) 0 ( 푄20 ) +푇푗 318 − (푏 ) 3 0 푖 (3) 0 푇푗 (3) (3) 3 ( 푄20 ) + 푇푗 푒 + ( 푄20 ) ≤ ( 푄20 ) (푀 20)

(3) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 into itself 319

The operator 풜(3) is a contraction with respect to the metric 320

1 1 2 2 푑 퐺23 , 푇23 , 퐺23 , 푇23 =

3 3 1 2 −(푀 20) 푡 1 2 −(푀 20) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

www.ijmer.com 2065 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Indeed if we denote 321

(3) Definition of 퐺 23 , 푇 23 : 퐺23 , 푇23 = 풜 퐺23 , 푇23

It results 322

1 2 푡 1 2 3 3 3 −( 푀20) 푠 20 ( 푀20) 푠 20 퐺20 − 퐺푖 ≤ 0 (푎20 ) 퐺21 − 퐺21 푒 푒 푑푠 20 +

푡 1 2 3 3 ′ 3 −( 푀20) 푠 20 −( 푀20) 푠 20 0 {(푎20 ) 퐺20 − 퐺20 푒 푒 + 323 3 3 ′′ 3 1 1 2 −( 푀20) 푠 20 ( 푀20) 푠 20 (푎20 ) 푇21 , 푠 20 퐺20 − 퐺20 푒 푒 +

3 3 2 ′′ 3 1 ′′ 3 2 −( 푀20) 푠 20 ( 푀20) 푠 20 퐺20 |(푎20 ) 푇21 , 푠 20 − (푎20 ) 푇21 ,푠 20 | 푒 푒 }푑푠 20

Where 푠 20 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows

3 퐺 1 − 퐺 2 푒−( 푀20) 푡 ≤ 324 1 3 ′ 3 3 3 3 1 1 2 2 3 (푎20 ) + (푎20 ) + ( 퐴20 ) + ( 푃20 ) ( 푘20 ) 푑 퐺23 , 푇23 ; 퐺23 , 푇23 ( 푀20)

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

′′ 3 ′′ 3 Remark 1: The fact that we supposed (푎20 ) and (푏20 ) depending also on t can be considered as not 325 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 3 3 3 ( 푀20) 푡 3 ( 푀20) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃20 ) 푒 푎푛푑 ( 푄20) 푒 respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices to ′′ 3 ′′ 3 consider that (푎푖 ) and (푏푖 ) , 푖 = 20,21,22 depend only on T21 and respectively on 퐺23 (푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 326 From 19 to 24 it results

푡 ′ 3 ′′ 3 0 − 0 (푎푖 ) −(푎푖 ) 푇21 푠 20 ,푠 20 푑푠 20 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0

′ 3 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 3 , ( 푀 ) 3 푎푛푑 ( 푀 ) 3 : 327 20 1 20 2 20 3

Remark 3: if 퐺20 is bounded, the same property have also 퐺21 푎푛푑 퐺22 . indeed if

푑퐺 퐺 < ( 푀 ) 3 it follows 21 ≤ ( 푀 ) 3 − (푎′ ) 3 퐺 and by integrating 20 20 푑푡 20 1 21 21

퐺 ≤ ( 푀 ) 3 = 퐺0 + 2(푎 ) 3 ( 푀 ) 3 /(푎′ ) 3 21 20 2 21 21 20 1 21 In the same way , one can obtain

퐺 ≤ ( 푀 ) 3 = 퐺0 + 2(푎 ) 3 ( 푀 ) 3 /(푎′ ) 3 22 20 3 22 22 20 2 22

If 퐺21 표푟 퐺22 is bounded, the same property follows for 퐺20 , 퐺22 and 퐺20 , 퐺21 respectively.

Remark 4: If 퐺20 푖푠 bounded, from below, the same property holds for 퐺21 푎푛푑 퐺22 . The proof is analogous 328 with the preceding one. An analogous property is true if 퐺21 is bounded from below.

′′ 3 ′ 3 Remark 5: If T20 is bounded from below and lim푡→∞ ((푏푖 ) 퐺23 푡 , 푡) = (푏21 ) then 푇21 → ∞. 329

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3 Definition of 푚 and 휀3 :

Indeed let 푡3 be so that for 푡 > 푡3 330

3 ′′ 3 3 (푏21 ) − (푏푖 ) 퐺23 푡 , 푡 < 휀3, 푇20 (푡) > 푚

푑푇 Then 21 ≥ (푎 ) 3 푚 3 − 휀 푇 which leads to 331 푑푡 21 3 21

(푎 ) 3 푚 3 1 21 −휀3푡 0 −휀3푡 −휀3푡 푇21 ≥ 1 − 푒 + 푇21 푒 If we take t such that 푒 = it results 휀3 2

3 3 (푎21) 푚 2 푇21 ≥ , 푡 = 푙표푔 By taking now 휀3 sufficiently small one sees that T21 is unbounded. The 2 휀3 ′′ 3 ′ 3 same property holds for 푇22 if lim푡→∞ (푏22 ) 퐺23 푡 , 푡 = (푏22 )

We now state a more precise theorem about the behaviors at infinity of the solutions

332

4 4 (푎푖) (푏푖) 333 It is now sufficient to take (4) , (4) < 1 and to choose ( 푀 24 ) ( 푀 24 )

(4) (4) ( P 24 ) and ( Q 24 ) large to have

(4) 0 ( 푃24 ) +퐺푗 334 − (푎 ) 4 0 푖 4 (4) 0 퐺푗 (4) 4 ( 푃24 ) + ( 푃24 ) + 퐺푗 푒 ≤ ( 푃24 ) (푀 24)

(4) 0 ( 푄24 ) +푇푗 335 − (푏 ) 4 0 푖 (4) 0 푇푗 (4) (4) 4 ( 푄24 ) + 푇푗 푒 + ( 푄24 ) ≤ ( 푄24 ) (푀 24)

(4) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying IN to itself 336

The operator 풜(4) is a contraction with respect to the metric 337

1 1 2 2 푑 퐺27 , 푇27 , 퐺27 , 푇27 =

4 4 1 2 −(푀 24) 푡 1 2 −(푀 24) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote

(4) Definition of 퐺27 , 푇27 : 퐺27 , 푇27 = 풜 ( 퐺27 , 푇27 )

It results

1 2 푡 1 2 4 4 4 −( 푀24) 푠 24 ( 푀24) 푠 24 퐺24 − 퐺푖 ≤ 0 (푎24 ) 퐺25 − 퐺25 푒 푒 푑푠 24 +

푡 1 2 4 4 ′ 4 −( 푀24) 푠 24 −( 푀24) 푠 24 0 {(푎24 ) 퐺24 − 퐺24 푒 푒 +

4 4 ′′ 4 1 1 2 −( 푀24) 푠 24 ( 푀24) 푠 24 (푎24 ) 푇25 , 푠 24 퐺24 − 퐺24 푒 푒 +

4 4 2 ′′ 4 1 ′′ 4 2 −( 푀24) 푠 24 ( 푀24) 푠 24 퐺24 |(푎24 ) 푇25 , 푠 24 − (푎24 ) 푇25 ,푠 24 | 푒 푒 }푑푠 24

Where 푠 24 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows

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4 1 2 −( 푀24 ) 푡 339 퐺27 − 퐺27 푒 ≤ 1 4 ′ 4 4 4 4 1 1 2 2 4 (푎24 ) + (푎24 ) + ( 퐴24 ) + ( 푃24 ) ( 푘24 ) 푑 퐺27 , 푇27 ; 퐺27 , 푇27 ( 푀24)

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

′′ 4 ′′ 4 Remark 1: The fact that we supposed (푎24 ) and (푏24 ) depending also on t can be considered as not 340 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 4 4 4 ( 푀24) 푡 4 ( 푀24) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃24 ) 푒 푎푛푑 ( 푄24 ) 푒 respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices ′′ 4 ′′ 4 to consider that (푎푖 ) and (푏푖 ) , 푖 = 24,25,26 depend only on T25 and respectively on 퐺27 (푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 341

From GLOBAL EQUATIONS it results

푡 ′ 4 ′′ 4 0 − 0 (푎푖 ) −(푎푖 ) 푇25 푠 24 ,푠 24 푑푠 24 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0

′ 4 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 4 , ( 푀 ) 4 푎푛푑 ( 푀 ) 4 : 342 24 1 24 2 24 3

Remark 3: if 퐺24 is bounded, the same property have also 퐺25 푎푛푑 퐺26 . indeed if

푑퐺 퐺 < ( 푀 ) 4 it follows 25 ≤ ( 푀 ) 4 − (푎′ ) 4 퐺 and by integrating 24 24 푑푡 24 1 25 25

퐺 ≤ ( 푀 ) 4 = 퐺0 + 2(푎 ) 4 ( 푀 ) 4 /(푎′ ) 4 25 24 2 25 25 24 1 25

In the same way , one can obtain

퐺 ≤ ( 푀 ) 4 = 퐺0 + 2(푎 ) 4 ( 푀 ) 4 /(푎′ ) 4 26 24 3 26 26 24 2 26

If 퐺25 표푟 퐺26 is bounded, the same property follows for 퐺24 , 퐺26 and 퐺24 , 퐺25 respectively.

Remark 4: If 퐺24 푖푠 bounded, from below, the same property holds for 퐺25 푎푛푑 퐺26 . The proof is analogous 343 with the preceding one. An analogous property is true if 퐺25 is bounded from below.

′′ 4 ′ 4 Remark 5: If T24 is bounded from below and lim푡→∞ ((푏푖 ) ( 퐺27 푡 , 푡)) = (푏25 ) then 푇25 → ∞. 344

4 Definition of 푚 and 휀4 :

Indeed let 푡4 be so that for 푡 > 푡4

4 ′′ 4 4 (푏25 ) − (푏푖 ) ( 퐺27 푡 , 푡) < 휀4, 푇24 (푡) > 푚

푑푇 Then 25 ≥ (푎 ) 4 푚 4 − 휀 푇 which leads to 345 푑푡 25 4 25

(푎 ) 4 푚 4 1 25 −휀4푡 0 −휀4푡 −휀4푡 푇25 ≥ 1 − 푒 + 푇25 푒 If we take t such that 푒 = it results 휀4 2

4 4 (푎25) 푚 2 푇25 ≥ , 푡 = 푙표푔 By taking now 휀4 sufficiently small one sees that T25 is unbounded. The 2 휀4 www.ijmer.com 2068 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

′′ 4 ′ 4 same property holds for 푇26 if lim푡→∞ (푏26 ) 퐺27 푡 , 푡 = (푏26)

We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS inequalities hold also for 퐺29 , 퐺30 , 푇28 , 푇29 , 푇30

346

5 5 (푎푖) (푏푖) 347 It is now sufficient to take (5) , (5) < 1 and to choose ( 푀 28 ) ( 푀 28 )

(5) (5) ( P 28 ) and ( Q 28 ) large to have

(5) 0 ( 푃28 ) +퐺푗 348 − (푎 ) 5 0 푖 5 (5) 0 퐺푗 (5) 5 ( 푃28 ) + ( 푃28 ) + 퐺푗 푒 ≤ ( 푃28 ) (푀 28)

(5) 0 ( 푄28 ) +푇푗 349 − (푏 ) 5 0 푖 (5) 0 푇푗 (5) (5) 5 ( 푄28 ) + 푇푗 푒 + ( 푄28 ) ≤ ( 푄28 ) (푀 28)

(5) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 into itself 350

The operator 풜(5) is a contraction with respect to the metric 351

1 1 2 2 푑 퐺31 , 푇31 , 퐺31 , 푇31 =

5 5 1 2 −(푀 28) 푡 1 2 −(푀 28) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote

(5) Definition of 퐺31 , 푇31 : 퐺31 , 푇31 = 풜 퐺31 , 푇31

It results

1 2 푡 1 2 5 5 5 −( 푀28) 푠 28 ( 푀28) 푠 28 퐺28 − 퐺푖 ≤ 0 (푎28 ) 퐺29 − 퐺29 푒 푒 푑푠 28 +

푡 1 2 5 5 ′ 5 −( 푀28) 푠 28 −( 푀28) 푠 28 0 {(푎28 ) 퐺28 − 퐺28 푒 푒 +

5 5 ′′ 5 1 1 2 −( 푀28) 푠 28 ( 푀28) 푠 28 (푎28 ) 푇29 , 푠 28 퐺28 − 퐺28 푒 푒 +

5 5 2 ′′ 5 1 ′′ 5 2 −( 푀28) 푠 28 ( 푀28) 푠 28 퐺28 |(푎28 ) 푇29 , 푠 28 − (푎28 ) 푇29 ,푠 28 | 푒 푒 }푑푠 28

Where 푠 28 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows 352

5 1 2 −( 푀28 ) 푡 353 퐺31 − 퐺31 푒 ≤ 1 5 ′ 5 5 5 5 1 1 2 2 5 (푎28 ) + (푎28) + ( 퐴28 ) + ( 푃28 ) ( 푘28 ) 푑 퐺31 , 푇31 ; 퐺31 , 푇31 ( 푀28)

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

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′′ 5 ′′ 5 Remark 1: The fact that we supposed (푎28 ) and (푏28 ) depending also on t can be considered as not 354 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 5 5 5 ( 푀28) 푡 5 ( 푀28) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃28 ) 푒 푎푛푑 ( 푄28 ) 푒 respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices ′′ 5 ′′ 5 to consider that (푎푖 ) and (푏푖 ) , 푖 = 28,29,30 depend only on T29 and respectively on 퐺31 (푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 355

From GLOBAL EQUATIONS it results

푡 ′ 5 ′′ 5 0 − 0 (푎푖 ) −(푎푖 ) 푇29 푠 28 ,푠 28 푑푠 28 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0

′ 5 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 5 , ( 푀 ) 5 푎푛푑 ( 푀 ) 5 : 356 28 1 28 2 28 3

Remark 3: if 퐺28 is bounded, the same property have also 퐺29 푎푛푑 퐺30 . indeed if

푑퐺 퐺 < ( 푀 ) 5 it follows 29 ≤ ( 푀 ) 5 − (푎′ ) 5 퐺 and by integrating 28 28 푑푡 28 1 29 29

퐺 ≤ ( 푀 ) 5 = 퐺0 + 2(푎 ) 5 ( 푀 ) 5 /(푎′ ) 5 29 28 2 29 29 28 1 29

In the same way , one can obtain

퐺 ≤ ( 푀 ) 5 = 퐺0 + 2(푎 ) 5 ( 푀 ) 5 /(푎′ ) 5 30 28 3 30 30 28 2 30

If 퐺29 표푟 퐺30 is bounded, the same property follows for 퐺28 , 퐺30 and 퐺28 , 퐺29 respectively.

Remark 4: If 퐺28 푖푠 bounded, from below, the same property holds for 퐺29 푎푛푑 퐺30 . The proof is analogous 357 with the preceding one. An analogous property is true if 퐺29 is bounded from below.

