A Class of Convergent Series with Golden Ratio Based on Fibonacci Sequence

Lainiotis filter, golden section and Fibonacci sequence

The relation between the discrete time Lainiotis filter on the one side and the golden section and the Fibonacci sequence on the other is established. As far as the random walk system is concerned, the relation between the Lainiotis filter and the golden section is derived through the Riccati equation since the steady state estimation error covariance is related to the golden section. The relat...

The Fibonacci Sequence in Successive Partitions of a Golden Triangle

A Golden Triangle is a triangle with two of its sides in the ratio (J>:1, where is the Fibonacci Ratio, i.e., (J> = ^(1 + /J) ^ 1.618. Let AABC be a triangle whose sides are a,b, and o and let a/b = k > 1. Bicknell and Hoggatt [1] have shown that (1) a triangle with a side equal to b can be removed from AABC to leave a triangle similar to AABC if and only if k = cj), and (2) a triangle simi...

Fibonacci sequence, golden section, Kalman filter and optimal control

A connection between the Kalman filter and the Fibonacci sequence is developed. More precisely it is shown that, for a scalar random walk system in which the two noise sources (process and measurement noise) have equal variance, the Kalman filter’s estimate turns out to be a convex linear combination of the a priori estimate and of the measurements with coefficients suitably related to the Fibo...

The Golden-fibonacci Equivalence

We shall refer to A and B as the large and the small Golden Ratios, respectively, and shall in general simply refer to these and their powers collectively as Golden Numbers. Likewise, the ratio between the neighboring Fibonacci Numbers un+i/un will be called the large Fibonacci Ratio. Here, "large" means that the suffices n + 1 > n, without inference to the values of the u s or their ratio. Its...

Gait recognition based on the golden ratio

Gait is known to be an effective behavioral biometric trait for the identification of individuals. However, clothing has a dramatic influence on the recognition rate. Researchers have attempted to deal with this issue of clothing by segmenting parts of the gait images based on anatomical proportions. However, the clothing proportion is not the same as the anatomical proportion, as clothing is d...

A New Class of Banach Sequence Spaces