Parikh Fuzzy Vector for Finite Words of Rectangular Hilbert Space Filling Curve

: Parikh Fuzzy vector for finite words of Rectangular Hilbert Space Filling Curve is introduced. Recurrence relations for this vector and its complement vector are produced. It is shown that the components of the Parikh Fuzzy vector are equally distributed at the limiting level. Hence, it is observed that the Parikh Fuzzy vector tends to a constant vector as n tends to infinity. Moreover, it is also valid to any other kinds of Space Filling Curves like Lebesgue Space Filling Curve, Peano Curve and Moore Curve


Introduction
Space Filling Curves are applied to visit each cell of a multidimensional grid exactly once.These concepts of traversal are very useful in image processing, data organization and in reducing dimensions of multidimensional data.Generally Space Filling Curves fill a square using iterative process.For a particular case, Rectangular Space Filling curves fill a rectangle by using recursive progression.
In [1] the concept of fuzzy basis of fuzzy vector space is studied.The authors of [2] studied the nature of binary alphabets in Parikh matrix mapping.Combinations and selections on words are explained in [3].The authors of [4] provided the notion of Parikh prime words.The concept of Parikh factor matrix is introduced in [5].
The properties of recurrence relations of Parikh vectors for finite words are discussed in [6] and [8].The authors of [7] formed different representations of fuzzy vectors.Finite words for Space Filling Curves are investigated in [9] and [10].
Finite words for Rectangular Hilbert Space Filling Curves (RHSFC) are specified from [5] in the second section.Parikh fuzzy vectors for these words are defined in third section.Also properties of Parikh fuzzy vector for finite words of Rectangular Hilbert Space Filling Curve are discussed in the third section.Finally the limiting case of the Parikh fuzzy vector is analyzed.

RHSFC And Finite Words
The construction of the Rectangular Hilbert Space Filling Curve is observed from [5] and n th finite iteration of this curve is described by the String Wn.

Parikh Fuzzy Vector Of Wn
Let the alphabet Σ of Wn is ordered by Then the Parikh Fuzzy vector of Wn is given by Parikh Fuzzy vector pf (Wn) of Wn can be recursively written as ( ) ( ) ) The recurrence equation is linear non-homogeneous non-autonomous equation with variable coefficients.

RECURRENCE RELATION FOR COMPLEMENT PARIKH FUZZY VECTOR cf (Wn) OF pf (Wn)
The complement of pf (Wn) is given by ) .

UPPER BOUND OF pf (Wn)
The largest element in the fuzzy vector 'a' is called its upper bound.where ) ,.... , , (

), ( ), ( LOWER BOUND OF pf (Wn)
The smallest element in the fuzzy vector 'a' is called as its lower bound.tend to their limiting value 0.125 faster than the other letters.. Therefore, the occurrences of letters of Wn are equally probably distributed as n tends to infinity.Moreover, it can be applicable to any formation of finite words for any Space Filling Curve.That is, if the finite words are formed with k letters, then the probability of occurrences of these letters are equal to 1/k at its limiting case.Hence the Parikh Fuzzy vector tends to a constant vector as n tends to infinity.

Theoretical View For Limiting Value Of Pf (Wn)
The limiting value of pf (Wn) can be found by applying limit  → n to pf (Wn).Firstly the limit value of ) (u p n W can be found as follows.

Conclusion
Parikh Fuzzy vector of a word over an ordered alphabet with finite number of letters was introduced.Parikh Fuzzy vectors are computed correspondingly for finite words of Rectangular Hilbert Space Filling Curve.It is observed that, this vector tends to a constant vector as n tends to infinity.Additionally, this nature is also true for other kinds of Space Filling Curve.Also some of the properties of these vectors were analyzed.

Further Research
Some more properties of Fuzzy Parikh Vectors have to be discussed further.

Table 1 .
Probabilites of occurrences of the letters n