′′ 5 ′ 5 Remark 5: If T28 is bounded from below and lim푡→∞ ((푏푖 ) ( 퐺31 푡 , 푡)) = (푏29) then 푇29 → ∞. 358

5 Definition of 푚 and 휀5 :

Indeed let 푡5 be so that for 푡 > 푡5

5 ′′ 5 5 (푏29) − (푏푖 ) ( 퐺31 푡 , 푡) < 휀5, 푇28 (푡) > 푚

359

푑푇 Then 29 ≥ (푎 ) 5 푚 5 − 휀 푇 which leads to 360 푑푡 29 5 29

(푎 ) 5 푚 5 1 29 −휀5푡 0 −휀5푡 −휀5푡 푇29 ≥ 1 − 푒 + 푇29 푒 If we take t such that 푒 = it results 휀5 2

5 5 (푎29) 푚 2 푇29 ≥ , 푡 = 푙표푔 By taking now 휀5 sufficiently small one sees that T29 is unbounded. The 2 휀5 ′′ 5 ′ 5 same property holds for 푇30 if lim푡→∞ (푏30 ) 퐺31 푡 , 푡 = (푏30)

We now state a more precise theorem about the behaviors at infinity of the solutions

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Analogous inequalities hold also for 퐺33 , 퐺34 , 푇32 , 푇33 , 푇34

361

6 6 (푎푖) (푏푖) 362 It is now sufficient to take (6) , (6) < 1 and to choose ( 푀 32 ) ( 푀 32 )

(6) (6) ( P 32 ) and ( Q 32 ) large to have

(6) 0 ( 푃32 ) +퐺푗 363 − (푎 ) 6 0 푖 6 (6) 0 퐺푗 (6) 6 ( 푃32 ) + ( 푃32 ) + 퐺푗 푒 ≤ ( 푃32 ) (푀 32)

(6) 0 ( 푄32 ) +푇푗 364 − (푏 ) 6 0 푖 (6) 0 푇푗 (6) (6) 6 ( 푄32 ) + 푇푗 푒 + ( 푄32 ) ≤ ( 푄32 ) (푀 32)

(6) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 into itself 365

The operator 풜(6) is a contraction with respect to the metric 366

1 1 2 2 푑 퐺35 , 푇35 , 퐺35 , 푇35 =

6 6 1 2 −(푀 32) 푡 1 2 −(푀 32) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote

(6) Definition of 퐺35 , 푇35 : 퐺35 , 푇35 = 풜 퐺35 , 푇35

It results

1 2 푡 1 2 6 6 6 −( 푀32) 푠 32 ( 푀32) 푠 32 퐺32 − 퐺푖 ≤ 0 (푎32 ) 퐺33 − 퐺33 푒 푒 푑푠 32 +

푡 1 2 6 6 ′ 6 −( 푀32) 푠 32 −( 푀32) 푠 32 0 {(푎32 ) 퐺32 − 퐺32 푒 푒 +

6 6 ′′ 6 1 1 2 −( 푀32) 푠 32 ( 푀32) 푠 32 (푎32 ) 푇33 , 푠 32 퐺32 − 퐺32 푒 푒 +

6 6 2 ′′ 6 1 ′′ 6 2 −( 푀32) 푠 32 ( 푀32) 푠 32 퐺32 |(푎32 ) 푇33 , 푠 32 − (푎32 ) 푇33 ,푠 32 | 푒 푒 }푑푠 32

Where 푠 32 represents integrand that is integrated over the interval 0, t 367

From the hypotheses it follows

′ 1 ′′ 1 1 ′ 1 ′′ 1 (1) 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 13,14,15

′′ 1 ′′ 1 (2)The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 1 1 Definition of (푝푖 ) , (푟푖 ) :

′′ 1 1 (1) 푎푖 (푇14 , 푡) ≤ (푝푖 ) ≤ ( 퐴13 )

′′ 1 1 ′ 1 (1) 푏푖 (퐺, 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵13 )

′′ 1 1 (3) 푙푖푚푇2→∞ 푎푖 푇14 , 푡 = (푝푖 ) ′′ 1 1 limG→∞ 푏푖 퐺, 푡 = (푟푖 )

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(1) (1) Definition of ( 퐴 13 ) ,( 퐵 13 ) :

(1) (1) 1 1 Where ( 퐴 13 ) ,( 퐵 13 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 13,14,15

They satisfy Lipschitz condition: (1) ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 |(푎푖 ) 푇14 , 푡 − (푎푖 ) 푇14 , 푡 | ≤ ( 푘13 ) |푇14 − 푇14 |푒

(1) ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 |(푏푖 ) 퐺 , 푡 − (푏푖 ) 퐺, 푇 | < ( 푘13 ) ||퐺 − 퐺 ||푒

′′ 1 ′ ′′ 1 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇14 , 푡 and(푎푖 ) 푇14 , 푡 . 푇14 , 푡 and (1) (1) ′′ 1 푇14 , 푡 are points belonging to the interval ( 푘13 ) , ( 푀13 ) . It is to be noted that (푎푖 ) 푇14 , 푡 is uniformly continuous. In (1) ′′ 1 the eventuality of the fact, that if ( 푀13 ) = 1 then the function (푎푖 ) 푇14 , 푡 , the first augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous.

(1) (1) Definition of ( 푀 13 ) ,( 푘13 ) :

(1) (1) (Z) ( 푀 13 ) , ( 푘13 ) , are positive constants

1 1 (푎푖) (푏푖) (1) , (1) < 1 ( 푀 13 ) ( 푀 13 )

(1) (1) Definition of ( 푃 13 ) , ( 푄 13 ) :

(1) (1) (1) (1) (1) (1) (AA) There exists two constants ( 푃 13 ) and ( 푄 13 ) which together with ( 푀 13 ) , ( 푘13 ) , (퐴 13 ) 푎푛푑 ( 퐵 13 ) and the 1 ′ 1 1 ′ 1 1 1 constants (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 13,14,15, satisfy the inequalities

1 1 ′ 1 (1) (1) (1) (1) [ (푎푖 ) + (푎푖 ) + ( 퐴13 ) + ( 푃13 ) ( 푘13 ) ] < 1 ( 푀 13 )

1 1 ′ 1 (1) (1) (1) (1) [ (푏푖 ) + (푏푖 ) + ( 퐵13 ) + ( 푄13 ) ( 푘13 ) ] < 1 ( 푀 13 )

Analogous inequalities hold also for 퐺37 , 퐺38 , 푇36 , 푇37 , 푇38 368

7 7 (푎푖) (푏푖) It is now sufficient to take (7) , (7) < 7 and to choose ( 푀 36 ) ( 푀 36 )

(7) (7) ( P 36 ) and ( Q 36 ) large to have

(7) 0 ( 푃36 ) +퐺푗 369 − (푎 ) 7 0 푖 7 (7) 0 퐺푗 (7) 7 ( 푃36 ) + ( 푃36 ) + 퐺푗 푒 ≤ ( 푃36 ) (푀 36)

(7) 0 ( 푄36 ) +푇푗 370 − (푏 ) 7 0 푖 (7) 0 푇푗 (7) (7) 7 ( 푄36 ) + 푇푗 푒 + ( 푄36 ) ≤ ( 푄36 ) (푀 36)

371

www.ijmer.com 2072 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 The operator 풜(7) is a contraction with respect to the metric 372 1 1 2 2 푑 퐺39 , 푇39 , 퐺39 , 푇39 =

7 7 1 2 −(푀 36) 푡 1 2 −(푀 36) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote

Definition of 퐺39 , 푇39 :

(7) 퐺39 , 푇39 = 풜 ( 퐺39 , 푇39 )

It results

1 2 푡 1 2 7 7 7 −( 푀36) 푠 36 ( 푀36) 푠 36 퐺36 − 퐺푖 ≤ 0 (푎36 ) 퐺37 − 퐺37 푒 푒 푑푠 36 +

푡 1 2 7 7 ′ 7 −( 푀36) 푠 36 −( 푀36) 푠 36 0 {(푎36 ) 퐺36 − 퐺36 푒 푒 +

7 7 ′′ 7 1 1 2 −( 푀36) 푠 36 ( 푀36) 푠 36 (푎36 ) 푇37 , 푠 36 퐺36 − 퐺36 푒 푒 +

7 7 2 ′′ 7 1 ′′ 7 2 −( 푀36) 푠 36 ( 푀36) 푠 36 퐺36 |(푎36 ) 푇37 , 푠 36 − (푎36 ) 푇37 ,푠 36 | 푒 푒 }푑푠 36

Where 푠 36 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows

373 7 1 2 −( 푀36) 푡 퐺39 − 퐺39 푒 ≤

1 7 ′ 7 7 7 7 1 1 2 2 7 (푎36 ) + (푎36 ) + ( 퐴36 ) + ( 푃36 ) ( 푘36 ) 푑 퐺39 , 푇39 ; 퐺39 , 푇39 ( 푀36)

And analogous inequalities for 퐺푖 푎푛푑 푇푖 . Taking into account the hypothesis (37,35,36) the result follows 374 ′′ 7 ′′ 7 Remark 1: The fact that we supposed (푎36 ) and (푏36 ) depending also on t can be considered as not 375 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 7 7 7 ( 푀36) 푡 7 ( 푀36) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃36 ) 푒 푎푛푑 ( 푄36) 푒 respectively of ℝ+. If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices to ′′ 7 ′′ 7 consider that (푎푖 ) and (푏푖 ) , 푖 = 36,37,38 depend only on T37 and respectively on 퐺39 (푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 376 From 79 to 36 it results 푡 ′ 7 ′′ 7 0 − 0 (푎푖) −(푎푖 ) 푇37 푠 36 ,푠 36 푑푠 36 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0 ′ 7 0 −(푏푖) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 7 , ( 푀 ) 7 푎푛푑 ( 푀 ) 7 : 377 36 1 36 2 36 3

Remark 3: if 퐺36 is bounded, the same property have also 퐺37 푎푛푑 퐺38 . indeed if

푑퐺 퐺 < ( 푀 ) 7 it follows 37 ≤ ( 푀 ) 7 − (푎′ ) 7 퐺 and by integrating 36 36 푑푡 36 1 37 37

퐺 ≤ ( 푀 ) 7 = 퐺0 + 2(푎 ) 7 ( 푀 ) 7 /(푎′ ) 7 37 36 2 37 37 36 1 37

In the same way , one can obtain

www.ijmer.com 2073 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 퐺 ≤ ( 푀 ) 7 = 퐺0 + 2(푎 ) 7 ( 푀 ) 7 /(푎′ ) 7 38 36 3 38 38 36 2 38

If 퐺37 표푟 퐺38 is bounded, the same property follows for 퐺36 , 퐺38 and 퐺36 , 퐺37 respectively.

Remark 7: If 퐺36 푖푠 bounded, from below, the same property holds for 퐺37 푎푛푑 퐺38 . The proof is analogous 378 with the preceding one. An analogous property is true if 퐺37 is bounded from below.

′′ 7 ′ 7 Remark 5: If T36 is bounded from below and lim푡→∞((푏푖 ) ( 퐺39 푡 , 푡)) = (푏37 ) then 푇37 → ∞. 379 7 Definition of 푚 and 휀7 : Indeed let 푡7 be so that for 푡 > 푡7 7 ′′ 7 7 (푏37 ) − (푏푖 ) ( 퐺39 푡 , 푡) < 휀7, 푇36 (푡) > 푚

푑푇 Then 37 ≥ (푎 ) 7 푚 7 − 휀 푇 which leads to 380 푑푡 37 7 37 (푎 ) 7 푚 7 1 37 −휀7푡 0 −휀7푡 −휀7푡 푇37 ≥ 1 − 푒 + 푇37 푒 If we take t such that 푒 = it results 휀7 2

7 7 (푎37) 푚 2 푇37 ≥ , 푡 = 푙표푔 By taking now 휀7 sufficiently small one sees that T37 is unbounded. The 2 휀7 ′′ 7 ′ 7 same property holds for 푇38 if lim푡→∞(푏38 ) 퐺39 푡 , 푡 = (푏38 )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 72

(7) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying GLOBAL 381 EQUATIONS AND ITS CONCOMITANT CONDITIONALITIES into itself 382

The operator 풜(7) is a contraction with respect to the metric 383

1 1 2 2 푑 퐺39 , 푇39 , 퐺39 , 푇39 =

7 7 1 2 −(푀 36) 푡 1 2 −(푀 36) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+

Indeed if we denote

Definition of 퐺39 , 푇39 :

(7) 퐺39 , 푇39 = 풜 ( 퐺39 , 푇39 )

It results

1 2 푡 1 2 7 7 7 −( 푀36) 푠 36 ( 푀36) 푠 36 퐺36 − 퐺푖 ≤ 0 (푎36 ) 퐺37 − 퐺37 푒 푒 푑푠 36 +

푡 1 2 7 7 ′ 7 −( 푀36) 푠 36 −( 푀36) 푠 36 0 {(푎36 ) 퐺36 − 퐺36 푒 푒 +

7 7 ′′ 7 1 1 2 −( 푀36) 푠 36 ( 푀36) 푠 36 (푎36 ) 푇37 , 푠 36 퐺36 − 퐺36 푒 푒 +

7 7 2 ′′ 7 1 ′′ 7 2 −( 푀36) 푠 36 ( 푀36) 푠 36 퐺36 |(푎36 ) 푇37 , 푠 36 − (푎36 ) 푇37 ,푠 36 | 푒 푒 }푑푠 36

Where 푠 36 represents integrand that is integrated over the interval 0, t

From the hypotheses it follows

384

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7 1 2 −( 푀36) 푡 퐺39 − 퐺39 푒 ≤ 1 7 ′ 7 7 7 7 1 1 2 2 7 (푎36 ) + (푎36 ) + ( 퐴36 ) + ( 푃36 ) ( 푘36 ) 푑 퐺39 , 푇39 ; 퐺39 , 푇39 ( 푀36)

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

′′ 7 ′′ 7 Remark 1: The fact that we supposed (푎36 ) and (푏36 ) depending also on t can be considered as not 385 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 7 7 7 ( 푀36) 푡 7 ( 푀36) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃36 ) 푒 푎푛푑 ( 푄36 ) 푒 respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it suffices ′′ 7 ′′ 7 to consider that (푎푖 ) and (푏푖 ) , 푖 = 36,37,38 depend only on T37 and respectively on 퐺39 (푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 386

From CONCATENATED GLOBAL EQUATIONS it results

푡 ′ 7 ′′ 7 0 − 0 (푎푖 ) −(푎푖 ) 푇37 푠 36 ,푠 36 푑푠 36 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0

′ 7 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0

Definition of ( 푀 ) 7 , ( 푀 ) 7 푎푛푑 ( 푀 ) 7 : 387 36 1 36 2 36 3

Remark 3: if 퐺36 is bounded, the same property have also 퐺37 푎푛푑 퐺38 . indeed if

푑퐺 퐺 < ( 푀 ) 7 it follows 37 ≤ ( 푀 ) 7 − (푎′ ) 7 퐺 and by integrating 36 36 푑푡 36 1 37 37

퐺 ≤ ( 푀 ) 7 = 퐺0 + 2(푎 ) 7 ( 푀 ) 7 /(푎′ ) 7 37 36 2 37 37 36 1 37

In the same way , one can obtain

퐺 ≤ ( 푀 ) 7 = 퐺0 + 2(푎 ) 7 ( 푀 ) 7 /(푎′ ) 7 38 36 3 38 38 36 2 38

If 퐺37 표푟 퐺38 is bounded, the same property follows for 퐺36 , 퐺38 and 퐺36 , 퐺37 respectively.

Remark 7: If 퐺36 푖푠 bounded, from below, the same property holds for 퐺37 푎푛푑 퐺38 . The proof is analogous 388 with the preceding one. An analogous property is true if 퐺37 is bounded from below.

′′ 7 ′ 7 Remark 5: If T36 is bounded from below and lim푡→∞ ((푏푖 ) ( 퐺39 푡 , 푡)) = (푏37 ) then 푇37 → ∞. 389

7 Definition of 푚 and 휀7 :

Indeed let 푡7 be so that for 푡 > 푡7

7 ′′ 7 7 (푏37 ) − (푏푖 ) ( 퐺39 푡 , 푡) < 휀7, 푇36 (푡) > 푚

푑푇 Then 37 ≥ (푎 ) 7 푚 7 − 휀 푇 which leads to 390 푑푡 37 7 37

(푎 ) 7 푚 7 1 37 −휀7푡 0 −휀7푡 −휀7푡 푇37 ≥ 1 − 푒 + 푇37 푒 If we take t such that 푒 = it results 휀7 2

7 7 (푎37) 푚 2 푇37 ≥ , 푡 = 푙표푔 By taking now 휀7 sufficiently small one sees that T37 is unbounded. The 2 휀7 www.ijmer.com 2075 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

′′ 7 ′ 7 same property holds for 푇38 if lim푡→∞ (푏38 ) 퐺39 푡 , 푡 = (푏38 )

We now state a more precise theorem about the behaviors at infinity of the solutions

2 ′ 2 ′ 2 ′′ 2 ′′ 2 2 −(σ2) ≤ −(푎16 ) + (푎17 ) − (푎16 ) T17 , 푡 + (푎17 ) T17 , 푡 ≤ −(σ1) 391

2 ′ 2 ′ 2 ′′ 2 ′′ 2 2 −(τ2) ≤ −(푏16 ) + (푏17) − (푏16 ) 퐺19 , 푡 − (푏17 ) 퐺19 , 푡 ≤ −(τ1) 392

2 2 2 2 Definition of (휈1 ) , (ν2) , (푢1) , (푢2) : 393

2 2 2 2 By (휈1) > 0 , (ν2) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots 394

2 2 2 2 2 2 395 (a) of the equations (푎17 ) 휈 + (σ1) 휈 − (푎16 ) = 0

2 2 2 2 2 2 396 and (푏14 ) 푢 + (τ1) 푢 − (푏16 ) = 0 and

2 2 2 2 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 397

2 2 2 2 By (휈1 ) > 0 , (ν 2) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the 398

2 2 2 2 2 2 399 roots of the equations (푎17 ) 휈 + (σ2) 휈 − (푎16 ) = 0

2 2 2 2 2 2 400 and (푏17 ) 푢 + (τ2) 푢 − (푏16 ) = 0

2 2 2 2 Definition of (푚1) , (푚2) , (휇1) , (휇2) :- 401

2 2 2 2 (b) If we define (푚1) , (푚2) , (휇1) , (휇2) by 402

2 2 2 2 2 2 (푚2) = (휈0 ) , (푚1) = (휈1 ) , 풊풇 (휈0 ) < (휈1) 403

2 2 2 2 2 2 2 (푚2) = (휈1 ) , (푚1) = (휈1 ) , 풊풇 (휈1) < (휈0 ) < (휈1 ) , 404

0 2 G16 and (휈0 ) = 0 G17

2 2 2 2 2 2 ( 푚2) = (휈1 ) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0 ) 405 and analogously 406

