The Conditional Sequence Information Function

Problem statement: A great deal of attention has been given to the theory of information. It has found its applications in science especially in the area of Biothecnology. Previous studies on the subject has been limited to either conditional or s equence problem solving. This study combines both conditional and sequence properties of the informat ion function. By doing this, researchers can find solutions to more problems that are applicable in r eal life. Approach: First, properties of the dynamical systems and the information function defined by Sha nnon has been provided, Then, the conditional sequence information function of a dynamical system together with its proof was presented. Result: This new function now exists now and ready for the use i n many real life problems such as finding solutions to DNA sequence with conditional sickness. Conclusion: This new function created opens a new avenue to researches in solving more complex problems by usin g its developed properties.


INTRODUCTION
Currently many researchers are investigating bioinformation using very complex mathematical functions. Of particular interest to us is the information function defined by Shannon (1948). Both Khinchin (1957) and Shannon (1948) have investigated the properties of this information function. Brown (1976) and Gray and Davidson (1970) defined the entropy function of dynamical systems and investigated its properties. Tok (1986) defined the fuzzy information function and investigated its properties. Moreover, Newton (1970a;1970b); Walters (1975;2000) and Guzide (1990) (in Turkish) defined conditional sequence entropy and sequence entropy functions and investigated their properties.
In this study, we give the definition of a conditional sequence information function and prove that it exists.
First we give the some properties of dynamical systems necessary to our discussion and states the sequence information function and list its properties. The conditional sequence entropy defined by Zhang (1993). We then define the conditional sequence information function and finish with a proof of its existence.
Dynamical systems and information function: We will give very important background support and definition of dynamical system and information Function.
Definition 1: A measure-preserving dynamical system is defined as a probability space and a measurepreserving transformation on it. In more detail, it is a system (X, A, µ,T) with the following structure: • X is a set • A is aσ-algebra over X • µ (A)→[0,1] is a probability measure, so that µ (X) = 1 • T:X→Y is a measurable transformation which preserves the measure µ Definition 2: Consider two dynamical systems (X, A, µ, T) and (Y, B, υ, S). Then a mapping φ: X→Y is a homomorphism of dynamical systems if it satisfies the following three properties: The system (Y, B, υ, S) is then called a factor of (X, A, µ, T).
The map ϕ is an isomorphism of dynamical systems if, in addition, there exists another mapping ψ: Y→X that is also a homomorphism, which satisfies: • For µ-almost all x∈X, one has x = ψ (ϕx) • For υ-almost all y∈Y, one has y = ϕ (ψy) Definition 3: • Consider a dynamical system (X, A, µ,T) and P is a σ-measure on X.
Now we are ready to give conditional information function definition.
Definition 5: Consider a dynamical system (X, A, µ,T)and P∈Z where x X ∀ ∈ . I (T, P, x) Γ is called the sequence information function of P under T and exits based on Theorem 1.
≥ . Need to investigate σ-addition of n n 1 (a ) ≥ . Let t 1 = 0 and {t i } i be sequence integer number where n 1 i n m + ≤ ≤ + . From proof of sequence and conditional information function and using same proof of sequence information function Lemma 1 in (Guzide, 1990). First investigation is:

MATERIALS AND METHODS
This problem only use previous functions related dynamical systems. So far only certain level problems are solved by previous functions. Now researchers more advanced and complex functions to solve complex problems.

RESULTS AND DISCUSSION
The conditional information function can be used for real life problem with conditional and sequence information together.
For future research, the investigations will be properties of new conditional sequence function.
With new function the researcher will invastigate both conditional and sequence case of problems. Next step one research and proof the properties of The Conditional Sequence Information Function.

CONCLUSION
New function will be provided real life problem to solve more complex problem. Also this function has properties we did not investigate yet. Researcher will use this function and properties for complex real life problems.