The Retention of Information in the Presence of Increasing Entropy Using Lie Algebras Defines Fibonacci-Type Sequences

In the general linear Lie algebra of continuous linear transformations in n dimensions, we show that unequal Abelian scaling transformations on the components of a vector can stabilize the system information in the presence of Markov component transformations on the vector, which, alone, would lead to increasing entropy. The more interesting results follow from seeking Diophantine (integer) solutions, with the result that the system can be stabilized with constant information for each of a set of entropy rates (k=1,2,3,… ). The first of these—the simplest—where k=1, results in the Fibonacci sequence, with information determined by the olden mean, and Fibonacci interpolating functions. Other interesting results include the fact that a new set of higher order generalized Fibonacci sequences, functions, golden means, and geometric patterns emerges for k=2, 3, … Specifically, we define the kth order golden mean as Φk=k/2+(k/2)2+1 for k =1, 2, 3, ….. One can easily observe that one can form a right triangle with sides of 1 and k/2 and that this will give a hypotenuse of (k/2)2+1. Thus, the sum of the k/2 side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden means for any value of k relative to the third side with a value of unity. The sequential powers of the matrix (k2+1,k,k,1) for any integer value of k provide a generalized Fibonacci sequence. Also, using the general equation expressed as Φk=k2+(k/2)2+1 for k =1,2,3, …, one can easily prove that Φk=k+1/Φk which is a generalization of the familiar equation expressed as Φ=1+1/Φ. We suggest that one could look for these new ratios and patterns in nature, with the possibility that all of these systems are connected with the retention of information in the presence of increasing entropy. Thus, we show that two components of the general linear Lie algebra (GL(n,R)), acting simultaneously with certain parameters, can stabilize the information content of a vector over time.