2 2 2 2 2 2 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

2 2 2 2 2 2 2 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) ,

0 2 T16 and (푢0) = 0 T17

2 2 2 2 2 2 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) 407 Then the solution satisfies the inequalities 408

2 2 2 0 (S1) −(푝16) t 0 (S1) t G16 e ≤ 퐺16 푡 ≤ G16 e

2 (푝푖 ) is defined 409

1 2 2 1 2 0 (S1) −(푝16 ) t 0 (S1) t 410 2 G16 e ≤ 퐺17 (푡) ≤ 2 G16 e (푚1) (푚2)

(푎 ) 2 G0 2 2 2 2 18 16 (S1) −(푝16 ) t −(S2) t 0 −(S2) t 411 ( 2 2 2 2 e − e + G18 e ≤ G18 (푡) ≤ (푚1) (S1) −(푝16) −(S2)

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(푎 ) 2 G0 2 ′ 2 ′ 2 18 16 (S1) t −(푎18) t 0 −(푎18) t 2 2 ′ 2 [e − e ] + G18 e ) (푚2) (S1) −(푎18)

2 2 2 0 (R1) 푡 0 (R1) +(푟16) 푡 412 T16 e ≤ 푇16 (푡) ≤ T16 e

1 2 1 2 2 0 (R1) 푡 0 (R1) +(푟16 ) 푡 413 2 T16 e ≤ 푇16 (푡) ≤ 2 T16 e (휇1) (휇2)

(푏 ) 2 T0 2 ′ 2 ′ 2 18 16 (R1) 푡 −(푏18) 푡 0 −(푏18) 푡 414 2 2 ′ 2 e − e + T18 e ≤ 푇18 (푡) ≤ (휇1) (R1) −(푏18)

(푎 ) 2 T0 2 2 2 2 18 16 (R1) +(푟16) 푡 −(R2) 푡 0 −(R2) 푡 2 2 2 2 e − e + T18 e (휇2) (R1) +(푟16) +(R2)

2 2 2 2 Definition of (S1) , (S2) , (R1) , (R2) :- 415

2 2 2 ′ 2 Where (S1) = (푎16 ) (푚2) − (푎16 ) 416

2 2 2 (S2) = (푎18 ) − (푝18 )

2 2 1 ′ 2 (푅1) = (푏16 ) (휇2) − (푏16 ) 417

2 ′ 2 2 (R2) = (푏18) − (푟18 ) 418

Behavior of the solutions 419

If we denote and define

3 3 3 3 Definition of (휎1) , (휎2) , (휏1) , (휏2) :

3 3 3 3 (a) 휎1) , (휎2) , (휏1) , (휏2) four constants satisfying

3 ′ 3 ′ 3 ′′ 3 ′′ 3 3 −(휎2) ≤ −(푎20 ) + (푎21 ) − (푎20 ) 푇21 , 푡 + (푎21 ) 푇21 , 푡 ≤ −(휎1)

3 ′ 3 ′ 3 ′′ 3 ′′ 3 3 −(휏2) ≤ −(푏20 ) + (푏21) − (푏20) 퐺, 푡 − (푏21) 퐺23 , 푡 ≤ −(휏1)

3 3 3 3 Definition of (휈1 ) , (휈2) , (푢1) , (푢2) : 420

3 3 3 3 (b) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations 3 3 2 3 3 3 (푎21 ) 휈 + (휎1) 휈 − (푎20 ) = 0

3 3 2 3 3 3 and (푏21 ) 푢 + (휏1) 푢 − (푏20 ) = 0 and

3 3 3 3 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the

3 3 2 3 3 3 roots of the equations (푎21 ) 휈 + (휎2) 휈 − (푎20 ) = 0

3 3 2 3 3 3 and (푏21 ) 푢 + (휏2) 푢 − (푏20 ) = 0

3 3 3 3 Definition of (푚1) , (푚2) , (휇1) , (휇2) :- 421

3 3 3 3 (c) If we define (푚1) , (푚2) , (휇1) , (휇2) by

3 3 3 3 3 3 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

3 3 3 3 3 3 3 (푚2) = (휈1) , (푚1) = (휈1 ) , 풊풇 (휈1 ) < (휈0 ) < (휈1 ) ,

0 3 퐺20 and (휈0 ) = 0 퐺21

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3 3 3 3 3 3 ( 푚2) = (휈1) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0) and analogously 422

3 3 3 3 3 3 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

0 3 3 3 3 3 3 3 3 푇20 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) , and (푢0) = 0 푇21

3 3 3 3 3 3 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) Then the solution satisfies the inequalities

3 3 3 0 (푆1) −(푝20) 푡 0 (푆1) 푡 퐺20 푒 ≤ 퐺20 (푡) ≤ 퐺20 푒

3 (푝푖 ) is defined 423

1 3 3 1 3 0 (푆1) −(푝20) 푡 0 (푆1) 푡 424 3 퐺20 푒 ≤ 퐺21 (푡) ≤ 3 퐺20 푒 (푚1) (푚2)

(푎 ) 3 퐺0 3 3 3 3 22 20 (푆1) −(푝20 ) 푡 −(푆2) 푡 0 −(푆2) 푡 425 ( 3 3 3 3 푒 − 푒 + 퐺22 푒 ≤ 퐺22(푡) ≤ (푚1) (푆1) −(푝20 ) −(푆2) (푎 ) 3 퐺0 3 ′ 3 ′ 3 22 20 (푆1) 푡 −(푎22) 푡 0 −(푎22) 푡 3 3 ′ 3 [푒 − 푒 ] + 퐺22 푒 ) (푚2) (푆1) −(푎22)

3 3 3 0 (푅1) 푡 0 (푅1) +(푟20) 푡 426 푇20 푒 ≤ 푇20 (푡) ≤ 푇20 푒

1 3 1 3 3 0 (푅1) 푡 0 (푅1) +(푟20 ) 푡 427 3 푇20 푒 ≤ 푇20 (푡) ≤ 3 푇20 푒 (휇1) (휇2)

(푏 ) 3 푇0 3 ′ 3 ′ 3 22 20 (푅1) 푡 −(푏22 ) 푡 0 −(푏22) 푡 428 3 3 ′ 3 푒 − 푒 + 푇22 푒 ≤ 푇22 (푡) ≤ (휇1) (푅1) −(푏22)

(푎 ) 3 푇0 3 3 3 3 22 20 (푅1) +(푟20) 푡 −(푅2) 푡 0 −(푅2) 푡 3 3 3 3 푒 − 푒 + 푇22 푒 (휇2) (푅1) +(푟20 ) +(푅2)

3 3 3 3 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 429

3 3 3 ′ 3 Where (푆1) = (푎20 ) (푚2) − (푎20 )

3 3 3 (푆2) = (푎22 ) − (푝22 )

3 3 3 ′ 3 (푅1) = (푏20 ) (휇2) − (푏20)

3 ′ 3 3 (푅2) = (푏22) − (푟22 ) 430 431

432 If we denote and define

4 4 4 4 Definition of (휎1) , (휎2) ,(휏1) , (휏2) :

4 4 4 4 (d) (휎1) , (휎2) ,(휏1) , (휏2) four constants satisfying

4 ′ 4 ′ 4 ′′ 4 ′′ 4 4 −(휎2) ≤ −(푎24 ) + (푎25 ) − (푎24 ) 푇25 , 푡 + (푎25 ) 푇25 , 푡 ≤ −(휎1)

4 ′ 4 ′ 4 ′′ 4 ′′ 4 4 −(휏2) ≤ −(푏24 ) + (푏25) − (푏24) 퐺27 , 푡 − (푏25) 퐺27 , 푡 ≤ −(휏1)

4 4 4 4 4 4 Definition of (휈1 ) , (휈2 ) , (푢1) , (푢2) , 휈 , 푢 : 433

4 4 4 4 (e) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations www.ijmer.com 2078 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

4 4 2 4 4 4 (푎25 ) 휈 + (휎1) 휈 − (푎24 ) = 0 4 4 2 4 4 4 and (푏25) 푢 + (휏1) 푢 − (푏24 ) = 0 and

4 4 4 4 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 434 435 4 4 4 4 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the 2 roots of the equations (푎 ) 4 휈 4 + (휎 ) 4 휈 4 − (푎 ) 4 = 0 25 2 24 4 4 2 4 4 4 and (푏25 ) 푢 + (휏2) 푢 − (푏24 ) = 0 436 4 4 4 4 4 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0 ) :-

4 4 4 4 (f) If we define (푚1) , (푚2) , (휇1) , (휇2) by

4 4 4 4 4 4 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

4 4 4 4 4 4 4 (푚2) = (휈1 ) , (푚1) = (휈1 ) , 풊풇 (휈4) < (휈0 ) < (휈1 ) , 0 4 퐺24 and (휈0 ) = 0 퐺25

4 4 4 4 4 4 ( 푚2) = (휈4) , (푚1) = (휈0 ) , 풊풇 (휈4 ) < (휈0) and analogously 437 438 4 4 4 4 4 4 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

4 4 4 4 4 4 4 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) , 0 4 푇24 and (푢0) = 0 푇25

4 4 4 4 4 4 4 4 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) where (푢1) , (푢 1) are defined respectively

Then the solution satisfies the inequalities 439 440 4 4 4 0 (푆1) −(푝24 ) 푡 0 (푆1) 푡 441 퐺24 푒 ≤ 퐺24 푡 ≤ 퐺24 푒 442 4 443 where (푝푖 ) is defined 444 445

1 4 4 1 4 0 (푆1) −(푝24) 푡 0 (푆1) 푡 446 4 퐺24 푒 ≤ 퐺25 푡 ≤ 4 퐺24 푒 (푚1) (푚2) 447

(푎 ) 4 퐺0 4 4 4 4 26 24 (푆1) −(푝24 ) 푡 −(푆2) 푡 0 −(푆2) 푡 448 4 4 4 4 푒 − 푒 + 퐺26 푒 ≤ 퐺26 푡 ≤ (푚1) (푆1) −(푝24 ) −(푆2) (푎26)4퐺240(푚2)4(푆1)4−(푎26′)4푒(푆1)4푡−푒−(푎26′)4푡+ 퐺260푒−(푎26′)4푡

4 4 4 0 (푅1) 푡 0 (푅1) +(푟24) 푡 449 푇24 푒 ≤ 푇24 푡 ≤ 푇24 푒

1 4 1 4 4 0 (푅1) 푡 0 (푅1) +(푟24 ) 푡 450 4 푇24 푒 ≤ 푇24 (푡) ≤ 4 푇24 푒 (휇1) (휇2)

(푏 ) 4 푇0 4 ′ 4 ′ 4 26 24 (푅1) 푡 −(푏26 ) 푡 0 −(푏26) 푡 451 4 4 ′ 4 푒 − 푒 + 푇26 푒 ≤ 푇26 (푡) ≤ (휇1) (푅1) −(푏26)

(푎 ) 4 푇0 4 4 4 4 26 24 (푅1) +(푟24) 푡 −(푅2) 푡 0 −(푅2) 푡 4 4 4 4 푒 − 푒 + 푇26 푒 (휇2) (푅1) +(푟24 ) +(푅2)

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4 4 4 4 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 452

4 4 4 ′ 4 Where (푆1) = (푎24 ) (푚2) − (푎24 )

4 4 4 (푆2) = (푎26 ) − (푝26 )

4 4 4 ′ 4 (푅1) = (푏24) (휇2) − (푏24 )

4 ′ 4 4 453 (푅2) = (푏26) − (푟26 )

Behavior of the solutions 454 If we denote and define

5 5 5 5 Definition of (휎1) , (휎2) ,(휏1) , (휏2) :

5 5 5 5 (g) (휎1) , (휎2) ,(휏1) , (휏2) four constants satisfying

5 ′ 5 ′ 5 ′′ 5 ′′ 5 5 −(휎2) ≤ −(푎28 ) + (푎29) − (푎28 ) 푇29 , 푡 + (푎29) 푇29 , 푡 ≤ −(휎1)

5 ′ 5 ′ 5 ′′ 5 ′′ 5 5 −(휏2) ≤ −(푏28 ) + (푏29) − (푏28) 퐺31 , 푡 − (푏29) 퐺31 , 푡 ≤ −(휏1)

5 5 5 5 5 5 Definition of (휈1 ) , (휈2 ) , (푢1) , (푢2) , 휈 , 푢 : 455

5 5 5 5 (h) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations 5 5 2 5 5 5 (푎29) 휈 + (휎1) 휈 − (푎28 ) = 0 5 5 2 5 5 5 and (푏29) 푢 + (휏1) 푢 − (푏28 ) = 0 and

5 5 5 5 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 456

5 5 5 5 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the 5 5 2 5 5 5 roots of the equations (푎29) 휈 + (휎2) 휈 − (푎28 ) = 0 5 5 2 5 5 5 and (푏29) 푢 + (휏2) 푢 − (푏28 ) = 0 5 5 5 5 5 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0 ) :-

5 5 5 5 (i) If we define (푚1) , (푚2) , (휇1) , (휇2) by

5 5 5 5 5 5 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

5 5 5 5 5 5 5 (푚2) = (휈1 ) , (푚1) = (휈1 ) , 풊풇 (휈1 ) < (휈0 ) < (휈1 ) , 0 5 퐺28 and (휈0 ) = 0 퐺29

5 5 5 5 5 5 ( 푚2) = (휈1) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0) and analogously 457

5 5 5 5 5 5 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

5 5 5 5 5 5 5 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) , 0 5 푇28 and (푢0) = 0 푇29

5 5 5 5 5 5 5 5 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) where (푢1) , (푢 1) are defined respectively

Then the solution satisfies the inequalities 458

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5 5 5 0 (푆1) −(푝28) 푡 0 (푆1) 푡 퐺28 푒 ≤ 퐺28 (푡) ≤ 퐺28 푒

5 where (푝푖 ) is defined 1 5 5 1 5 0 (푆1) −(푝28) 푡 0 (푆1) 푡 459 5 퐺28 푒 ≤ 퐺29 (푡) ≤ 5 퐺28 푒 (푚5) (푚2)

460 (푎 ) 5 퐺0 5 5 5 5 30 28 (푆1) −(푝28 ) 푡 −(푆2) 푡 0 −(푆2) 푡 461 5 5 5 5 푒 − 푒 + 퐺30 푒 ≤ 퐺30 푡 ≤ (푚1) (푆1) −(푝28 ) −(푆2) (푎30)5퐺280(푚2)5(푆1)5−(푎30′)5푒(푆1)5푡−푒−(푎30′)5푡+ 퐺300푒−(푎30′)5푡

5 5 5 0 (푅1) 푡 0 (푅1) +(푟28) 푡 462 푇28 푒 ≤ 푇28 (푡) ≤ 푇28 푒

1 5 1 5 5 0 (푅1) 푡 0 (푅1) +(푟28 ) 푡 463 5 푇28 푒 ≤ 푇28 (푡) ≤ 5 푇28 푒 (휇1) (휇2)

(푏 ) 5 푇0 5 ′ 5 ′ 5 30 28 (푅1) 푡 −(푏30 ) 푡 0 −(푏30) 푡 464 5 5 ′ 5 푒 − 푒 + 푇30 푒 ≤ 푇30 (푡) ≤ (휇1) (푅1) −(푏30)

(푎 ) 5 푇0 5 5 5 5 30 28 (푅1) +(푟28) 푡 −(푅2) 푡 0 −(푅2) 푡 5 5 5 5 푒 − 푒 + 푇30 푒 (휇2) (푅1) +(푟28 ) +(푅2)

5 5 5 5 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 465

5 5 5 ′ 5 Where (푆1) = (푎28 ) (푚2) − (푎28 )

5 5 5 (푆2) = (푎30 ) − (푝30 )

5 5 5 ′ 5 (푅1) = (푏28) (휇2) − (푏28 )

5 ′ 5 5 (푅2) = (푏30) − (푟30 )

Behavior of the solutions 466 If we denote and define

6 6 6 6 Definition of (휎1) , (휎2) ,(휏1) , (휏2) :

6 6 6 6 (j) (휎1) , (휎2) ,(휏1) , (휏2) four constants satisfying

6 ′ 6 ′ 6 ′′ 6 ′′ 6 6 −(휎2) ≤ −(푎32 ) + (푎33 ) − (푎32 ) 푇33 , 푡 + (푎33 ) 푇33 , 푡 ≤ −(휎1)

