# Properties Related To Kolakoski Array Biology Essay

Certain subdivisions of mathematics constitute a major beginning of beginning for formal linguistic communication theory. The term mathematical linguistic communication theory describes mathematical ( algebraic ) facets of formal linguistic communication theory – the chief accent is on the mathematical theory instead than any applications. Hence our motive comes from formal linguistic communication theory and hence the combinative facet will be stressed more than algebraic facet. Formal linguistic communications with particular combinative and structural belongingss are exploited in information processing or information transmittal [ 1 ] .Moreover the theory has now developed into many waies and has generated a quickly turning literature The theory

of words is deeply connected to legion different Fieldss of mathematics and its applications. In this paper we extended some particular combinative one dimensional word

belongingss to two dimensional arrays [ 3,4,5 ] .

In subdivision 2 we see some basic definitions. In subdivision 3 we define Row ( Column ) Kolakoski array, Kolakoski array and we give some belongingss related to Kolakoski array. In add-on to that we presented recursive expression for particular nodes.

## 2 BASIC DEFINITIONS

Let ? be a finite alphabet. The set of all words over ? is denoted by ?* . The empty word is denoted by l. We write ?+ = ?*- { cubic decimeter } .

An infinite word tungsten over a finite alphabet ? is a function from positive whole numbers into ? . We write w = a1 a2… Army Intelligence… where ai vitamin E & A ; lb ; . The set of all infinite words over & As ; lb ; is denoted & A ; lb ; ? . An infinite word tungsten is finally periodic if w = uvw.

An infinite word tungsten over a binary alphabet { a, B } which is non finally periodic and for any positive whole number N, the figure of its factors of length N, gx ( n ) = n + 1, is called a Sturmian word [ 2,3,4 ] .The Fibonacci word degree Fahrenheit which is an of import illustration of a Sturmian word is the fixed point of a morphism J: B* ® B* where B = { a, B } and J ( a ) = Bachelor of Arts, J ( B ) = a i.e. f = jw ( a ) . In fact the first few symbols of the Fibonacci word is abaababaabaababaababa … [ 4,5 ] .

An ( mxn ) array A = ( aij ) ( mxn ) over an alphabet ? is a rectangular agreement of symbols of ? with m rows and n columns. The size of the array A is the ordered brace ( mxn ) . The set of all arrays over ? is denoted by ?** . The empty array is besides denoted by ? . We adopt the convention that for an ( mxn ) array A = ( aij ) ( mxn ) the bottom most row is the first row and the left most column is the first column. Besides we write ?++ = ?** – { ? } . A factor or subarray of an array Angstrom is besides an array which is a portion of A ( formed by canceling certain back-to-back rows and certain back-to-back columns of A ) .

An infinite array U has an infinite figure of rows and infinite figure of columns. The aggregation of all infinite arrays over ? is denoted by ??? .

Row and column concatenation of arrays in ?** are partial operations. For row catenation of two arrays A and B, denoted by AqB, the figure of columns in A and B should be equal and for column concatenation AfB of A and B, the figure of rows in A and B should be equal. The aggregation of all arrays with finite figure of rows and infinite figure of columns is denoted by ? *? and the aggregation of all arrays with infinite figure of rows and a finite figure of columns is denoted by ??* .

An array tungsten vitamin E & A ; lb ; ?? is said to be row finally periodic ( Figure 1 ) if there exist two arrays u and V in ?*? such that w = uq vq? , where vq? denotes infinite figure of row concatenation of V with 5 itself. An array tungsten ? ?? is said to be column finally periodic ( Figure 2 ) if there exist two arrays u and V of ??* such that

tungsten = u degree Fahrenheit vf & A ; Eacute ; , where vf & A ; Eacute ; denotes infinite figure of column concatenation of V with 5 itself.

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Figure 2

The celebrated Kolakoski word? { 1,2 } tungsten which consists of back-to-back blocks of 1 ‘s and 2 ‘s such that the length of each block is either 1 or 2, and the length of the ith block is equal to the ith missive of. The word is an illustration of a ego reading infinite word. = 221121221221121122121121. . . . . . . . . . . . is an illustration of a Kolakoski word [ 7 ] .

## 3. KOLAKOSKI ARRAY

We define Row ( Column ) Kolakoski array and Kolakoski array and present some belongingss of Kolakoski array

## DEFINITION:3.1

Let A= ( aij ) be an infinite array. The elements of A are called vertices of A. The vertex at the ( I, I ) Thursday place is called as ith node of A. A way from an1 to a1n along the vertices an2an3…ann-1annan-1n… a2n is called right angle way. If all the vertices of right angle way are equal so the way is called bed of width 1.If two back-to-back beds have same vertices so it is called a bed of width 2.

The Row ( Column ) Kolakoski array W consists of back-to-back blocks of 1 ‘s and 2 ‘s such that in every row ( column ) the length of each block is either 1 or 2, and the length of the ith block in the jth row ( column ) is equal to ( i, J ) Thursday vertex of W.

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Row Kolakoski array

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Figure 2

Column Kolakoski array

The Kolakoski array W consists of back-to-back beds of 1 ‘s and 2 ‘s such that the breadth of each bed is either 1 or 2, and the breadth of the ith bed is equal to the ith node of W. This is an illustration of a ego reading infinite array. .

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Figure 3

Kolakoski array

## Remark:3.2

( I ) Kolakoski array is a symmetric array.

( two ) The way fall ining the vertices along the chief diagonal is a Kolakoski word.

( three ) The waies fall ining the vertices along right ( left ) diagonals and the predating vertices in the first row ( column ) are Kolakoski words.

( four ) Kolakoski array is non finally periodic.

We seek more belongingss in this array which has many applications.

Let be Kolakoski array. The n-th node is termed as particular node and is denoted by Kn

We will now deduce a recursive expression for Kn. Let

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Table 1: Kn and kn

## Lemma 3.3

Proof: We foremost notice that

The left inequality holds by definition and the right 1 is valid, since if

we get a contradiction to the minimality of kn-1. So, as the first instance, we consider

which implies kn = kn-1 + 1 = . In the 2nd instance leads to.

We notice that Lemma 3.3 holds in general for every sequence, whose merely values are 1 and 2

## Lemma 3.4

## Proof:

The undermentioned well known building produces an array which is indistinguishable to Kolakosai array K. Start with Kk1 and matching bed 2, continue with Kk2 and matching bed 1s, followed by Kk3 and matching bed 2 and so on. In this building, after kn-1 stairss two instances can look, as described in the cogent evidence of Lemma 3.2 The first possibility is that which means that we have constructed n-1 nodes of the array. Therefore, by building Kn must be different from Kn-1 connoting. In the 2nd instance that, it is necessary that, for if otherwise beliing the miniality of kn-1. So our building has added 2 equal Numberss at the kn-1th measure, such that Kn = Kn-1 and eventually. The 2nd equality follows by initiation.

Corollary 3.5 is an deduction of Lemma 3.4

## Corollary 3.5

severally.

Corollary 3.6 utilizations Lemma 3.3 and Corollary 3.5

## Corollary 3.6

Corollary 3.7 follows from Corollary 3.6

## Corollary 3.7

## Hence we have

Theorem 3.8 For we have

## 5.CONCLUSION

In this paper we defined self reading infinite arrays and we investigate interesting belongingss Kolakoski arrays.