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Sequential Ambiguity Elements

On Monday, May 26th/03, I posted the following question to which a solution was provided by Remus Nicoara, PhD (IP). I thank Remus for his assistance and solution. I'm interested in knowing what the solution would look like if for each term, the y elements are randomly determined.

Original Question
In a specific sequence that I am to create, a1 has the elements x1 and y1, a2 has the elements x2 and y2, a3 has the elements x3 and y3, The relationship between each term cannot be bijective, however has a bounded range of finite whole numbers. I am to create a rule such that f(x1,y2)=x2 and f(x2,y3) = x3. The rule must be such that after the sequence has been created, if I am given x3,y3,x2 and y1, then x1 and y2 can be determined i.e. y2 = g(x2,x3)

Remus Nicoara's solution

I will use the notation x_n for x sub n.
Let's define y1 to be any positive integer, y_(n+1)=y_n + 1 (for instance y_n is 2,3,4,5,...) and x1=1, then define:
x_n=x_(n-1)+100 modulo y_n (so f(x,y)=x+100 mod y). (Just in case you do not know, x modulo y, or x mod y, means the remainder of the division of x by y.)
So x2=x1+100 mod y2=101 mod 2 = 1, x3=x2+100 mod y3=101 mod 3 = 2, y4=102 mod 4 = 2, y5=102 mod 5 = 2, y6=102 mod 6 = 0, y7=100 mod 7 = 2, y8=102 mod 8 = 6, y9=106 mod 9 = 7 and so on.
Then f is not bijective because different numbers can have the same value modulo yn, but it has finite range because any number mod y_n takes values between 0 and y_n - 1.
Also, if x3,y3,x2,y1 are given we may first determine y2 as the average of y1 and y3 (or just substract 1 from y3) and then we may find x1=x2-100 mod y2 = 2-100 mod 3 =-98 mod 3 = 1(we are sure that this is x1 because x1 was at most y1, more generally xn is at most yn because it is the reminder of a division by yn, and yn is less than 100...when yn becomes bigger then 100 then the sequence becomes constant so everything is still ok).

So all conditions are satisfied!

Solution Preview

Please see attached file.

Problem 4322 (Answer)

Given any set of numbers such as {xn}={ x1, x2, .... XM } for which a paired relation in found such that x1 is related to y1 as x2 is related to y2 as .... As xn is related to yn. Where (yj) are randomly determined and (xj;yj) are assumjed to be regionalized variables.

A functional relation needs to be found so that given any xj,yj and yj+1 we have that:
...

Solution Summary

The sequential ambiguity elements is examined.

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