6 ′ 6 ′ 6 ′′ 6 ′′ 6 6 −(휏2) ≤ −(푏32 ) + (푏33) − (푏32) 퐺35 , 푡 − (푏33) 퐺35 , 푡 ≤ −(휏1)

6 6 6 6 6 6 Definition of (휈1 ) , (휈2 ) , (푢1) , (푢2) , 휈 , 푢 : 467

6 6 6 6 (k) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations 6 6 2 6 6 6 (푎33 ) 휈 + (휎1) 휈 − (푎32 ) = 0 6 6 2 6 6 6 and (푏33) 푢 + (휏1) 푢 − (푏32 ) = 0 and

6 6 6 6 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 468

6 6 6 6 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the 2 roots of the equations (푎 ) 6 휈 6 + (휎 ) 6 휈 6 − (푎 ) 6 = 0 33 2 32 6 6 2 6 6 6 and (푏33 ) 푢 + (휏2) 푢 − (푏32 ) = 0 6 6 6 6 6 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0 ) :-

6 6 6 6 (l) If we define (푚1) , (푚2) , (휇1) , (휇2) by

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6 6 6 6 6 6 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

6 6 6 6 6 6 6 (푚2) = (휈1 ) , (푚1) = (휈6 ) , 풊풇 (휈1 ) < (휈0 ) < (휈1 ) , 470 0 6 퐺32 and (휈0 ) = 0 퐺33

6 6 6 6 6 6 ( 푚2) = (휈1) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0) and analogously 471

6 6 6 6 6 6 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

6 6 6 6 6 6 6 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) , 0 6 푇32 and (푢0) = 0 푇33

6 6 6 6 6 6 6 6 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) where (푢1) , (푢 1) are defined respectively

Then the solution satisfies the inequalities 472

6 6 6 0 (푆1) −(푝32) 푡 0 (푆1) 푡 퐺32 푒 ≤ 퐺32 (푡) ≤ 퐺32 푒

6 where (푝푖 ) is defined 1 6 6 1 6 0 (푆1) −(푝32) 푡 0 (푆1) 푡 473 6 퐺32 푒 ≤ 퐺33 (푡) ≤ 6 퐺32 푒 (푚1) (푚2)

(푎 ) 6 퐺0 6 6 6 6 34 32 (푆1) −(푝32 ) 푡 −(푆2) 푡 0 −(푆2) 푡 474 6 6 6 6 푒 − 푒 + 퐺34 푒 ≤ 퐺34 푡 ≤ (푚1) (푆1) −(푝32 ) −(푆2) (푎34)6퐺320(푚2)6(푆1)6−(푎34′)6푒(푆1)6푡−푒−(푎34′)6푡+ 퐺340푒−(푎34′)6푡

6 6 6 0 (푅1) 푡 0 (푅1) +(푟32) 푡 475 푇32 푒 ≤ 푇32 (푡) ≤ 푇32 푒

1 6 1 6 6 0 (푅1) 푡 0 (푅1) +(푟32 ) 푡 476 6 푇32 푒 ≤ 푇32 (푡) ≤ 6 푇32 푒 (휇1) (휇2)

(푏 ) 6 푇0 6 ′ 6 ′ 6 34 32 (푅1) 푡 −(푏34 ) 푡 0 −(푏34) 푡 477 6 6 ′ 6 푒 − 푒 + 푇34 푒 ≤ 푇34 (푡) ≤ (휇1) (푅1) −(푏34)

(푎 ) 6 푇0 6 6 6 6 34 32 (푅1) +(푟32) 푡 −(푅2) 푡 0 −(푅2) 푡 6 6 6 6 푒 − 푒 + 푇34 푒 (휇2) (푅1) +(푟32 ) +(푅2)

6 6 6 6 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 478

6 6 6 ′ 6 Where (푆1) = (푎32 ) (푚2) − (푎32 )

6 6 6 (푆2) = (푎34 ) − (푝34 )

6 6 6 ′ 6 (푅1) = (푏32 ) (휇2) − (푏32 )

6 ′ 6 6 (푅2) = (푏34) − (푟34 ) 479

If we denote and define

7 7 7 7 Definition of (휎1) , (휎2) ,(휏1) , (휏2) :

7 7 7 7 (m) (휎1) , (휎2) ,(휏1) , (휏2) four constants satisfying

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7 ′ 7 ′ 7 ′′ 7 ′′ 7 7 −(휎2) ≤ −(푎36 ) + (푎37 ) − (푎36 ) 푇37 , 푡 + (푎37 ) 푇37 , 푡 ≤ −(휎1)

7 ′ 7 ′ 7 ′′ 7 ′′ 7 7 −(휏2) ≤ −(푏36 ) + (푏37) − (푏36) 퐺39 , 푡 − (푏37) 퐺39 , 푡 ≤ −(휏1)

7 7 7 7 7 7 Definition of (휈1 ) , (휈2 ) , (푢1) , (푢2) , 휈 , 푢 : 480

7 7 7 7 (n) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations 7 7 2 7 7 7 (푎37 ) 휈 + (휎1) 휈 − (푎36 ) = 0 481 7 7 2 7 7 7 and (푏37) 푢 + (휏1) 푢 − (푏36 ) = 0 and

7 7 7 7 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 482

7 7 7 7 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the

7 7 2 7 7 7 roots of the equations (푎37 ) 휈 + (휎2) 휈 − (푎36 ) = 0

7 7 2 7 7 7 and (푏37 ) 푢 + (휏2) 푢 − (푏36 ) = 0

7 7 7 7 7 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0 ) :-

7 7 7 7 (o) If we define (푚1) , (푚2) , (휇1) , (휇2) by

7 7 7 7 7 7 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

7 7 7 7 7 7 7 (푚2) = (휈1 ) , (푚1) = (휈1 ) , 풊풇 (휈1 ) < (휈0 ) < (휈1 ) ,

0 7 퐺36 and (휈0 ) = 0 퐺37

7 7 7 7 7 7 ( 푚2) = (휈1) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0) and analogously 483

7 7 7 7 7 7 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1)

7 7 7 7 7 7 7 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) ,

0 7 푇36 and (푢0) = 0 푇37

7 7 7 7 7 7 7 7 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) where (푢1) , (푢 1) are defined respectively

Then the solution satisfies the inequalities 484

7 7 7 퐺0 푒 (푆1) −(푝36 ) 푡 ≤ 퐺 (푡) ≤ 퐺0 푒(푆1) 푡 36 36 36 where (푝 ) 7 is defined 푖 www.ijmer.com 2083 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 485 1 7 7 1 7 0 (푆1) −(푝36) 푡 0 (푆1) 푡 486 7 퐺36 푒 ≤ 퐺37 (푡) ≤ 7 퐺36 푒 (푚7) (푚2)

( 487 (푎 ) 7 퐺0 7 7 7 7 38 36 (푆1) −(푝36) 푡 −(푆2) 푡 0 −(푆2) 푡 7 7 7 7 푒 − 푒 + 퐺38 푒 ≤ 퐺38 (푡) ≤ (푚1) (푆1) −(푝36) −(푆2) (푎 ) 7 퐺0 7 ′ 7 ′ 7 38 36 (푆1) 푡 −(푎38) 푡 0 −(푎38) 푡 7 7 ′ 7 [푒 − 푒 ] + 퐺38 푒 ) (푚2) (푆1) −(푎38) 7 7 7 0 (푅1) 푡 0 (푅1) +(푟36) 푡 488 푇36 푒 ≤ 푇36 (푡) ≤ 푇36 푒

1 7 1 7 7 0 (푅1) 푡 0 (푅1) +(푟36 ) 푡 489 7 푇36 푒 ≤ 푇36 (푡) ≤ 7 푇36 푒 (휇1) (휇2)

(푏 ) 7 푇0 7 ′ 7 ′ 7 38 36 (푅1) 푡 −(푏38 ) 푡 0 −(푏38) 푡 490 7 7 ′ 7 푒 − 푒 + 푇38 푒 ≤ 푇38 (푡) ≤ (휇1) (푅1) −(푏38)

(푎 ) 7 푇0 7 7 7 7 38 36 (푅1) +(푟36) 푡 −(푅2) 푡 0 −(푅2) 푡 7 7 7 7 푒 − 푒 + 푇38 푒 (휇2) (푅1) +(푟36 ) +(푅2)

7 7 7 7 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 491 7 7 7 ′ 7 Where (푆1) = (푎36 ) (푚2) − (푎36 ) 7 7 7 (푆2) = (푎38 ) − (푝38 ) 7 7 7 ′ 7 (푅1) = (푏36) (휇2) − (푏36 ) 7 ′ 7 7 (푅2) = (푏38) − (푟38 )

From GLOBAL EQUATIONS we obtain 492

푑휈 7 = (푎 ) 7 − (푎′ ) 7 − (푎′ ) 7 + (푎′′ ) 7 푇 , 푡 − 푑푡 36 36 37 36 37 ′′ 7 7 7 7 (푎37 ) 푇37 , 푡 휈 − (푎37 ) 휈

퐺 Definition of 휈 7 :- 휈 7 = 36 퐺37

It follows 2 푑휈 7 − (푎 ) 7 휈 7 + (휎 ) 7 휈 7 − (푎 ) 7 ≤ ≤ 37 2 36 푑푡 7 7 2 7 7 7 − (푎37 ) 휈 + (휎1) 휈 − (푎36 ) From which one obtains

7 7 Definition of (휈1 ) , (휈0) :-

0 7 퐺36 7 7 (a) For 0 < (휈0 ) = 0 < (휈1 ) < (휈1 ) 퐺37

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 +(퐶) 7 (휈 ) 7 푒 37 1 0 (휈 ) 7 −(휈 ) 7 휈 7 (푡) ≥ 1 2 , (퐶) 7 = 1 0 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 −(휈 ) 7 1+(퐶) 7 푒 37 1 0 0 2

7 7 7 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1)

In the same manner , we get 493

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 +(퐶 ) 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 −(휈 ) 7 휈 7 (푡) ≤ 1 2 , (퐶 ) 7 = 1 0 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 −(휈 ) 7 1+(퐶 ) 7 푒 37 1 2 0 2

7 7 7 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈1 )

0 7 7 퐺36 7 494 (b) If 0 < (휈1 ) < (휈0 ) = 0 < (휈1 ) we find like in the previous case, 퐺37 www.ijmer.com 2084 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 ≤ 1 2 ≤ 휈 7 푡 ≤ 1 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 1+ 퐶 7 푒 37 1 2

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 1 2 ≤ (휈 ) 7 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 1 1+ 퐶 7 푒 37 1 2 0 7 7 7 퐺36 495 (c) If 0 < (휈1 ) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain 퐺37

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 ≤ 휈 7 푡 ≤ 1 2 ≤ (휈 ) 7 1 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 0 1+ 퐶 7 푒 37 1 2

And so with the notation of the first part of condition (c) , we have Definition of 휈 7 푡 :-

7 7 7 7 퐺36 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺37 푡 In a completely analogous way, we obtain Definition of 푢 7 푡 :-

7 7 7 7 푇36 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇37 푡

Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : ′′ 7 ′′ 7 7 7 7 7 7 7 If (푎36 ) = (푎37 ) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1) = (휈1 ) if in addition (휈0) = (휈1) 7 7 7 7 then 휈 푡 = (휈0 ) and as a consequence 퐺36 (푡) = (휈0) 퐺37 (푡) this also defines (휈0 ) for the special case . ′′ 7 ′′ 7 7 7 Analogously if (푏36) = (푏37) , 푡푕푒푛 (휏1) = (휏2) and then 7 7 7 7 7 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇36 (푡) = (푢0) 푇37 (푡) This is an important consequence 7 7 7 of the relation between (휈1) and (휈1 ) , and definition of (푢0) .

We can prove the following 496 ′′ 7 ′′ 7 496A If (푎푖 ) 푎푛푑 (푏푖 ) are independent on 푡 , and the conditions ′ 7 ′ 7 7 7 496B (푎36 ) (푎37 ) − 푎36 푎37 < 0 (푎′ ) 7 (푎′ ) 7 − 푎 7 푎 7 + 푎 7 푝 7 + (푎′ ) 7 푝 7 + 푝 7 푝 7 > 0 496C 36 37 36 37 36 36 37 37 36 37 497C

497D ′ 7 ′ 7 7 7 (푏36 ) (푏37 ) − 푏36 푏37 > 0 , 497E 497F ′ 7 ′ 7 7 7 ′ 7 7 ′ 7 7 7 7 (푏36 ) (푏37 ) − 푏36 푏37 − (푏36) 푟37 − (푏37 ) 푟37 + 푟36 푟37 < 0 497G

7 7 푤푖푡푕 푝36 , 푟37 as defined are satisfied , then the system WITH THE SATISFACTION OF THE FOLLOWING PROPERTIES HAS A SOLUTION AS DERIVED BELOW.

Particular case : 498

′′ 2 ′′ 2 2 2 2 2 2 2 If (푎16 ) = (푎17 ) , 푡푕푒푛 (σ1) = (σ2) and in this case (휈1) = (휈1 ) if in addition (휈0 ) = (휈1) 2 2 2 then 휈 푡 = (휈0 ) and as a consequence 퐺16 (푡) = (휈0 ) 퐺17 (푡)

′′ 2 ′′ 2 2 2 Analogously if (푏16 ) = (푏17 ) , 푡푕푒푛 (τ1) = (τ2) and then

2 2 2 2 2 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇16 (푡) = (푢0) 푇17 (푡) This is an important consequence 2 2 of the relation between (휈1 ) and (휈1 ) 499

From GLOBAL EQUATIONS we obtain 500

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푑휈 3 = (푎 ) 3 − (푎′ ) 3 − (푎′ ) 3 + (푎′′ ) 3 푇 , 푡 − (푎′′ ) 3 푇 , 푡 휈 3 − (푎 ) 3 휈 3 푑푡 20 20 21 20 21 21 21 21

퐺20 501 Definition of 휈 3 :- 휈 3 = 퐺21 It follows 2 푑휈 3 2 − (푎 ) 3 휈 3 + (휎 ) 3 휈 3 − (푎 ) 3 ≤ ≤ − (푎 ) 3 휈 3 + (휎 ) 3 휈 3 − (푎 ) 3 21 2 20 푑푡 21 1 20 502 From which one obtains 0 3 퐺20 3 3 (a) For 0 < (휈0) = 0 < (휈1 ) < (휈1 ) 퐺21

− 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 +(퐶) 3 (휈 ) 3 푒 21 1 0 (휈 ) 3 −(휈 ) 3 휈 3 (푡) ≥ 1 2 , (퐶) 3 = 1 0 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 −(휈 ) 3 1+(퐶) 3 푒 21 1 0 0 2 3 3 3 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1) In the same manner , we get 503 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 +(퐶 ) 3 (휈 ) 3 푒 21 1 2 (휈 ) 3 −(휈 ) 3 휈 3 (푡) ≤ 1 2 , (퐶 ) 3 = 1 0 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 −(휈 ) 3 1+(퐶 ) 3 푒 21 1 2 0 2 3 Definition of (휈1 ) :- 3 3 3 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈1 ) 0 3 3 퐺20 3 504 (b) If 0 < (휈1 ) < (휈0) = 0 < (휈1 ) we find like in the previous case, 퐺21

− 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 + 퐶 3 (휈 ) 3 푒 21 1 2 (휈 ) 3 ≤ 1 2 ≤ 휈 3 푡 ≤ 1 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 1+ 퐶 3 푒 21 1 2 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 + 퐶 3 (휈 ) 3 푒 21 1 2 1 2 ≤ (휈 ) 3 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 1 1+ 퐶 3 푒 21 1 2

0 3 3 3 퐺20 505 (c) If 0 < (휈1 ) ≤ (휈1 ) ≤ (휈0 ) = 0 , we obtain 퐺21 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 (휈 ) 3 + 퐶 3 (휈 ) 3 푒 21 1 2 (휈 ) 3 ≤ 휈 3 푡 ≤ 1 2 ≤ (휈 ) 3 1 − 푎 3 (휈 ) 3 −(휈 ) 3 푡 0 1+ 퐶 3 푒 21 1 2 And so with the notation of the first part of condition (c) , we have Definition of 휈 3 푡 :- 3 3 3 3 퐺20 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺21 푡 In a completely analogous way, we obtain Definition of 푢 3 푡 :- 3 3 3 3 푇20 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇21 푡 Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : ′′ 3 ′′ 3 3 3 3 3 3 3 If (푎20 ) = (푎21 ) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1 ) = (휈1 ) if in addition (휈0 ) = (휈1 ) 3 3 3 then 휈 푡 = (휈0 ) and as a consequence 퐺20 (푡) = (휈0) 퐺21 (푡) ′′ 3 ′′ 3 3 3 Analogously if (푏20 ) = (푏21 ) ,푡푕푒푛 (휏1) = (휏2) and then 3 3 3 3 3 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇20 (푡) = (푢0) 푇21 (푡) This is an important consequence 3 3 of the relation between (휈1 ) and (휈1 ) 506 : From GLOBAL EQUATIONS we obtain 507

푑휈 4 = (푎 ) 4 − (푎′ ) 4 − (푎′ ) 4 + (푎′′ ) 4 푇 , 푡 − (푎′′ ) 4 푇 , 푡 휈 4 − (푎 ) 4 휈 4 푑푡 24 24 25 24 25 25 25 25

퐺 Definition of 휈 4 :- 휈 4 = 24 508 퐺25

It follows 2 푑휈 4 2 − (푎 ) 4 휈 4 + (휎 ) 4 휈 4 − (푎 ) 4 ≤ ≤ − (푎 ) 4 휈 4 + (휎 ) 4 휈 4 − (푎 ) 4 25 2 24 푑푡 25 4 24 www.ijmer.com 2086 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 From which one obtains

4 4 Definition of (휈1 ) , (휈0) :-

0 4 퐺24 4 4 (d) For 0 < (휈0 ) = 0 < (휈1 ) < (휈1 ) 퐺25

− 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 + 퐶 4 (휈 ) 4 푒 25 1 0 (휈 ) 4 −(휈 ) 4 휈 4 푡 ≥ 1 2 , 퐶 4 = 1 0 − 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 −(휈 ) 4 4+ 퐶 4 푒 25 1 0 0 2

4 4 4 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1)

In the same manner , we get 509

− 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 + 퐶 4 (휈 ) 4 푒 25 1 2 (휈 ) 4 −(휈 ) 4 휈 4 푡 ≤ 1 2 , (퐶 ) 4 = 1 0 − 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 −(휈 ) 4 4+ 퐶 4 푒 25 1 2 0 2

4 4 4 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈1 )

0 4 4 퐺24 4 510 (e) If 0 < (휈1 ) < (휈0 ) = 0 < (휈1 ) we find like in the previous case, 퐺25

− 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 + 퐶 4 (휈 ) 4 푒 25 1 2 (휈 ) 4 ≤ 1 2 ≤ 휈 4 푡 ≤ 1 − 푎 4 (휈 ) 4 −(휈 ) 4 푡 1+ 퐶 4 푒 25 1 2

− 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 + 퐶 4 (휈 ) 4 푒 25 1 2 1 2 ≤ (휈 ) 4 − 푎 4 (휈 ) 4 −(휈 ) 4 푡 1 1+ 퐶 4 푒 25 1 2 511 0 4 4 4 퐺24 512 (f) If 0 < (휈1 ) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain 퐺25

− 푎 4 (휈 ) 4 −(휈 ) 4 푡 (휈 ) 4 + 퐶 4 (휈 ) 4 푒 25 1 2 (휈 ) 4 ≤ 휈 4 푡 ≤ 1 2 ≤ (휈 ) 4 1 − 푎 4 (휈 ) 4 −(휈 ) 4 푡 0 1+ 퐶 4 푒 25 1 2

And so with the notation of the first part of condition (c) , we have

Definition of 휈 4 푡 :-

4 4 4 4 퐺24 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺25 푡

In a completely analogous way, we obtain

Definition of 푢 4 푡 :-

4 4 4 4 푇24 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇25 푡

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

Particular case :

If (푎′′ ) 4 = (푎′′ ) 4 , 푡푕푒푛 (휎 ) 4 = (휎 ) 4 and in this case (휈 ) 4 = (휈 ) 4 if in addition (휈 ) 4 = (휈 ) 4 24 25 1 2 1 1 0 1 then 휈 4 푡 = (휈 ) 4 and as a consequence 퐺 (푡) = (휈 ) 4 퐺 (푡) this also defines (휈 ) 4 for the special 0 24 0 25 0 513 case .

′′ 4 ′′ 4 4 4 Analogously if (푏24) = (푏25) , 푡푕푒푛 (휏1) = (휏2) and then 4 4 4 4 4 (푢1) = (푢 4) if in addition (푢0) = (푢1) then 푇24 (푡) = (푢0) 푇25 (푡) This is an important consequence 4 4 4 of the relation between (휈1) and (휈1 ) , and definition of (푢0) . 514

www.ijmer.com 2087 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 From GLOBAL EQUATIONS we obtain 515

푑휈 5 = (푎 ) 5 − (푎′ ) 5 − (푎′ ) 5 + (푎′′ ) 5 푇 , 푡 − (푎′′ ) 5 푇 , 푡 휈 5 − (푎 ) 5 휈 5 푑푡 28 28 29 28 29 29 29 29

퐺 Definition of 휈 5 :- 휈 5 = 28 퐺29

It follows 2 푑휈 5 2 − (푎 ) 5 휈 5 + (휎 ) 5 휈 5 − (푎 ) 5 ≤ ≤ − (푎 ) 5 휈 5 + (휎 ) 5 휈 5 − (푎 ) 5 29 2 28 푑푡 29 1 28

From which one obtains

5 5 Definition of (휈1 ) , (휈0) :-

0 5 퐺28 5 5 (g) For 0 < (휈0 ) = 0 < (휈1 ) < (휈1 ) 퐺29

− 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 +(퐶) 5 (휈 ) 5 푒 29 1 0 (휈 ) 5 −(휈 ) 5 휈 5 (푡) ≥ 1 2 , (퐶) 5 = 1 0 − 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 −(휈 ) 5 5+(퐶) 5 푒 29 1 0 0 2

5 5 5 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1)

In the same manner , we get 516

− 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 +(퐶 ) 5 (휈 ) 5 푒 29 1 2 (휈 ) 5 −(휈 ) 5 휈 5 (푡) ≤ 1 2 , (퐶 ) 5 = 1 0 − 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 −(휈 ) 5 5+(퐶 ) 5 푒 29 1 2 0 2

5 5 5 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈 5)

0 5 5 퐺28 5 517 (h) If 0 < (휈1 ) < (휈0 ) = 0 < (휈1 ) we find like in the previous case, 퐺29

− 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 + 퐶 5 (휈 ) 5 푒 29 1 2 (휈 ) 5 ≤ 1 2 ≤ 휈 5 푡 ≤ 1 − 푎 5 (휈 ) 5 −(휈 ) 5 푡 1+ 퐶 5 푒 29 1 2

− 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 + 퐶 5 (휈 ) 5 푒 29 1 2 1 2 ≤ (휈 ) 5 − 푎 5 (휈 ) 5 −(휈 ) 5 푡 1 1+ 퐶 5 푒 29 1 2 0 5 5 5 퐺28 518 (i) If 0 < (휈1 ) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain 퐺29

− 푎 5 (휈 ) 5 −(휈 ) 5 푡 (휈 ) 5 + 퐶 5 (휈 ) 5 푒 29 1 2 (휈 ) 5 ≤ 휈 5 푡 ≤ 1 2 ≤ (휈 ) 5 1 5 5 5 0 5 − 푎29 (휈 1) −(휈 2) 푡 1+ 퐶 푒 519

And so with the notation of the first part of condition (c) , we have Definition of 휈 5 푡 :-

5 5 5 5 퐺28 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺29 푡 In a completely analogous way, we obtain Definition of 푢 5 푡 :-

5 5 5 5 푇28 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇29 푡

www.ijmer.com 2088 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem.

Particular case :

′′ 5 ′′ 5 5 5 5 5 5 5 If (푎28 ) = (푎29) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1) = (휈1 ) if in addition (휈0 ) = (휈5) 5 5 5 5 then 휈 푡 = (휈0 ) and as a consequence 퐺28 (푡) = (휈0) 퐺29(푡) this also defines (휈0) for the special case .

′′ 5 ′′ 5 5 5 Analogously if (푏28) = (푏29) ,푡푕푒푛 (휏1) = (휏2) and then 5 5 5 5 5 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇28 (푡) = (푢0) 푇29 (푡) This is an important consequence 5 5 5 of the relation between (휈1) and (휈1 ) , and definition of (푢0) .

520 we obtain 521

푑휈 6 = (푎 ) 6 − (푎′ ) 6 − (푎′ ) 6 + (푎′′ ) 6 푇 , 푡 − (푎′′ ) 6 푇 , 푡 휈 6 − (푎 ) 6 휈 6 푑푡 32 32 33 32 33 33 33 33

퐺 Definition of 휈 6 :- 휈 6 = 32 퐺33

It follows 2 푑휈 6 2 − (푎 ) 6 휈 6 + (휎 ) 6 휈 6 − (푎 ) 6 ≤ ≤ − (푎 ) 6 휈 6 + (휎 ) 6 휈 6 − (푎 ) 6 33 2 32 푑푡 33 1 32

From which one obtains

6 6 Definition of (휈1 ) , (휈0) :-

0 6 퐺32 6 6 (j) For 0 < (휈0 ) = 0 < (휈1 ) < (휈1 ) 퐺33

− 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 +(퐶) 6 (휈 ) 6 푒 33 1 0 (휈 ) 6 −(휈 ) 6 휈 6 (푡) ≥ 1 2 , (퐶) 6 = 1 0 − 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 −(휈 ) 6 1+(퐶) 6 푒 33 1 0 0 2

6 6 6 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1)

In the same manner , we get 522

− 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 +(퐶 ) 6 (휈 ) 6 푒 33 1 2 (휈 ) 6 −(휈 ) 6 523 휈 6 (푡) ≤ 1 2 , (퐶 ) 6 = 1 0 − 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 −(휈 ) 6 1+(퐶 ) 6 푒 33 1 2 0 2

6 6 6 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈1 )

0 6 6 퐺32 6 524 (k) If 0 < (휈1 ) < (휈0 ) = 0 < (휈1 ) we find like in the previous case, 퐺33

− 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 + 퐶 6 (휈 ) 6 푒 33 1 2 (휈 ) 6 ≤ 1 2 ≤ 휈 6 푡 ≤ 1 − 푎 6 (휈 ) 6 −(휈 ) 6 푡 1+ 퐶 6 푒 33 1 2

− 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 + 퐶 6 (휈 ) 6 푒 33 1 2 1 2 ≤ (휈 ) 6 − 푎 6 (휈 ) 6 −(휈 ) 6 푡 1 1+ 퐶 6 푒 33 1 2 0 6 6 6 퐺32 525 (l) If 0 < (휈1 ) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain 퐺33

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− 푎 6 (휈 ) 6 −(휈 ) 6 푡 (휈 ) 6 + 퐶 6 (휈 ) 6 푒 33 1 2 (휈 ) 6 ≤ 휈 6 푡 ≤ 1 2 ≤ (휈 ) 6 1 − 푎 6 (휈 ) 6 −(휈 ) 6 푡 0 1+ 퐶 6 푒 33 1 2

And so with the notation of the first part of condition (c) , we have Definition of 휈 6 푡 :-

6 6 6 6 퐺32 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺33 푡 In a completely analogous way, we obtain Definition of 푢 6 푡 :-

6 6 6 6 푇32 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇33 푡

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

Particular case :

′′ 6 ′′ 6 6 6 6 6 6 6 If (푎32 ) = (푎33 ) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1) = (휈1 ) if in addition (휈0) = (휈1) 6 6 6 6 then 휈 푡 = (휈0 ) and as a consequence 퐺32 (푡) = (휈0) 퐺33 (푡) this also defines (휈0 ) for the special case . ′′ 6 ′′ 6 6 6 Analogously if (푏32) = (푏33) , 푡푕푒푛 (휏1) = (휏2) and then 6 6 6 6 6 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇32 (푡) = (푢0) 푇33 (푡) This is an important consequence 6 6 6 of the relation between (휈1) and (휈1 ) , and definition of (푢0) . 526 Behavior of the solutions 527

If we denote and define

7 7 7 7 Definition of (휎1) , (휎2) , (휏1) , (휏2) :

7 7 7 7 (p) (휎1) , (휎2) ,(휏1) , (휏2) four constants satisfying

7 ′ 7 ′ 7 ′′ 7 ′′ 7 7 −(휎2) ≤ −(푎36 ) + (푎37 ) − (푎36 ) 푇37 , 푡 + (푎37 ) 푇37 , 푡 ≤ −(휎1)

7 ′ 7 ′ 7 ′′ 7 ′′ 7 7 −(휏2) ≤ −(푏36 ) + (푏37) − (푏36) 퐺39 , 푡 − (푏37) 퐺39 , 푡 ≤ −(휏1)

7 7 7 7 7 7 Definition of (휈1 ) , (휈2) , (푢1) , (푢2) , 휈 , 푢 : 528 7 7 7 7 (q) By (휈1) > 0 , (휈2 ) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations 7 7 2 7 7 7 529 (푎37 ) 휈 + (휎1) 휈 − (푎36 ) = 0 7 7 2 7 7 7 and (푏37 ) 푢 + (휏1) 푢 − (푏36 ) = 0 and 7 7 7 7 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 530. 7 7 7 7 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the

7 7 2 7 7 7 roots of the equations (푎37 ) 휈 + (휎2) 휈 − (푎36 ) = 0

7 7 2 7 7 7 and (푏37 ) 푢 + (휏2) 푢 − (푏36 ) = 0

7 7 7 7 7 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0 ) :-

7 7 7 7 (r) If we define (푚1) , (푚2) , (휇1) , (휇2) by

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7 7 7 7 7 7 (푚2) = (휈0 ) , (푚1) = (휈1) , 풊풇 (휈0) < (휈1 )

7 7 7 7 7 7 7 (푚2) = (휈1) , (푚1) = (휈1 ) , 풊풇 (휈1 ) < (휈0 ) < (휈1 ) ,

0 7 퐺36 and (휈0 ) = 0 퐺37

7 7 7 7 7 7 ( 푚2) = (휈1) , (푚1) = (휈0 ) , 풊풇 (휈1 ) < (휈0)

and analogously 531 7 7 7 7 7 7 (휇2) = (푢0) , (휇1) = (푢1) , 풊풇 (푢0) < (푢1) 7 7 7 7 7 7 7 (휇2) = (푢1) , (휇1) = (푢 1) , 풊풇 (푢1) < (푢0) < (푢 1) , 0 7 푇36 and (푢0) = 0 푇37 7 7 7 7 7 7 7 7 ( 휇2) = (푢1) , (휇1) = (푢0) , 풊풇 (푢 1) < (푢0) where (푢1) , (푢 1) are defined by 59 and 67 respectively

Then the solution of GLOBAL EQUATIONS satisfies the inequalities 532 7 7 7 0 (푆1) −(푝36 ) 푡 0 (푆1) 푡 퐺36 푒 ≤ 퐺36 (푡) ≤ 퐺36 푒 7 where (푝푖 ) is defined

1 7 7 1 7 0 (푆1) −(푝36) 푡 0 (푆1) 푡 533 7 퐺36 푒 ≤ 퐺37 (푡) ≤ 7 퐺36 푒 (푚7) (푚2)

( 534 (푎 ) 7 퐺0 7 7 7 7 38 36 (푆1) −(푝36) 푡 −(푆2) 푡 0 −(푆2) 푡 7 7 7 7 푒 − 푒 + 퐺38 푒 ≤ 퐺38 (푡) ≤ (푚1) (푆1) −(푝36) −(푆2) (푎 ) 7 퐺0 7 ′ 7 ′ 7 38 36 (푆1) 푡 −(푎38) 푡 0 −(푎38) 푡 7 7 ′ 7 [푒 − 푒 ] + 퐺38 푒 ) (푚2) (푆1) −(푎38)

7 7 7 0 (푅1) 푡 0 (푅1) +(푟36) 푡 535 푇36 푒 ≤ 푇36 (푡) ≤ 푇36 푒

1 7 1 7 7 0 (푅1) 푡 0 (푅1) +(푟36 ) 푡 536 7 푇36 푒 ≤ 푇36 (푡) ≤ 7 푇36 푒 (휇1) (휇2)

(푏 ) 7 푇0 7 ′ 7 ′ 7 38 36 (푅1) 푡 −(푏38 ) 푡 0 −(푏38) 푡 537 7 7 ′ 7 푒 − 푒 + 푇38 푒 ≤ 푇38 (푡) ≤ (휇1) (푅1) −(푏38) (푎 ) 7 푇0 7 7 7 7 38 36 (푅1) +(푟36) 푡 −(푅2) 푡 0 −(푅2) 푡 7 7 7 7 푒 − 푒 + 푇38 푒 (휇2) (푅1) +(푟36 ) +(푅2)

7 7 7 7 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 538 Where (푆 ) 7 = (푎 ) 7 (푚 ) 7 − (푎′ ) 7 1 36 2 36

7 7 7 (푆2) = (푎38 ) − (푝38 )

7 7 7 ′ 7 (푅1) = (푏36) (휇2) − (푏36 )

7 ′ 7 7 539 (푅2) = (푏38) − (푟38 )

From CONCATENATED GLOBAL EQUATIONS we obtain 540

푑휈 7 = (푎 ) 7 − (푎′ ) 7 − (푎′ ) 7 + (푎′′ ) 7 푇 , 푡 − 푑푡 36 36 37 36 37

′′ 7 7 7 7 (푎37 ) 푇37 , 푡 휈 − (푎37 ) 휈

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퐺 Definition of 휈 7 :- 휈 7 = 36 퐺37

It follows

2 푑휈 7 − (푎 ) 7 휈 7 + (휎 ) 7 휈 7 − (푎 ) 7 ≤ ≤ 37 2 36 푑푡

7 7 2 7 7 7 − (푎37 ) 휈 + (휎1) 휈 − (푎36 )

From which one obtains

7 7 Definition of (휈1 ) , (휈0 ) :-

0 7 퐺36 7 7 (m) For 0 < (휈0 ) = 0 < (휈1) < (휈1 ) 퐺37

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 +(퐶) 7 (휈 ) 7 푒 37 1 0 (휈 ) 7 −(휈 ) 7 휈 7 (푡) ≥ 1 2 , (퐶) 7 = 1 0 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 −(휈 ) 7 1+(퐶) 7 푒 37 1 0 0 2

7 7 7 it follows (휈0 ) ≤ 휈 (푡) ≤ (휈1)

In the same manner , we get 541 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 +(퐶 ) 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 −(휈 ) 7 휈 7 (푡) ≤ 1 2 , (퐶 ) 7 = 1 0 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 −(휈 ) 7 1+(퐶 ) 7 푒 37 1 2 0 2 7 7 7 From which we deduce (휈0 ) ≤ 휈 (푡) ≤ (휈1 )

0 7 7 퐺36 7 542 (n) If 0 < (휈1 ) < (휈0) = 0 < (휈1 ) we find like in the previous case, 퐺37

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 ≤ 1 2 ≤ 휈 7 푡 ≤ 1 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 1+ 퐶 7 푒 37 1 2

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 1 2 ≤ (휈 ) 7 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 1 1+ 퐶 7 푒 37 1 2

0 7 7 7 퐺36 543 (o) If 0 < (휈1) ≤ (휈1 ) ≤ (휈0 ) = 0 , we obtain 퐺37

− 푎 7 (휈 ) 7 −(휈 ) 7 푡 (휈 ) 7 + 퐶 7 (휈 ) 7 푒 37 1 2 (휈 ) 7 ≤ 휈 7 푡 ≤ 1 2 ≤ (휈 ) 7 1 − 푎 7 (휈 ) 7 −(휈 ) 7 푡 0 1+ 퐶 7 푒 37 1 2

And so with the notation of the first part of condition (c) , we have Definition of 휈 7 푡 :-

7 7 7 7 퐺36 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺37 푡 In a completely analogous way, we obtain Definition of 푢 7 푡 :-

7 7 7 7 푇36 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇37 푡

Now, using this result and replacing it in CONCATENATED GLOBAL EQUATIONS we get easily the result www.ijmer.com 2092 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 stated in the theorem. Particular case : ′′ 7 ′′ 7 7 7 7 7 7 7 If (푎36 ) = (푎37 ) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1 ) = (휈1 ) if in addition (휈0 ) = (휈1 ) 7 7 7 7 then 휈 푡 = (휈0 ) and as a consequence 퐺36 (푡) = (휈0) 퐺37 (푡) this also defines (휈0) for the special case . ′′ 7 ′′ 7 7 7 Analogously if (푏36 ) = (푏37 ) ,푡푕푒푛 (휏1) = (휏2) and then 7 7 7 7 7 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇36 (푡) = (푢0) 푇37 (푡) This is an important consequence 7 7 7 of the relation between (휈1 ) and (휈1 ) , and definition of (푢0) .

1 ′ 1 ′′ 1 푏14 푇13 − [(푏14) − (푏14 ) 퐺 ]푇14 = 0 544 1 ′ 1 ′′ 1 푏15 푇14 − [(푏15) − (푏15 ) 퐺 ]푇15 = 0 545 has a unique positive solution , which is an equilibrium solution for the system 546 2 ′ 2 ′′ 2 푎16 퐺17 − (푎16 ) + (푎16 ) 푇17 퐺16 = 0 547 2 ′ 2 ′′ 2 푎17 퐺16 − (푎17 ) + (푎17 ) 푇17 퐺17 = 0 548 2 ′ 2 ′′ 2 푎18 퐺17 − (푎18 ) + (푎18 ) 푇17 퐺18 = 0 549 2 ′ 2 ′′ 2 푏16 푇17 − [(푏16) − (푏16 ) 퐺19 ]푇16 = 0 550 2 ′ 2 ′′ 2 푏17 푇16 − [(푏17) − (푏17 ) 퐺19 ]푇17 = 0 551 2 ′ 2 ′′ 2 푏18 푇17 − [(푏18) − (푏18 ) 퐺19 ]푇18 = 0 552 has a unique positive solution , which is an equilibrium solution for 553 3 ′ 3 ′′ 3 푎20 퐺21 − (푎20 ) + (푎20 ) 푇21 퐺20 = 0 554 3 ′ 3 ′′ 3 푎21 퐺20 − (푎21 ) + (푎21 ) 푇21 퐺21 = 0 555 3 ′ 3 ′′ 3 푎22 퐺21 − (푎22 ) + (푎22 ) 푇21 퐺22 = 0 556 3 ′ 3 ′′ 3 푏20 푇21 − [(푏20 ) − (푏20 ) 퐺23 ]푇20 = 0 557 3 ′ 3 ′′ 3 푏21 푇20 − [(푏21 ) − (푏21 ) 퐺23 ]푇21 = 0 558 3 ′ 3 ′′ 3 푏22 푇21 − [(푏22 ) − (푏22 ) 퐺23 ]푇22 = 0 559 has a unique positive solution , which is an equilibrium solution 560 4 ′ 4 ′′ 4 푎24 퐺25 − (푎24 ) + (푎24 ) 푇25 퐺24 = 0 561

4 ′ 4 ′′ 4 푎25 퐺24 − (푎25 ) + (푎25 ) 푇25 퐺25 = 0 563 4 ′ 4 ′′ 4 푎26 퐺25 − (푎26 ) + (푎26 ) 푇25 퐺26 = 0 564

4 ′ 4 ′′ 4 푏24 푇25 − [(푏24 ) − (푏24 ) 퐺27 ]푇24 = 0 565

4 ′ 4 ′′ 4 푏25 푇24 − [(푏25 ) − (푏25 ) 퐺27 ]푇25 = 0 566

4 ′ 4 ′′ 4 푏26 푇25 − [(푏26 ) − (푏26 ) 퐺27 ]푇26 = 0 567 has a unique positive solution , which is an equilibrium solution for the system 568

5 ′ 5 ′′ 5 푎28 퐺29 − (푎28 ) + (푎28 ) 푇29 퐺28 = 0 569

5 ′ 5 ′′ 5 푎29 퐺28 − (푎29) + (푎29) 푇29 퐺29 = 0 570

5 ′ 5 ′′ 5 푎30 퐺29 − (푎30 ) + (푎30 ) 푇29 퐺30 = 0 571

5 ′ 5 ′′ 5 푏28 푇29 − [(푏28) − (푏28 ) 퐺31 ]푇28 = 0 572

5 ′ 5 ′′ 5 푏29 푇28 − [(푏29) − (푏29) 퐺31 ]푇29 = 0 573

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5 ′ 5 ′′ 5 푏30 푇29 − [(푏30) − (푏30 ) 퐺31 ]푇30 = 0 574 has a unique positive solution , which is an equilibrium solution for the system 575

6 ′ 6 ′′ 6 푎32 퐺33 − (푎32 ) + (푎32 ) 푇33 퐺32 = 0 576

6 ′ 6 ′′ 6 푎33 퐺32 − (푎33 ) + (푎33 ) 푇33 퐺33 = 0 577

6 ′ 6 ′′ 6 푎34 퐺33 − (푎34 ) + (푎34 ) 푇33 퐺34 = 0 578

6 ′ 6 ′′ 6 푏32 푇33 − [(푏32 ) − (푏32 ) 퐺35 ]푇32 = 0 579

6 ′ 6 ′′ 6 푏33 푇32 − [(푏33 ) − (푏33 ) 퐺35 ]푇33 = 0 580

6 ′ 6 ′′ 6 푏34 푇33 − [(푏34 ) − (푏34 ) 퐺35 ]푇34 = 0 584 has a unique positive solution , which is an equilibrium solution for the system 582

7 ′ 7 ′′ 7 푎36 퐺37 − (푎36 ) + (푎36 ) 푇37 퐺36 = 0 583

7 ′ 7 ′′ 7 푎37 퐺36 − (푎37 ) + (푎37 ) 푇37 퐺37 = 0 584

7 ′ 7 ′′ 7 푎38 퐺37 − (푎38 ) + (푎38 ) 푇37 퐺38 = 0 585 586

7 ′ 7 ′′ 7 푏36 푇37 − [(푏36 ) − (푏36 ) 퐺39 ]푇36 = 0 587

7 ′ 7 ′′ 7 푏37 푇36 − [(푏37 ) − (푏37 ) 퐺39 ]푇37 = 0 588

7 ′ 7 ′′ 7 푏38 푇37 − [(푏38 ) − (푏38 ) 퐺39 ]푇38 = 0 589 has a unique positive solution , which is an equilibrium solution for the system 560

(a) Indeed the first two equations have a nontrivial solution 퐺36 , 퐺37 if

퐹 푇39 = ′ 7 ′ 7 7 7 ′ 7 ′′ 7 ′ 7 ′′ 7 (푎36 ) (푎37 ) − 푎36 푎37 + (푎36 ) (푎37 ) 푇37 + (푎37 ) (푎36 ) 푇37 + ′′ 7 ′′ 7 (푎36 ) 푇37 (푎37 ) 푇37 = 0

∗ Definition and uniqueness of T37 :- 561 ′′ 7 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇37 being increasing, it follows that there ∗ ∗ exists a unique 푇37 for which 푓 푇37 = 0. With this value , we obtain from the three first equations

7 7 푎36 퐺37 푎38 퐺37 퐺36 = ′ 7 ′′ 7 ∗ , 퐺38 = ′ 7 ′′ 7 ∗ (푎36 ) +(푎36) 푇37 (푎38) +(푎38) 푇37

(e) By the same argument, the equations( SOLUTIONAL) admit solutions 퐺36 , 퐺37 if

′ 7 ′ 7 7 7 휑 퐺39 = (푏36 ) (푏37 ) − 푏36 푏37 −

′ 7 ′′ 7 ′ 7 ′′ 7 ′′ 7 ′′ 7 (푏36 ) (푏37 ) 퐺39 + (푏37 ) (푏36 ) 퐺39 +(푏36 ) 퐺39 (푏37) 퐺39 = 0

www.ijmer.com 2094 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

Where in 퐺39 퐺36 , 퐺37 , 퐺38 , 퐺36 , 퐺38 must be replaced by their values from 96. It is easy to see that φ is a 562 decreasing function in 퐺37 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists ∗ ∗ a unique 퐺37 such that 휑 퐺 = 0

Finally we obtain the unique solution OF THE SYSTEM

∗ ∗ ∗ ∗ 퐺37 given by 휑 퐺39 = 0 , 푇37 given by 푓 푇37 = 0 and

7 ∗ 7 ∗ ∗ 푎36 퐺37 ∗ 푎38 퐺37 퐺36 = ′ 7 ′′ 7 ∗ , 퐺38 = ′ 7 ′′ 7 ∗ (푎36) +(푎36) 푇37 (푎38) +(푎38) 푇37

7 ∗ 7 ∗ ∗ 푏36 푇37 ∗ 푏38 푇37 563 푇36 = ′ 7 ′′ 7 ∗ , 푇38 = ′ 7 ′′ 7 ∗ (푏36) −(푏36) 퐺39 (푏38 ) −(푏38) 퐺39

∗ Definition and uniqueness of T21 :- 564 ′′ 1 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇21 being increasing, it follows that there ∗ ∗ exists a unique 푇21 for which 푓 푇21 = 0. With this value , we obtain from the three first equations 565 3 3 푎20 퐺21 푎22 퐺21 퐺20 = ′ 3 ′′ 3 ∗ , 퐺22 = ′ 3 ′′ 3 ∗ (푎20 ) +(푎20) 푇21 (푎22 ) +(푎22) 푇21

∗ Definition and uniqueness of T25 :- 566

′′ 4 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇25 being increasing, it follows that there ∗ ∗ exists a unique 푇25 for which 푓 푇25 = 0. With this value , we obtain from the three first equations

4 4 푎24 퐺25 푎26 퐺25 퐺24 = ′ 4 ′′ 4 ∗ , 퐺26 = ′ 4 ′′ 4 ∗ (푎24 ) +(푎24) 푇25 (푎26) +(푎26) 푇25

∗ Definition and uniqueness of T29 :- 567

′′ 5 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇29 being increasing, it follows that there ∗ ∗ exists a unique 푇29 for which 푓 푇29 = 0. With this value , we obtain from the three first equations

5 5 푎28 퐺29 푎30 퐺29 퐺28 = ′ 5 ′′ 5 ∗ , 퐺30 = ′ 5 ′′ 5 ∗ (푎28 ) +(푎28) 푇29 (푎30) +(푎30) 푇29

∗ Definition and uniqueness of T33 :- 568

′′ 6 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇33 being increasing, it follows that there ∗ ∗ exists a unique 푇33 for which 푓 푇33 = 0. With this value , we obtain from the three first equations

6 6 푎32 퐺33 푎34 퐺33 퐺32 = ′ 6 ′′ 6 ∗ , 퐺34 = ′ 6 ′′ 6 ∗ (푎32 ) +(푎32) 푇33 (푎34) +(푎34) 푇33

(f) By the same argument, the equations 92,93 admit solutions 퐺13 , 퐺14 if 569

′ 1 ′ 1 1 1 휑 퐺 = (푏13 ) (푏14) − 푏13 푏14 −

′ 1 ′′ 1 ′ 1 ′′ 1 ′′ 1 ′′ 1 (푏13 ) (푏14 ) 퐺 + (푏14) (푏13) 퐺 +(푏13 ) 퐺 (푏14 ) 퐺 = 0

Where in 퐺 퐺13 , 퐺14 , 퐺15 , 퐺13 , 퐺15 must be replaced by their values from 96. It is easy to see that φ is a decreasing function in 퐺14 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists a ∗ ∗ unique 퐺14 such that 휑 퐺 = 0

(g) By the same argument, the equations 92,93 admit solutions 퐺16 , 퐺17 if 570

′ 2 ′ 2 2 2 φ 퐺19 = (푏16) (푏17) − 푏16 푏17 −

www.ijmer.com 2095 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645

′ 2 ′′ 2 ′ 2 ′′ 2 ′′ 2 ′′ 2 (푏16 ) (푏17 ) 퐺19 + (푏17 ) (푏16) 퐺19 +(푏16 ) 퐺19 (푏17 ) 퐺19 = 0

Where in 퐺19 퐺16 , 퐺17 , 퐺18 , 퐺16 , 퐺18 must be replaced by their values from 96. It is easy to see that φ is a 571 decreasing function in 퐺17 taking into account the hypothesis φ 0 > 0 , 휑 ∞ < 0 it follows that there exists a ∗ ∗ unique G14 such that φ 퐺19 = 0

(a) By the same argument, the concatenated equations admit solutions 퐺20 , 퐺21 if 572

′ 3 ′ 3 3 3 휑 퐺23 = (푏20) (푏21) − 푏20 푏21 −

′ 3 ′′ 3 ′ 3 ′′ 3 ′′ 3 ′′ 3 (푏20 ) (푏21 ) 퐺23 + (푏21 ) (푏20 ) 퐺23 +(푏20 ) 퐺23 (푏21 ) 퐺23 = 0

Where in 퐺23 퐺20 , 퐺21 , 퐺22 , 퐺20 , 퐺22 must be replaced by their values from 96. It is easy to see that φ is a decreasing function in 퐺21 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists a unique 퐺∗ such that 휑 퐺 ∗ = 0 21 23 573

(b) By the same argument, the equations of modules admit solutions 퐺24 , 퐺25 if 574

′ 4 ′ 4 4 4 휑 퐺27 = (푏24) (푏25) − 푏24 푏25 −

′ 4 ′′ 4 ′ 4 ′′ 4 ′′ 4 ′′ 4 (푏24 ) (푏25 ) 퐺27 + (푏25 ) (푏24 ) 퐺27 +(푏24 ) 퐺27 (푏25 ) 퐺27 = 0

Where in 퐺27 퐺24 , 퐺25 , 퐺26 , 퐺24 , 퐺26 must be replaced by their values from 96. It is easy to see that φ is a decreasing function in 퐺25 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists ∗ ∗ a unique 퐺25 such that 휑 퐺27 = 0

(c) By the same argument, the equations (modules) admit solutions 퐺28 , 퐺29 if 575

′ 5 ′ 5 5 5 휑 퐺31 = (푏28) (푏29) − 푏28 푏29 −

′ 5 ′′ 5 ′ 5 ′′ 5 ′′ 5 ′′ 5 (푏28 ) (푏29) 퐺31 + (푏29) (푏28 ) 퐺31 +(푏28 ) 퐺31 (푏29) 퐺31 = 0

Where in 퐺31 퐺28 , 퐺29, 퐺30 , 퐺28 , 퐺30 must be replaced by their values from 96. It is easy to see that φ is a decreasing function in 퐺29 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists ∗ ∗ a unique 퐺29 such that 휑 퐺31 = 0

(d) By the same argument, the equations (modules) admit solutions 퐺32 , 퐺33 if 578

579 ′ 6 ′ 6 6 6 휑 퐺35 = (푏32) (푏33) − 푏32 푏33 − 580 ′ 6 ′′ 6 ′ 6 ′′ 6 ′′ 6 ′′ 6 (푏32 ) (푏33 ) 퐺35 + (푏33 ) (푏32 ) 퐺35 +(푏32 ) 퐺35 (푏33 ) 퐺35 = 0 581

Where in 퐺35 퐺32 , 퐺33 , 퐺34 , 퐺32 , 퐺34 must be replaced by their values It is easy to see that φ is a decreasing function in 퐺33 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there exists a unique ∗ ∗ 퐺33 such that 휑 퐺 = 0

Finally we obtain the unique solution of 89 to 94 582

∗ ∗ ∗ ∗ 퐺14 given by 휑 퐺 = 0 , 푇14 given by 푓 푇14 = 0 and

1 ∗ 1 ∗ ∗ 푎13 퐺14 ∗ 푎15 퐺14 퐺13 = ′ 1 ′′ 1 ∗ , 퐺15 = ′ 1 ′′ 1 ∗ (푎13 ) +(푎13) 푇14 (푎15) +(푎15) 푇14

1 ∗ 1 ∗ ∗ 푏13 푇14 ∗ 푏15 푇14 푇13 = ′ 1 ′′ 1 ∗ , 푇15 = ′ 1 ′′ 1 ∗ (푏13) −(푏13) 퐺 (푏15 ) −(푏15) 퐺

www.ijmer.com 2096 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution 583

∗ ∗ ∗ ∗ G17 given by φ 퐺19 = 0 , T17 given by 푓 T17 = 0 and 584

2 ∗ 2 ∗ ∗ a16 G17 ∗ a18 G17 585 G16 = ′ 2 ′′ 2 ∗ , G18 = ′ 2 ′′ 2 ∗ (a16) +(a16) T17 (a18) +(a18) T17

2 ∗ 2 ∗ ∗ b16 T17 ∗ b18 T17 586 T16 = ′ 2 ′′ 2 ∗ , T18 = ′ 2 ′′ 2 ∗ (b16) −(b16) 퐺19 (b18) −(b18) 퐺19

Obviously, these values represent an equilibrium solution 587

Finally we obtain the unique solution 588

∗ ∗ ∗ ∗ 퐺21 given by 휑 퐺23 = 0 , 푇21 given by 푓 푇21 = 0 and

3 ∗ 3 ∗ ∗ 푎20 퐺21 ∗ 푎22 퐺21 퐺20 = ′ 3 ′′ 3 ∗ , 퐺22 = ′ 3 ′′ 3 ∗ (푎20) +(푎20) 푇21 (푎22) +(푎22) 푇21

3 ∗ 3 ∗ ∗ 푏20 푇21 ∗ 푏22 푇21 푇20 = ′ 3 ′′ 3 ∗ , 푇22 = ′ 3 ′′ 3 ∗ (푏20) −(푏20) 퐺23 (푏22) −(푏22) 퐺23

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution 589

∗ ∗ ∗ 퐺25 given by 휑 퐺27 = 0 , 푇25 given by 푓 푇25 = 0 and

4 ∗ 4 ∗ ∗ 푎24 퐺25 ∗ 푎26 퐺25 퐺24 = ′ 4 ′′ 4 ∗ , 퐺26 = ′ 4 ′′ 4 ∗ (푎24) +(푎24) 푇25 (푎26) +(푎26) 푇25

4 ∗ 4 ∗ ∗ 푏24 푇25 ∗ 푏26 푇25 590 푇24 = ′ 4 ′′ 4 ∗ , 푇26 = ′ 4 ′′ 4 ∗ (푏24) −(푏24) 퐺27 (푏26 ) −(푏26) 퐺27

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution 591

∗ ∗ ∗ ∗ 퐺29 given by 휑 퐺31 = 0 , 푇29 given by 푓 푇29 = 0 and

5 ∗ 5 ∗ ∗ 푎28 퐺29 ∗ 푎30 퐺29 퐺28 = ′ 5 ′′ 5 ∗ , 퐺30 = ′ 5 ′′ 5 ∗ (푎28) +(푎28) 푇29 (푎30) +(푎30) 푇29

5 ∗ 5 ∗ ∗ 푏28 푇29 ∗ 푏30 푇29 592 푇28 = ′ 5 ′′ 5 ∗ , 푇30 = ′ 5 ′′ 5 ∗ (푏28) −(푏28) 퐺31 (푏30 ) −(푏30) 퐺31

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution 593

∗ ∗ ∗ ∗ 퐺33 given by 휑 퐺35 = 0 , 푇33 given by 푓 푇33 = 0 and

6 ∗ 6 ∗ ∗ 푎32 퐺33 ∗ 푎34 퐺33 퐺32 = ′ 6 ′′ 6 ∗ , 퐺34 = ′ 6 ′′ 6 ∗ (푎32) +(푎32) 푇33 (푎34) +(푎34) 푇33

6 ∗ 6 ∗ ∗ 푏32 푇33 ∗ 푏34 푇33 594 푇32 = ′ 6 ′′ 6 ∗ , 푇34 = ′ 6 ′′ 6 ∗ (푏32) −(푏32) 퐺35 (푏34 ) −(푏34) 퐺35

Obviously, these values represent an equilibrium solution

www.ijmer.com 2097 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 ASYMPTOTIC STABILITY ANALYSIS 595

′′ 1 ′′ 1 Theorem 4: If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 퐶 1 ( ℝ ) then the above equilibrium point is asymptotically stable. + Proof: Denote

Definition of 픾 , 핋 :- 푖 푖 퐺 = 퐺∗ + 픾 , 푇 = 푇∗ + 핋 푖 푖 푖 푖 푖 푖 596 ′′ 1 ′′ 1 휕(푎14) ∗ 1 휕(푏푖 ) ∗ 푇14 = 푞14 , 퐺 = 푠푖푗 휕푇14 휕퐺푗

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

푑픾 13 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 598 푑푡 13 13 13 13 14 13 13 14

푑픾 14 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 599 푑푡 14 14 14 14 13 14 14 14

푑픾 15 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 600 푑푡 15 15 15 15 14 15 15 14

푑핋 13 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 601 푑푡 13 13 13 13 14 푗=13 13 푗 13 푗

푑핋 14 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 602 푑푡 14 14 14 14 13 푗=13 14 (푗 ) 14 푗

푑핋 15 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 603 푑푡 15 15 15 15 14 푗=13 15 (푗 ) 15 푗

′′ 2 ′′ 2 If the conditions of the previous theorem are satisfied and if the functions (a푖 ) and (b푖 ) Belong to 604 2 C ( ℝ+) then the above equilibrium point is asymptotically stable

Denote 605

Definition of 픾푖 , 핋푖 :-

∗ ∗ G푖 = G푖 + 픾푖 , T푖 = T푖 + 핋푖 606

′′ 2 ′′ 2 ∂(푎17) ∗ 2 ∂(푏푖 ) ∗ 607 T17 = 푞17 , 퐺19 = 푠푖푗 ∂T17 ∂G푗 taking into account equations (global)and neglecting the terms of power 2, we obtain 608 d픾 16 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 609 dt 16 16 16 16 17 16 16 17 d픾 17 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 610 dt 17 17 17 17 16 17 17 17 d픾 18 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 611 dt 18 18 18 18 17 18 18 17 d핋 16 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 612 dt 16 16 16 16 17 푗=16 16 푗 16 푗 d핋 17 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 613 dt 17 17 17 17 16 푗=16 17 (푗 ) 17 푗 d핋 18 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 614 dt 18 18 18 18 17 푗=16 18 (푗 ) 18 푗

′′ 3 ′′ 3 If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 615 퐶 3 ( ℝ ) then the above equilibrium point is asymptotically stabl +

www.ijmer.com 2098 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Denote 616

Definition of 픾푖 , 핋푖 :-

∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖

′′ 3 ′′ 3 휕(푎21) ∗ 3 휕(푏푖 ) ∗ 푇21 = 푞21 , 퐺23 = 푠푖푗 휕푇21 휕퐺푗

Then taking into account equations (global) and neglecting the terms of power 2, we obtain 617

푑픾 20 = − (푎′ ) 3 + 푝 3 픾 + 푎 3 픾 − 푞 3 퐺∗ 핋 618 푑푡 20 20 20 20 21 20 20 21

푑픾 21 = − (푎′ ) 3 + 푝 3 픾 + 푎 3 픾 − 푞 3 퐺∗ 핋 619 푑푡 21 21 21 21 20 21 21 21

푑픾 22 = − (푎′ ) 3 + 푝 3 픾 + 푎 3 픾 − 푞 3 퐺∗ 핋 6120 푑푡 22 22 22 22 21 22 22 21

푑핋 20 = − (푏′ ) 3 − 푟 3 핋 + 푏 3 핋 + 22 푠 푇∗ 픾 621 푑푡 20 20 20 20 21 푗=20 20 푗 20 푗

푑핋 21 = − (푏′ ) 3 − 푟 3 핋 + 푏 3 핋 + 22 푠 푇∗ 픾 622 푑푡 21 21 21 21 20 푗=20 21 (푗 ) 21 푗

푑핋 22 = − (푏′ ) 3 − 푟 3 핋 + 푏 3 핋 + 22 푠 푇∗ 픾 623 푑푡 22 22 22 22 21 푗=20 22 (푗 ) 22 푗

′′ 4 ′′ 4 If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 624 4 퐶 ( ℝ+) then the above equilibrium point is asymptotically stabl

Denote

Definition of 픾푖 , 핋푖 :- 625

∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖

′′ 4 ′′ 4 휕(푎25) ∗ 4 휕(푏푖 ) ∗ 푇25 = 푞25 , 퐺27 = 푠푖푗 휕푇25 휕퐺푗

Then taking into account equations (global) and neglecting the terms of power 2, we obtain 626

푑픾 24 = − (푎′ ) 4 + 푝 4 픾 + 푎 4 픾 − 푞 4 퐺∗ 핋 627 푑푡 24 24 24 24 25 24 24 25

푑픾 25 = − (푎′ ) 4 + 푝 4 픾 + 푎 4 픾 − 푞 4 퐺∗ 핋 628 푑푡 25 25 25 25 24 25 25 25

푑픾 26 = − (푎′ ) 4 + 푝 4 픾 + 푎 4 픾 − 푞 4 퐺∗ 핋 629 푑푡 26 26 26 26 25 26 26 25

푑핋 24 = − (푏′ ) 4 − 푟 4 핋 + 푏 4 핋 + 26 푠 푇∗ 픾 630 푑푡 24 24 24 24 25 푗=24 24 푗 24 푗

푑핋 25 = − (푏′ ) 4 − 푟 4 핋 + 푏 4 핋 + 26 푠 푇∗ 픾 631 푑푡 25 25 25 25 24 푗=24 25 푗 25 푗

푑핋 26 = − (푏′ ) 4 − 푟 4 핋 + 푏 4 핋 + 26 푠 푇∗ 픾 632 푑푡 26 26 26 26 25 푗=24 26 (푗 ) 26 푗

633

′′ 5 ′′ 5 If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 5 퐶 ( ℝ+) then the above equilibrium point is asymptotically stable

www.ijmer.com 2099 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-2028-2109 ISSN: 2249-6645 Denote

Definition of 픾푖 , 핋푖 :- 634

∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖

′′ 5 ′′ 5 휕(푎29) ∗ 5 휕(푏푖 ) ∗ 푇29 = 푞29 , 퐺31 = 푠푖푗 휕 푇29 휕퐺푗

Then taking into account equations (global) and neglecting the terms of power 2, we obtain 635

푑픾 28 = − (푎′ ) 5 + 푝 5 픾 + 푎 5 픾 − 푞 5 퐺∗ 핋 636 푑푡 28 28 28 28 29 28 28 29

푑픾 29 = − (푎′ ) 5 + 푝 5 픾 + 푎 5 픾 − 푞 5 퐺∗ 핋 637 푑푡 29 29 29 29 28 29 29 29

푑픾 30 = − (푎′ ) 5 + 푝 5 픾 + 푎 5 픾 − 푞 5 퐺∗ 핋 638 푑푡 30 30 30 30 29 30 30 29

푑핋 28 = − (푏′ ) 5 − 푟 5 핋 + 푏 5 핋 + 30 푠 푇∗ 픾 639 푑푡 28 28 28 28 29 푗=28 28 푗 28 푗

푑핋 29 = − (푏′ ) 5 − 푟 5 핋 + 푏 5 핋 + 30 푠 푇∗ 픾 640 푑푡 29 29 29 29 28 푗 =28 29 푗 29 푗

푑핋 30 = − (푏′ ) 5 − 푟 5 핋 + 푏 5 핋 + 30 푠 푇∗ 픾 641 푑푡 30 30 30 30 29 푗=28 30 (푗 ) 30 푗

′′ 6 ′′ 6 If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 642 6 퐶 ( ℝ+) then the above equilibrium point is asymptotically stable

Denote

Definition of 픾푖 , 핋푖 :- 643

∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖

′′ 6 ′′ 6 휕(푎33) ∗ 6 휕(푏푖 ) ∗ 푇33 = 푞33 , 퐺35 = 푠푖푗 휕푇33 휕퐺푗

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

푑픾 32 = − (푎′ ) 6 + 푝 6 픾 + 푎 6 픾 − 푞 6 퐺∗ 핋 645 푑푡 32 32 32 32 33 32 32 33

푑픾 33 = − (푎′ ) 6 + 푝 6 픾 + 푎 6 픾 − 푞 6 퐺∗ 핋 646 푑푡 33 33 33 33 32 33 33 33

푑픾 34 = − (푎′ ) 6 + 푝 6 픾 + 푎 6 픾 − 푞 6 퐺∗ 핋 647 푑푡 34 34 34 34 33 34 34 33

푑핋 32 = − (푏′ ) 6 − 푟 6 핋 + 푏 6 핋 + 34 푠 푇∗ 픾 648 푑푡 32 32 32 32 33 푗=32 32 푗 32 푗

푑핋 33 = − (푏′ ) 6 − 푟 6 핋 + 푏 6 핋 + 34 푠 푇∗ 픾 649 푑푡 33 33 33 33 32 푗=32 33 푗 33 푗

푑핋 34 = − (푏′ ) 6 − 푟 6 핋 + 푏 6 핋 + 34 푠 푇∗ 픾 650 푑푡 34 34 34 34 33 푗=32 34 (푗 ) 34 푗

Obviously, these values represent an equilibrium solution of 79,20,36,22,23, 651 ′′ 7 ′′ 7 If the conditions of the previous theorem are satisfied and if the functions (푎푖 ) 푎푛푑 (푏푖 ) Belong to 7 퐶 ( ℝ+) then the above equilibrium point is asymptotically stable.

Proof: Denote

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Definition of 픾푖 , 핋푖 :- 652 653 ∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖 ′′ 7 ′′ 7 휕(푎37) ∗ 7 휕(푏푖 ) ∗∗ 푇37 = 푞37 , 퐺39 = 푠푖푗 휕푇37 휕퐺푗

Then taking into account equations(SOLUTIONAL) and neglecting the terms of power 2, we obtain 654

655 푑픾 36 = − (푎′ ) 7 + 푝 7 픾 + 푎 7 픾 − 푞 7 퐺∗ 핋 656 푑푡 36 36 36 36 37 36 36 37 푑픾 37 = − (푎′ ) 7 + 푝 7 픾 + 푎 7 픾 − 푞 7 퐺∗ 핋 657 푑푡 37 37 37 37 36 37 37 37

푑픾 38 = − (푎′ ) 7 + 푝 7 픾 + 푎 7 픾 − 푞 7 퐺∗ 핋 658 푑푡 38 38 38 38 37 38 38 37

푑핋 36 = − (푏′ ) 7 − 푟 7 핋 + 푏 7 핋 + 38 푠 푇∗ 픾 659 푑푡 36 36 36 36 37 푗=36 36 푗 36 푗

푑핋 37 = − (푏′ ) 7 − 푟 7 핋 + 푏 7 핋 + 38 푠 푇∗ 픾 660 푑푡 37 37 37 37 36 푗=36 37 푗 37 푗

푑핋 38 = − (푏′ ) 7 − 푟 7 핋 + 푏 7 핋 + 38 푠 푇∗ 픾 661 푑푡 38 38 38 38 37 푗=36 38 (푗 ) 38 푗

2. The characteristic equation of this system is 1 1 1 1 ′ 1 1 ′ 휆 + (푏15) − 푟15 { 휆 + (푎15) + 푝15 1 1 1 1 1 ′ ∗ 1 ∗ 휆 + (푎13) + 푝13 푞14 퐺14 + 푎14 푞13 퐺13

1 1 ′ 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 14 푇14 + 푏14 푠 13 , 14 푇14

1 ′ 1 1 1 ∗ 1 1 ∗ + 휆 + (푎14 ) + 푝14 푞13 퐺13 + 푎13 푞14 퐺14

1 1 ′ 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 13 푇14 + 푏14 푠 13 , 13 푇13

2 1 1 1 1 1 ′ ′ 1 휆 + (푎13) + (푎14) + 푝13 + 푝14 휆 2 1 1 1 ′ ′ 1 1 1 휆 + (푏13) + (푏14) − 푟13 + 푟14 휆 1 2 ′ 1 ′ 1 1 1 1 1 + 휆 + (푎13 ) + (푎14 ) + 푝13 + 푝14 휆 푞15 퐺15 1 ′ 1 1 1 1 ∗ 1 1 1 ∗ + 휆 + (푎13 ) + 푝13 푎15 푞14 퐺14 + 푎14 푎15 푞13 퐺13 1 1 ′ 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 15 푇14 + 푏14 푠 13 , 15 푇13 } = 0

+ 2 2 2 2 ′ 2 2 ′ 휆 + (푏18) − 푟18 { 휆 + (푎18) + 푝18

2 2 2 2 2 ′ ∗ 2 ∗ 휆 + (푎16) + 푝16 푞17 G17 + 푎17 푞16 G16

2 2 ′ 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 17 T17 + 푏17 푠 16 , 17 T17

2 ′ 2 2 2 ∗ 2 2 ∗ + 휆 + (푎17 ) + 푝17 푞16 G16 + 푎16 푞17 G17

2 2 ′ 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 16 T17 + 푏17 푠 16 , 16 T16

2 2 2 2 2 2 ′ ′ 2 휆 + (푎16) + (푎17) + 푝16 + 푝17 휆

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2 2 ′ 2 ′ 2 2 2 2 휆 + (푏16) + (푏17 ) − 푟16 + 푟17 휆 2 2 ′ 2 ′ 2 2 2 2 2 + 휆 + (푎16 ) + (푎17 ) + 푝16 + 푝17 휆 푞18 G18 2 ′ 2 2 2 2 ∗ 2 2 2 ∗ + 휆 + (푎16 ) + 푝16 푎18 푞17 G17 + 푎17 푎18 푞16 G16

2 2 ′ 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 18 T17 + 푏17 푠 16 , 18 T16 } = 0

+ 3 3 3 3 ′ 3 3 ′ 휆 + (푏22) − 푟22 { 휆 + (푎22) + 푝22

3 3 3 3 3 ′ ∗ 3 ∗ 휆 + (푎20) + 푝20 푞21 퐺21 + 푎21 푞20 퐺20

3 3 ′ 3 ∗ 3 ∗ 휆 + (푏20) − 푟20 푠 21 , 21 푇21 + 푏21 푠 20 , 21 푇21

3 ′ 3 3 3 ∗ 3 1 ∗ + 휆 + (푎21 ) + 푝21 푞20 퐺20 + 푎20 푞21 퐺21

3 3 ′ 3 ∗ 3 ∗ 휆 + (푏20) − 푟20 푠 21 , 20 푇21 + 푏21 푠 20 , 20 푇20

2 3 3 3 3 3 ′ ′ 3 휆 + (푎20) + (푎21) + 푝20 + 푝21 휆 2 3 3 3 ′ ′ 3 3 3 휆 + (푏20) + (푏21) − 푟20 + 푟21 휆 3 2 ′ 3 ′ 3 3 3 3 3 + 휆 + (푎20 ) + (푎21 ) + 푝20 + 푝21 휆 푞22 퐺22 3 ′ 3 3 3 3 ∗ 3 3 3 ∗ + 휆 + (푎20 ) + 푝20 푎22 푞21 퐺21 + 푎21 푎22 푞20 퐺20 3 3 ′ 3 ∗ 3 ∗ 휆 + (푏20) − 푟20 푠 21 , 22 푇21 + 푏21 푠 20 , 22 푇20 } = 0

+ 4 4 4 4 ′ 4 4 ′ 휆 + (푏26) − 푟26 { 휆 + (푎26) + 푝26

4 4 4 4 4 ′ ∗ 4 ∗ 휆 + (푎24) + 푝24 푞25 퐺25 + 푎25 푞24 퐺24

4 4 ′ 4 ∗ 4 ∗ 휆 + (푏24) − 푟24 푠 25 , 25 푇25 + 푏25 푠 24 , 25 푇25

4 ′ 4 4 4 ∗ 4 4 ∗ + 휆 + (푎25 ) + 푝25 푞24 퐺24 + 푎24 푞25 퐺25

4 4 ′ 4 ∗ 4 ∗ 휆 + (푏24) − 푟24 푠 25 , 24 푇25 + 푏25 푠 24 , 24 푇24

2 4 4 4 4 4 ′ ′ 4 휆 + (푎24) + (푎25) + 푝24 + 푝25 휆 4 2 ′ 4 ′ 4 4 4 4 휆 + (푏24) + (푏25 ) − 푟24 + 푟25 휆 4 2 ′ 4 ′ 4 4 4 4 4 + 휆 + (푎24 ) + (푎25 ) + 푝24 + 푝25 휆 푞26 퐺26 4 ′ 4 4 4 4 ∗ 4 4 4 ∗ + 휆 + (푎24 ) + 푝24 푎26 푞25 퐺25 + 푎25 푎26 푞24 퐺24 4 4 ′ 4 ∗ 4 ∗ 휆 + (푏24) − 푟24 푠 25 , 26 푇25 + 푏25 푠 24 , 26 푇24 } = 0

+ 5 5 5 5 ′ 5 5 ′ 휆 + (푏30) − 푟30 { 휆 + (푎30) + 푝30

5 5 5 5 5 ′ ∗ 5 ∗ 휆 + (푎28) + 푝28 푞29 퐺29 + 푎29 푞28 퐺28

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5 5 ′ 5 ∗ 5 ∗ 휆 + (푏28) − 푟28 푠 29 , 29 푇29 + 푏29 푠 28 , 29 푇29

5 ′ 5 5 5 ∗ 5 5 ∗ + 휆 + (푎29) + 푝29 푞28 퐺28 + 푎28 푞29 퐺29

5 5 ′ 5 ∗ 5 ∗ 휆 + (푏28) − 푟28 푠 29 , 28 푇29 + 푏29 푠 28 , 28 푇28

2 5 5 5 5 5 ′ ′ 5 휆 + (푎28) + (푎29) + 푝28 + 푝29 휆 5 2 ′ 5 ′ 5 5 5 5 휆 + (푏28) + (푏29) − 푟28 + 푟29 휆 5 2 ′ 5 ′ 5 5 5 5 5 + 휆 + (푎28 ) + (푎29) + 푝28 + 푝29 휆 푞30 퐺30 5 ′ 5 5 5 5 ∗ 5 5 5 ∗ + 휆 + (푎28 ) + 푝28 푎30 푞29 퐺29 + 푎29 푎30 푞28 퐺28 5 5 ′ 5 ∗ 5 ∗ 휆 + (푏28) − 푟28 푠 29 , 30 푇29 + 푏29 푠 28 , 30 푇28 } = 0

+ 6 6 6 6 ′ 6 6 ′ 휆 + (푏34) − 푟34 { 휆 + (푎34) + 푝34

6 6 6 6 6 ′ ∗ 6 ∗ 휆 + (푎32) + 푝32 푞33 퐺33 + 푎33 푞32 퐺32

6 6 ′ 6 ∗ 6 ∗ 휆 + (푏32) − 푟32 푠 33 , 33 푇33 + 푏33 푠 32 , 33 푇33

6 ′ 6 6 6 ∗ 6 6 ∗ + 휆 + (푎33 ) + 푝33 푞32 퐺32 + 푎32 푞33 퐺33

6 6 ′ 6 ∗ 6 ∗ 휆 + (푏32) − 푟32 푠 33 , 32 푇33 + 푏33 푠 32 , 32 푇32

2 6 6 6 6 6 ′ ′ 6 휆 + (푎32) + (푎33) + 푝32 + 푝33 휆 6 2 ′ 6 ′ 6 6 6 6 휆 + (푏32) + (푏33 ) − 푟32 + 푟33 휆 6 2 ′ 6 ′ 6 6 6 6 6 + 휆 + (푎32 ) + (푎33 ) + 푝32 + 푝33 휆 푞34 퐺34 6 ′ 6 6 6 6 ∗ 6 6 6 ∗ + 휆 + (푎32 ) + 푝32 푎34 푞33 퐺33 + 푎33 푎34 푞32 퐺32 6 6 ′ 6 ∗ 6 ∗ 휆 + (푏32) − 푟32 푠 33 , 34 푇33 + 푏33 푠 32 , 34 푇32 } = 0 +

7 ′ 7 7 7 ′ 7 7 휆 + (푏38 ) − 푟38 { 휆 + (푎38 ) + 푝38 7 ′ 7 7 7 ∗ 7 7 ∗ 휆 + (푎36 ) + 푝36 푞37 퐺37 + 푎37 푞36 퐺36

7 ′ 7 7 ∗ 7 ∗ 휆 + (푏36) − 푟36 푠 37 , 37 푇37 + 푏37 푠 36 , 37 푇37 7 ′ 7 7 7 ∗ 7 7 ∗ + 휆 + (푎37 ) + 푝37 푞36 퐺36 + 푎36 푞37 퐺37 7 ′ 7 7 ∗ 7 ∗ 휆 + (푏36) − 푟36 푠 37 , 36 푇37 + 푏37 푠 36 , 36 푇36 7 2 ′ 7 ′ 7 7 7 7 휆 + (푎36 ) + (푎37 ) + 푝36 + 푝37 휆 7 2 ′ 7 ′ 7 7 7 7 휆 + (푏36) + (푏37 ) − 푟36 + 푟37 휆

7 2 ′ 7 ′ 7 7 7 7 7 + 휆 + (푎36 ) + (푎37 ) + 푝36 + 푝37 휆 푞38 퐺38 7 ′ 7 7 7 7 ∗ 7 7 7 ∗ + 휆 + (푎36 ) + 푝36 푎38 푞37 퐺37 + 푎37 푎38 푞36 퐺36 7 ′ 7 7 ∗ 7 ∗ 휆 + (푏36) − 푟36 푠 37 , 38 푇37 + 푏37 푠 36 , 38 푇36 } = 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 ≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers' conversion value of one gram TNT ≡ 1000 calories used.

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(23)^ Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average Cp of 25.8, 5.134 moles of metal, and 132 J.K-1 for the prototype. A variation of ±1.5 picograms is of course, much smaller than the actual uncertainty in the mass of the international prototype, which are ±2 micrograms.

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(53)Delamotte, Bertrand; A hint of renormalization, American Journal of Physics 72 (2004) pp. 170–184. Beautiful elementary introduction to the ideas, no prior knowledge of field theory being necessary. Full text available at: hep-th/0212049

(54)Baez, John; Renormalization Made Easy, (2005). A qualitative introduction to the subject.

(55)Blechman, Andrew E. ; Renormalization: Our Greatly Misunderstood Friend, (2002). Summary of a lecture; has more information about specific regularization and divergence-subtraction schemes.

(56)Cao, Tian Yu & Schweber, Silvian S. ; The Conceptual Foundations and the Philosophical Aspects of Renormalization Theory, Synthese, 97(1) (1993), 33–108.

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(57)Shirkov, Dmitry; Fifty Years of the Renormalization Group, C.E.R.N. Courrier 41(7) (2001). Full text available at: I.O.P Magazines.

(58)E. Elizalde; Zeta regularization techniques with Applications.

(59)N. N. Bogoliubov, D. V. Shirkov (1959): The Theory of Quantized Fields. New York, Interscience. The first text-book on the renormalization group theory.

(60)Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), ISBN 0-521-33859-X Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics.

(61)Zee, Anthony; Quantum Field Theory in a Nutshell, Princeton University Press (2003) ISBN 0-691-01019-6. Another excellent textbook on Q.F.T.

(62)Weinberg, Steven; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.

(63)Pokorski, Stefan; Gauge Field Theories, Cambridge University Press (1987) ISBN 0-521-47816-2.

(64)'t Hooft, Gerard; The Glorious Days of Physics – Renormalization of Gauge theories, lecture given at Erice (August/September 1998) by the Nobel laureate 1999 . Full text available at: hep-th/9812203.

(65)Rivasseau, Vincent; An introduction to renormalization, Poincaré Seminar (Paris, Oct. 12, 2002), published in: Duplantier, Bertrand; Rivasseau, Vincent (Eds.) ; Poincaré Seminar 2002, Progress in Mathematical Physics 30, Birkhäuser (2003) ISBN 3-7643-0579-7. Full text available in PostScript.

(66) Rivasseau, Vincent; From perturbative to constructive renormalization, Princeton University Press (1991) ISBN 0-691-08530-7. Full text available in PostScript.

(67) Iagolnitzer, Daniel & Magnen, J. ; Renormalization group analysis, Encyclopaedia of Mathematics, Kluwer Academic Publisher (1996). Full text available in PostScript and pdfhere.

(68) Scharf, Günter; Finite quantum electrodynamics: The casual approach, Springer Verlag Berlin Heidelberg New York (1995) ISBN 3-540-60142-2.

(69)A. S. Švarc (Albert Schwarz), Математические основы квантовой теории поля, (Mathematical aspects of quantum field theory), Atomizdat, Moscow, 1975. 368 pp.

(70) A. N. Vasil'ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004); ISBN 978-0-415-31002-4

(71) Nigel Goldenfeld ; Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics 85, West view Press (June, 1992) ISBN 0-201-55409-7. Covering the elementary aspects of the physics of phase‘s transitions and the renormalization group, this popular book emphasizes understanding and clarity rather than technical manipulations.

(72) Zinn-Justin, Jean; Quantum Field Theory and Critical Phenomena, Oxford University Press (4th edition – 2002) ISBN 0-19-850923-5. A masterpiece on applications of renormalization methods to the calculation of critical exponents in statistical mechanics, following Wilson's ideas (Kenneth Wilson was Nobel laureate 1982).

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(73)Dematte, L., Priami, C., and Romanel, A. The BlenX language, a tutorial. In M. Bernardo, P. Degano, and G. Zavattaro (Eds.): SFM 2008, LNCS 5016, pp. 313-365, Springer, 2008. Denning, P. J. Beyond computational thinking. Comm. ACM, pp. 28-30, June 2009. (74)Goldin, D., Smolka, S., and Wegner, P. Interactive Computation: The New Paradigm, Springer, 2006. (75)Hewitt, C., Bishop, P., and Steiger, R. A universal modular ACTOR formalism for artificial intelligence. In Proc. of the 3rd IJCAI, pp. 235-245, Stanford, USA, 1973. (76)Priami, C. Computational thinking in biology, Trans. on Comput. Syst. Biol. VIII, LNBI 4780, pp. 63-76, Springer, 2007. (77)Schor, P. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Press, Los Almitos, CA, 1994. (78)Turing, A. On computable numbers with an application to the Entscheidungsproblem. In Proc. London Mathematical Society 42, pp. 230-265, 1936. (79)Wing, J. Computational thinking. Comm. ACM, pp. 33-35, March 2006 ======AAcknowled gments: The introduction is a collection of information from various articles, Books, News Paper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,Article Abstracts, Research papers, Abstracts Of Research Papers, Stanford Encyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors, the internet including Wikipedia. We acknowledge all authors who have contributed to the same. In the eventuality of the fact that there has been any act of omission on the part of the authors, we regret with great deal of compunction, contrition, regret, 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 ‗Mathematical Models in Political Science‘--- 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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